init commit

.drone.yml

@@ -0,0 +1,198 @@
+kind: pipeline
+type: docker
+name: compile
+steps:
+- name: compile individual chapters
+  image: stpranger/latex-minted-latexmk
+  commands:
+  - chmod +x compile
+  - ./compile -d -b
+  - ./compile -d -b solution
+  when:
+    ref:
+      exclude:
+      - refs/tags/release_complete*
+
+- name: compile complete questionnaire
+  image: stpranger/latex-minted-latexmk
+  commands:
+  - chmod +x compile
+  - ./compile -d
+  - ./compile -d solution
+  when:
+    ref:
+      include:
+      - refs/tags/release_complete*
+
+- name: release complete questionnaire
+  image: plugins/gitea-release
+  settings:
+    base_url: https://git.pranger.xyz
+    api_key:
+      from_secret: gitea_token
+    files:
+    - questionnaire.pdf
+    - questionnaire_with_solutions.pdf
+  when:
+    ref:
+      include:
+      - refs/tags/release_complete*
+
+- name: release chapter smtzthree
+  image: plugins/gitea-release
+  settings:
+    base_url: https://git.pranger.xyz
+    api_key:
+      from_secret: gitea_token
+    files:
+    - questionnaire_smtzthree.pdf
+    - questionnaire_with_solutions_smtzthree.pdf
+  when:
+    ref:
+      include:
+      - refs/tags/release_smtzthree*
+
+- name: release chapter proplogic
+  image: plugins/gitea-release
+  settings:
+    base_url: https://git.pranger.xyz
+    api_key:
+      from_secret: gitea_token
+    files:
+    - questionnaire_proplogic.pdf
+    - questionnaire_with_solutions_proplogic.pdf
+  when:
+    ref:
+      include:
+      - refs/tags/release_proplogic*
+
+- name: release chapter satsolver
+  image: plugins/gitea-release
+  settings:
+    base_url: https://git.pranger.xyz
+    api_key:
+      from_secret: gitea_token
+    files:
+    - questionnaire_satsolver.pdf
+    - questionnaire_with_solutions_satsolver.pdf
+  when:
+    ref:
+      include:
+      - refs/tags/release_satsolver*
+
+- name: release chapter ndprop
+  image: plugins/gitea-release
+  settings:
+    base_url: https://git.pranger.xyz
+    api_key:
+      from_secret: gitea_token
+    files:
+    - questionnaire_ndprop.pdf
+    - questionnaire_with_solutions_ndprop.pdf
+  when:
+    ref:
+      include:
+      - refs/tags/release_ndprop*
+
+- name: release chapter predlogic
+  image: plugins/gitea-release
+  settings:
+    base_url: https://git.pranger.xyz
+    api_key:
+      from_secret: gitea_token
+    files:
+    - questionnaire_predlogic.pdf
+    - questionnaire_with_solutions_predlogic.pdf
+  when:
+    ref:
+      include:
+      - refs/tags/release_predlogic*
+
+- name: release chapter ndpred
+  image: plugins/gitea-release
+  settings:
+    base_url: https://git.pranger.xyz
+    api_key:
+      from_secret: gitea_token
+    files:
+    - questionnaire_ndpred.pdf
+    - questionnaire_with_solutions_ndpred.pdf
+  when:
+    ref:
+      include:
+      - refs/tags/release_ndpred*
+
+- name: release chapter smt
+  image: plugins/gitea-release
+  settings:
+    base_url: https://git.pranger.xyz
+    api_key:
+      from_secret: gitea_token
+    files:
+    - questionnaire_smt.pdf
+    - questionnaire_with_solutions_smt.pdf
+  when:
+    ref:
+      include:
+      - refs/tags/release_smt*
+
+- name: release chapter bdd
+  image: plugins/gitea-release
+  settings:
+    base_url: https://git.pranger.xyz
+    api_key:
+      from_secret: gitea_token
+    files:
+    - questionnaire_bdd.pdf
+    - questionnaire_with_solutions_bdd.pdf
+  when:
+    ref:
+      include:
+      - refs/tags/release_bdd*
+
+- name: release chapter eqchecking
+  image: plugins/gitea-release
+  settings:
+    base_url: https://git.pranger.xyz
+    api_key:
+      from_secret: gitea_token
+    files:
+    - questionnaire_eqchecking.pdf
+    - questionnaire_with_solutions_eqchecking.pdf
+  when:
+    ref:
+      include:
+      - refs/tags/release_eqchecking*
+
+- name: release chapter symbenc
+  image: plugins/gitea-release
+  settings:
+    base_url: https://git.pranger.xyz
+    api_key:
+      from_secret: gitea_token
+    files:
+    - questionnaire_symbenc.pdf
+    - questionnaire_with_solutions_symbenc.pdf
+  when:
+    ref:
+      include:
+      - refs/tags/release_symbenc*
+
+- name: release chapter temporal
+  image: plugins/gitea-release
+  settings:
+    base_url: https://git.pranger.xyz
+    api_key:
+      from_secret: gitea_token
+    files:
+    - questionnaire_temporal.pdf
+    - questionnaire_with_solutions_temporal.pdf
+  when:
+    ref:
+      include:
+      - refs/tags/release_temporal*
+trigger:
+  event:
+    include:
+    - push
+    - tag

.gitignore

@@ -0,0 +1,314 @@
+# source: https://github.com/github/gitignore/blob/main/TeX.gitignore
+## Core latex/pdflatex auxiliary files:
+*.aux
+*.lof
+*.log
+*.lot
+*.fls
+*.out
+*.toc
+*.fmt
+*.fot
+*.cb
+*.cb2
+.*.lb
+
+*.d
+
+## Intermediate documents:
+*.dvi
+*.xdv
+*-converted-to.*
+# these rules might exclude image files for figures etc.
+# *.ps
+# *.eps
+# *.pdf
+
+## Generated if empty string is given at "Please type another file name for output:"
+.pdf
+
+## Bibliography auxiliary files (bibtex/biblatex/biber):
+*.bbl
+*.bcf
+*.blg
+*-blx.aux
+*-blx.bib
+*.run.xml
+
+## Build tool auxiliary files:
+*.fdb_latexmk
+*.synctex
+*.synctex(busy)
+*.synctex.gz
+*.synctex.gz(busy)
+*.pdfsync
+
+## Build tool directories for auxiliary files
+# latexrun
+latex.out/
+
+## Auxiliary and intermediate files from other packages:
+# algorithms
+*.alg
+*.loa
+
+# achemso
+acs-*.bib
+
+# amsthm
+*.thm
+
+# beamer
+*.nav
+*.pre
+*.snm
+*.vrb
+
+# changes
+*.soc
+
+# comment
+*.cut
+
+# cprotect
+*.cpt
+
+# elsarticle (documentclass of Elsevier journals)
+*.spl
+
+# endnotes
+*.ent
+
+# fixme
+*.lox
+
+# feynmf/feynmp
+*.mf
+*.mp
+*.t[1-9]
+*.t[1-9][0-9]
+*.tfm
+
+#(r)(e)ledmac/(r)(e)ledpar
+*.end
+*.?end
+*.[1-9]
+*.[1-9][0-9]
+*.[1-9][0-9][0-9]
+*.[1-9]R
+*.[1-9][0-9]R
+*.[1-9][0-9][0-9]R
+*.eledsec[1-9]
+*.eledsec[1-9]R
+*.eledsec[1-9][0-9]
+*.eledsec[1-9][0-9]R
+*.eledsec[1-9][0-9][0-9]
+*.eledsec[1-9][0-9][0-9]R
+
+# glossaries
+*.acn
+*.acr
+*.glg
+*.glo
+*.gls
+*.glsdefs
+*.lzo
+*.lzs
+*.slg
+*.slo
+*.sls
+
+# uncomment this for glossaries-extra (will ignore makeindex's style files!)
+# *.ist
+
+# gnuplot
+*.gnuplot
+*.table
+
+# gnuplottex
+*-gnuplottex-*
+
+# gregoriotex
+*.gaux
+*.glog
+*.gtex
+
+# htlatex
+*.4ct
+*.4tc
+*.idv
+*.lg
+*.trc
+*.xref
+
+# hyperref
+*.brf
+
+# knitr
+*-concordance.tex
+# TODO Uncomment the next line if you use knitr and want to ignore its generated tikz files
+# *.tikz
+*-tikzDictionary
+
+# listings
+*.lol
+
+# luatexja-ruby
+*.ltjruby
+
+# makeidx
+*.idx
+*.ilg
+*.ind
+
+# minitoc
+*.maf
+*.mlf
+*.mlt
+*.mtc[0-9]*
+*.slf[0-9]*
+*.slt[0-9]*
+*.stc[0-9]*
+
+# minted
+_minted*
+*.pyg
+
+# morewrites
+*.mw
+
+# newpax
+*.newpax
+
+# nomencl
+*.nlg
+*.nlo
+*.nls
+
+# pax
+*.pax
+
+# pdfpcnotes
+*.pdfpc
+
+# sagetex
+*.sagetex.sage
+*.sagetex.py
+*.sagetex.scmd
+
+# scrwfile
+*.wrt
+
+# svg
+svg-inkscape/
+
+# sympy
+*.sout
+*.sympy
+sympy-plots-for-*.tex/
+
+# pdfcomment
+*.upa
+*.upb
+
+# pythontex
+*.pytxcode
+pythontex-files-*/
+
+# tcolorbox
+*.listing
+
+# thmtools
+*.loe
+
+# TikZ & PGF
+*.dpth
+*.md5
+*.auxlock
+
+# titletoc
+*.ptc
+
+# todonotes
+*.tdo
+
+# vhistory
+*.hst
+*.ver
+
+# easy-todo
+*.lod
+
+# xcolor
+*.xcp
+
+# xmpincl
+*.xmpi
+
+# xindy
+*.xdy
+
+# xypic precompiled matrices and outlines
+*.xyc
+*.xyd
+
+# endfloat
+*.ttt
+*.fff
+
+# Latexian
+TSWLatexianTemp*
+
+## Editors:
+# WinEdt
+*.bak
+*.sav
+
+# Texpad
+.texpadtmp
+
+# LyX
+*.lyx~
+
+# Kile
+*.backup
+
+# gummi
+.*.swp
+
+# KBibTeX
+*~[0-9]*
+
+# TeXnicCenter
+*.tps
+
+# auto folder when using emacs and auctex
+./auto/*
+*.el
+
+# expex forward references with \gathertags
+*-tags.tex
+
+# standalone packages
+*.sta
+
+# Makeindex log files
+*.lpz
+
+# xwatermark package
+*.xwm
+
+# REVTeX puts footnotes in the bibliography by default, unless the nofootinbib
+# option is specified. Footnotes are the stored in a file with suffix Notes.bib.
+# Uncomment the next line to have this generated file ignored.
+#*Notes.bib
+
+# Mac
+.DS_Store
+.nova
+
+# Do not allow pushing of compiled files
+*.pdf
+
+util/chapter.tex
+util/toggle.tex

bdds/0001.tex

@@ -0,0 +1,32 @@
+\item \lect Given the \textit{Binary Decision Diagram (BDD)} below, label and explain the different elements of the diagram.
+
+\begin{center}
+\begin{forest}
+for tree={circle, draw, no edge,
+            minimum size=2em,
+            inner sep=0pt,
+            s sep=2mm,
+            l sep=4mm}   
+[$f$, rectangle, draw,  tikz={\draw [line] () to (a.north);}
+    [$a$, name=a, tikz={\draw [line] () to (b.north east); \draw [dot=.5] () to (c2.north west);}
+        [$b$, name=b, tikz={\draw [dot=.5, bend left] () to (c1.north east); \draw [line, bend right] () to (c1.north west);}
+            [$c$, name=c1, s sep=20mm, tikz={\draw [line] () to (one.north); \draw [dot=.5] () to (e.north west);}
+                [,phantom]
+                [$e$, name=e, s sep=10mm, tikz={\draw [line, bend left] () to (one.north east); \draw [dot=.5, bend left] () to (one.east);}
+                    [$1$, rectangle, draw, name=one]					
+                    [$0$, rectangle, draw, name=zero]
+                ]
+            ]
+        ]
+        [$c$, name=c2, tikz={\draw [line, bend right] () to (e.north); \draw [dot=.5, bend left] () to (d.north);}
+            [,phantom]
+            [$d$, name=d, tikz={\draw [line] () to (e.north east); \draw[dot=.5] () to (zero.north);
+}
+                [,phantom]
+                [,phantom]							
+            ]
+        ]	
+    ]
+]
+\end{forest}
+\end{center}

bdds/0001_sol.tex

@@ -0,0 +1,38 @@
+\begin{center}
+\begin{forest}
+for tree={circle, draw, no edge,
+            minimum size=2em,
+            inner sep=0pt,
+            s sep=2mm,
+            l sep=4mm}
+[$f$, rectangle, draw,  tikz={\draw [line] () to (a.north);}
+    [$a$, name=a, tikz={\draw [line] () to node[pos=.7, above] {\scriptsize T \;} (b.north east); \draw [dot=.5] () to node[pos=.5, above] {\; \scriptsize E} (c2.north west);}
+        [$b$, name=b, tikz={\draw [dot=.5, bend left] () to node[pos=.5, right] {\scriptsize E \;} (c1.north east); \draw [line, bend right] () to node[pos=.5, left] {\; \scriptsize T} (c1.north west);}
+            [$c$, name=c1, s sep=20mm, tikz={\draw [line] () to node[pos=.7, left] {\scriptsize T \;} (one.north); \draw [dot=.5] () to node[pos=.5, above] {\; \scriptsize E} (e.north west);}
+                [,phantom]
+                [$e$, name=e, s sep=10mm, tikz={\draw [line, bend left] () to node[pos=.7, above] {\scriptsize T \;} (one.north east); \draw [dot=.5, bend left] () to node[pos=.2, below] {\; \scriptsize E} (one.east);}
+                    [$1$, rectangle, draw, name=one]
+                    [$0$, rectangle, draw, name=zero]
+                ]
+            ]
+        ]
+        [$c$, name=c2, tikz={\draw [line, bend right] () to node[pos=.3, above] {\scriptsize T \;} (e.north); \draw [dot=.5, bend left] () to node[pos=.5, above] {\; \scriptsize E} (d.north);}
+            [,phantom]
+            [$d$, name=d, tikz={\draw [line] () to node[pos=.7, above] {\scriptsize T \;} (e.north east); \draw[dot=.5] () to node[pos=.3, below] {\; \scriptsize E} (zero.north);
+}
+                [,phantom]
+                [,phantom]
+            ]
+        ]
+    ]
+]
+\end{forest}
+\end{center}
+
+A binary decision diagram represents a Boolean formula $f$. It is a DAG with two terminal nodes that are
+labelled with 0 and 1. The internal nodes are labelled with the Boolean variables
+of the formula (here a, b, c, d and e). Each internal node has exactly two outgoing
+edges: one edge labeled with a T (the then-edge), and another edge that
+is labeled with an E (the else-edge) or marked with a circle. There is a unique
+initial node called the function node labeled with $f$ that does not have any incoming edges and one outgoing edge to the internal variable node on the first
+level.

bdds/0002.tex

@@ -0,0 +1,33 @@
+\item \lect Given the \textit{Binary Decision Diagram (BDD)} below,
+  find a satisfying and a falsifying model of $f$.
+
+\begin{center}
+\begin{forest}
+for tree={circle, draw, no edge,
+            minimum size=2em,
+            inner sep=0pt,
+            s sep=2mm,
+            l sep=4mm}
+[$f$, rectangle, draw,  tikz={\draw [line] () to (a.north);}
+    [$a$, name=a, tikz={\draw [line] () to (b.north east); \draw [dot=.5] () to (c2.north west);}
+        [$b$, name=b, tikz={\draw [dot=.5, bend left] () to (c1.north east); \draw [line, bend right] () to (c1.north west);}
+            [$c$, name=c1, s sep=20mm, tikz={\draw [line] () to (one.north); \draw [dot=.5] () to (e.north west);}
+                [,phantom]
+                [$e$, name=e, s sep=10mm, tikz={\draw [line, bend left] () to (one.north east); \draw [dot=.5, bend left] () to (one.east);}
+                    [$1$, rectangle, draw, name=one]
+                    [$0$, rectangle, draw, name=zero]
+                ]
+            ]
+        ]
+        [$c$, name=c2, tikz={\draw [line, bend right] () to (e.north); \draw [dot=.5, bend left] () to (d.north);}
+            [,phantom]
+            [$d$, name=d, tikz={\draw [line] () to (e.north east); \draw[dot=.5] () to (zero.north);
+}
+                [,phantom]
+                [,phantom]
+            ]
+        ]
+    ]
+]
+\end{forest}
+\end{center}

bdds/0002_sol.tex

@@ -0,0 +1,4 @@
+\begin{enumerate}
+    \item $\mathcal{M}_1$ = \{a = $\top$, b = $\top$, c = $\top$\}
+    \item $\mathcal{M}_2$ = \{a = $\bot$, c = $\bot$, d = $\bot$\}
+\end{enumerate}

bdds/0003.tex

@@ -0,0 +1,32 @@
+\item \lect Given the following \textit{Binary Decision Diagram (BDD)} that represents the formula $f$. Compute its disjunctive normal form \DNF{f}.
+  \TODONOTE{THIS IS BULLCRAP.}
+\begin{center}
+\begin{forest}
+for tree={circle, draw, no edge,
+            minimum size=2em,
+            inner sep=0pt,
+            s sep=2mm,
+            l sep=4mm}
+[$f$, rectangle, draw,  tikz={\draw [line] () to (a.north);}
+    [$a$, name=a, tikz={\draw [line] () to (b.north east); \draw [dot=.5] () to (c2.north west);}
+        [$b$, name=b, tikz={\draw [dot=.5, bend left] () to (c1.north east); \draw [line, bend right] () to (c1.north west);}
+            [$c$, name=c1, s sep=20mm, tikz={\draw [line] () to (one.north); \draw [dot=.5] () to (e.north west);}
+                [,phantom]
+                [$e$, name=e, s sep=10mm, tikz={\draw [line, bend left] () to (one.north east); \draw [dot=.5, bend left] () to (one.east);}
+                    [$1$, rectangle, draw, name=one]
+                    [$0$, rectangle, draw, name=zero]
+                ]
+            ]
+        ]
+        [$c$, name=c2, tikz={\draw [line, bend right] () to (e.north); \draw [dot=.5, bend left] () to (d.north);}
+            [,phantom]
+            [$d$, name=d, tikz={\draw [line] () to (e.north east); \draw[dot=.5] () to (zero.north);
+}
+                [,phantom]
+                [,phantom]
+            ]
+        ]
+    ]
+]
+\end{forest}
+\end{center}

bdds/0003_sol.tex

@@ -0,0 +1,7 @@
+\begin{center}
+$f = (a \land b \land c) \lor (a \land \lnot b \land c) \lor (a \land b \land \lnot c \land e) \lor (a \land \lnot b \land \lnot c \land e) \lor 
+\newline 
+(a \land b \land \lnot c \land \lnot e) \lor (a \land \lnot b \land \lnot c \land \lnot e) \lor (\lnot a \land c \land e) \lor (\lnot a \land c \land \lnot e) \lor 
+\newline
+(\lnot a \land \lnot c \land d \land e) \lor (\lnot a \land \lnot c \land d \land \lnot e)$
+\end{center}

bdds/0004.tex

@@ -0,0 +1,29 @@
+\item \BDDToROBDD
+
+\begin{center}
+\begin{forest}
+for tree={circle, draw, no edge,
+            minimum size=2em,
+            inner sep=0pt,
+            s sep=2mm,
+            l sep=2mm}
+[$f$, rectangle, draw,  tikz={\draw [line] () to (a.north);}
+    [$a$, name=a, tikz={\draw [line] () to (b1.north east); \draw [dot=.5] () to (b2.north west);}
+        [$b$, name=b1, tikz={\draw [line] () to (c1.north); \draw [dot=.5] () to (c2.north);}
+            [$c$, name=c1, tikz={\draw [line, bend right] () to (one.west); \draw [dot=.5, bend right] () to (zero.west);}]
+            [$c$, l sep=15mm, name=c2, tikz={\draw [line, bend right] () to (one.north); \draw [dot=.5, bend right] () to (zero.north west);}
+                [$1$, rectangle, draw, name=one]
+                [,phantom]
+            ]
+        ]
+        [$b$, name=b2, tikz={\draw [line] () to (c3.north); \draw [dot=.5] () to (c4.north);}
+            [$c$, l sep=15mm, name=c3, tikz={\draw [line, bend left] (.south) to (one.east); \draw [dot=.3, bend left] (.south east) to (one.south east);}
+                [,phantom]
+                [$0$, rectangle, draw, name=zero]
+            ]
+            [$c$, name=c4, tikz={\draw [line, bend left] (.south) to (zero.north); \draw [dot=.5, bend left] () to (zero.north east);}]
+        ]
+    ]
+]
+\end{forest}
+\end{center}

bdds/0004_sol.tex

@@ -0,0 +1,17 @@
+\begin{center}
+\begin{forest}
+for tree={circle, draw, no edge,
+            minimum size=2em,
+            inner sep=0pt,
+            s sep=2mm,
+            l sep=2mm}
+[$f$, rectangle, draw,  tikz={\draw [line] () to (a.north);}
+    [$a$, name=a, tikz={\draw [line] () to (c1.north east); \draw [dot=.5] () to (b1.north west);}
+        [$c$, name=c1, tikz={\draw [line] () to (250:3); \draw [fulldot=.5] () to (265:3);}
+        ]
+        [$b$, name=b1, tikz={\draw [line] () to (270:3); \draw [fulldot=.5] () to (285:3);}
+        ]
+    ]
+]
+\end{forest}
+\end{center}

bdds/0005.tex

@@ -0,0 +1,19 @@
+\item \BDDToROBDD
+
+\begin{center}
+\begin{forest}
+for tree={circle, draw, no edge,
+            minimum size=2em,
+            inner sep=0pt,
+            s sep=5mm,
+            l sep=5mm}
+[$f$, rectangle, draw,  tikz={\draw [line] () to (a.north);}
+    [$a$, name=a, tikz={\draw [line, bend right] () to (b.north west); \draw [fulldot=.5, bend left] () to (b.north east);}
+        [$b$, name=b, tikz={\draw [line] () to (c1.north); \draw [dot=.5] () to (c2.north);}
+            [$c$, name=c1, tikz={\draw [line] () to (256:4.6); \draw [fulldot=.5] () to (268:4.5);}]
+            [$c$, name=c2, tikz={\draw [line] () to (272:4.5); \draw [fulldot=.5] () to (284:4.6);}]
+        ]
+    ]
+]
+\end{forest}
+\end{center}

bdds/0005_sol.tex

@@ -0,0 +1,15 @@
+\begin{center}
+\begin{forest}
+for tree={circle, draw, no edge,
+            minimum size=2em,
+            inner sep=0pt,
+            s sep=5mm,
+            l sep=5mm}   
+[$f$, rectangle, draw,  tikz={\draw [line] () to (a.north);}
+    [$a$, name=a, tikz={\draw [line, bend right] () to (c.north west); \draw [fulldot=.5, bend left] () to (c.north east);}
+        [$c$, name=c, tikz={\draw [line] () to (260:3.4); \draw [fulldot=.5] () to (280:3.4);}
+        ]
+    ]
+]
+\end{forest}
+\end{center}

bdds/0006.tex

@@ -0,0 +1,2 @@
+\item A reduced and ordered BDD is a canonical representation of a propositional formula.
+Explain the term \textit{canonical} in the context of propositional formulas and explain why reduced and ordered BDDs give canonical representations.

bdds/0006_sol.tex

@@ -0,0 +1,3 @@
+A canonical representation means, that for any two semantically equivalent formulas $f_1$ and $f_2$ and a fixed variable order, $f_1$ and $f_2$ are represented by the same reduced and ordered BDD.
+
+Since semantically equivalent formulas have the same set of satisfying models and a reduced and ordered BDD does not have redundancies, the representation of $f_1$ and $f_2$ are the same.

bdds/0007.tex

@@ -0,0 +1,34 @@
+\item \BDDToDNF
+
+
+\begin{center}
+\begin{forest}
+for tree={circle, draw, no edge,
+            minimum size=2em,
+            inner sep=0pt,
+            s sep=2mm,
+            l sep=1mm}
+[$f$, rectangle, draw,  tikz={\draw [line] () to (a.north);}
+    [$a$, name=a, tikz={\draw [line] () to (b.north east); \draw [fulldot=.5] () to (c2.north west);}
+        [$b$, name=b, tikz={\draw [dot=.5] () to (e.north); \draw [line] () to (c1.north);}
+            [$c$, name=c1, s sep=20mm, tikz={\draw [line] () to (240:4.2); \draw [fulldot=.5] () to (e.north west);}
+                [,phantom]
+                [$e$, name=e, s sep=10mm, tikz={\draw [fulldot=.5] () to (zero.north); \draw [line] () to (one.north);}
+                    [,phantom, name=one]
+                    [,phantom, name=zero]
+                ]
+            ]
+            [,phantom]
+        ]
+        [$c$, name=c2, tikz={\draw [line] () to (e.north); \draw [fulldot=.5] () to (d.north);}
+            [,phantom]
+            [$d$, name=d, tikz={\draw [line] () to (e.north east); \draw[dot=.5] () to (300:4.2);
+}
+                [,phantom]
+                [,phantom]
+            ]
+        ]
+    ]
+]
+\end{forest}
+\end{center}

bdds/0007_sol.tex

@@ -0,0 +1,5 @@
+\begin{center}
+$f = (a \land b \land c) \lor (a \land \lnot b \land e) \lor (a \land b \land \lnot c \land \lnot e) \lor (\lnot a \land \lnot c \land d) \lor 
+\newline
+(\lnot a \land c \land \lnot e) \lor (\lnot a \land \lnot c \land d \land e)$
+\end{center}

bdds/0008.tex

@@ -0,0 +1 @@
+\item \constructROBDD{f = ((a \land b) \lor \lnot a \lor(c \biimp d)}{\textit{alphabetic variable order}}

bdds/0008_sol.tex

@@ -0,0 +1,38 @@
+\begin{cofactors}
+$f_{a}$ \= $= b \lor(c \biimp d)$ \\
+\>$f_{ab}$ \= $= \true $\\
+\>$f_{a\lnot b}$ \= $= c \biimp d$ \\
+\>\>$f_{a\lnot bc}$ \= $= d$ \\
+\>\>\>$f_{a\lnot b c d}$ \= $= \true$ \\
+\>\>\>$f_{a\lnot bc\lnot d}$ \= $= \false$ \\
+\>\>$f_{a\lnot b\lnot c}$ \= $= \lnot d = f_{abc}$ \\
+$f_{\lnot a}$ \= $=\true $
+\end{cofactors}
+
+The final ROBDD:\\
+\begin{center}
+\begin{bdd}
+  \node[func node] (f)                     {$f$};
+  \node[cofactor] (a1) [below      of=f]   {$a$};
+  \node[phantom]  (aR) [below right of=a1] {};
+  \node[cofactor] (b1) [below left of=a1]  {$b$};
+  \node[phantom]  (bL) [below left of=b1]  {};
+  \node[cofactor] (c1) [below right of=b1] {$c$};
+  \node[cofactor] (d1) [below      of=c1]  {$d$};
+  \node[phantom]  (dL) [below left of=d1]  {};
+  \node[phantom]  (dR) [below right of=d1] {};
+
+  \funcEdge{f}{a1}
+  \thenEdge{a1}{b1}
+  \elseEdge{a1}{aR}
+
+  \thenEdge{b1}{bL}
+  \elseEdge{b1}{c1}
+
+  \thenEdge[bend right]{c1}{d1}
+  \negatedEdge[bend left]{c1}{d1}
+
+  \thenEdge{d1}{dL}
+  \elseEdge{d1}{dR}
+\end{bdd}
+\end{center}

bdds/0009.tex

@@ -0,0 +1,3 @@
+\item \constructROBDD{
+f = (r \land p) \lor (\neg r \land \neg p) \lor (s \land \neg r) \lor (\neg s \land r)\lor (\lnot r \land q)
+}{\textit{variable order} $p < q < r < s$}

bdds/0009_sol.tex

@@ -0,0 +1,50 @@
+\begin{cofactors}
+$f_{p}$ \= $= r \lor(s \land \lnot r) \lor (\lnot s \land  r) \lor (\lnot r \land q)$ \\
+\>$f_{pq}$ \= $= r\lor (s \land \lnot r) \lor (\lnot s \land  r)\lor \lnot r = \true $\\
+\>$f_{p\lnot q}$ \= $= r\lor (s \land \lnot r)\lor (\lnot s \land  r) $\\
+\>\>$f_{p\lnot qr}$ \= $= \true $\\
+\>\>$f_{p\lnot q\lnot r}$ \= $= s $\\
+\>\>\>$f_{p\lnot q\lnot rs}$ \= $= \true $\\
+\>\>\>$f_{p\lnot q\lnot r\lnot s}$ \= $= \false $\\
+$f_{\lnot p}$ \= $= \lnot r \lor (s \land \lnot r) \lor (\lnot s \land  r) \lor (\lnot r \land q) = \lnot r \lor (s \land \lnot r) \lor (\lnot s \land  r)$ \\
+\>$f_{\lnot pr}$ \= $= \lnot s = f_{p\lnot q\lnot r} $ \\
+\>$f_{\lnot p\lnot r}$ \= $= \true $
+\end{cofactors}
+
+The final ROBDD:\\
+\begin{center}
+\begin{bdd}[4em]
+  \node[func node] (f)                      {$f$};
+  \node[cofactor] (p1)  [below      of=f]   {$p$};
+
+  \node[cofactor] (q1)  [below left of=p1]  {$q$};
+  \node[cofactor] (r1)  [below right of=p1,yshift=-2.75em,xshift=1.75em] {$r$};
+
+  \node[cofactor] (r2)  [below right of=q1] {$r$};
+  \node[phantom]  (q1L) [below left of=q1]  {};
+
+  \node[phantom]  (r1R) [below right of=r1] {};
+
+  \node[cofactor] (s1)  [below right of=r2] {$s$};
+  \node[phantom]  (r2L) [below left of=r2]  {};
+
+  \node[phantom]  (s2L) [below left of=s1]  {};
+  \node[phantom]  (s2R) [below right of=s1] {};
+
+  \funcEdge{f}{p1}
+  \thenEdge{p1}{q1}
+  \negatedEdge{p1}{r1}
+
+  \thenEdge{q1}{q1L}
+  \elseEdge{q1}{r2}
+
+  \thenEdge{r2}{r2L}
+  \elseEdge{r2}{s1}
+
+  \thenEdge{r1}{s1}
+  \negatedEdge{r1}{r1R}
+
+  \thenEdge{s1}{s2L}
+  \negatedEdge{s1}{s2R}
+\end{bdd}
+\end{center}

bdds/0010.tex

@@ -0,0 +1,3 @@
+\item \constructROBDD{
+f = (r \land \neg p) \lor (\neg r \land p) \lor (s \land \neg r) \lor (\neg s \land r) \lor ( r \land q)
+}{\textit{variable order} $p < q < r < s$}

bdds/0010_sol.tex

@@ -0,0 +1,57 @@
+$$f = (r \land \lnot p) \lor (r \land \lnot p) \lor (s \land \lnot r) \lor (\lnot s \land r)\lor (r \land q),$$
+
+\begin{cofactors}
+$f_{p}$ \= $= \lnot r \lor(s \land \lnot r) \lor (\lnot s \land  r) \lor (r \land q)$ \\
+\>$f_{pq}$ \= $= \true$\\
+\>$f_{p\lnot q}$ \= $=  r \lor  (s \land \lnot r) \lor (\lnot s \land r)$\\
+\>\>$f_{p\lnot qr}$ \= $= \true$\\
+\>\>$f_{p\lnot q\lnot r}$ \= $= s$\\
+\>\>\>$f_{p\lnot q\lnot rs}$ \= $= \true$\\
+\>\>\>$f_{p\lnot q\lnot r\lnot s}$ \= $= \false$\\
+$f_{\lnot p}$ \= $= r \lor(s \land \lnot r) \lor (\lnot s \land  r) \lor (r \land q)$ \\
+\>$f_{\lnot pq}$ \= $= r \lor(s \land \lnot r) \lor (\lnot s \land  r)$ \\
+\>$f_{\lnot p\lnot q}$ \= $= r \lor(s \land \lnot r) \lor (\lnot s \land  r)$ \\
+\>\>$\Rightarrow q$ does not have an influence on the formula. These cofactors can be skipped.\\
+\>$f_{\lnot pr}$ \= $= \true$ \\
+\>$f_{\lnot p\lnot r}$ \= $= s = f_{p\lnot q\lnot r}$ \\
+\end{cofactors}
+
+The final ROBDD:\\
+\begin{center}
+\begin{bdd}[4em]
+  \node[func node] (f)                      {$f$};
+  \node[cofactor] (p1)  [below      of=f]   {$q$};
+
+  \node[cofactor] (q1)  [below left of=p1]  {$q$};
+  \node[cofactor] (r1)  [below right of=p1] {$r$};
+
+  \node[cofactor] (r2)  [below right of=q1,yshift=-0.8em,xshift=-1.5em] {$r$};
+  \node[phantom]  (q1L) [below left of=q1]  {};
+
+  \node[phantom]  (r1L) [below left of=r1] {};
+
+  \node[cofactor] (s1)  [below left of=r2] {$s$};
+  \node[phantom]  (r2L) [below left of=r2]  {};
+  \node[phantom]  (r2R) [below right of=r2]  {};
+
+  \node[phantom]  (s2L) [below left of=s1]  {};
+  \node[phantom]  (s2R) [below right of=s1] {};
+
+  \funcEdge{f}{p1}
+
+  \thenEdge{p1}{q1}
+  \elseEdge{p1}{r1}
+
+  \thenEdge{q1}{q1L}
+  \negatedEdge{q1}{r2}
+
+  \thenEdge{r2}{s1}
+  \negatedEdge{r2}{r2R}
+
+  \thenEdge{r1}{r1L}
+  \elseEdge[bend left]{r1}{s1}
+
+  \thenEdge{s1}{s2L}
+  \negatedEdge{s1}{s2R}
+\end{bdd}
+\end{center}

bdds/0011.tex

@@ -0,0 +1,5 @@
+\item Construct a Reduced Ordered Binary Decision Diagram (ROBDD) for the formula 
+
+$$f = (p \land q) \lor (r \land \lnot s) \lor (\lnot p \land t \land u)$$
+
+using \textit{alphabetic variable order}. Use complemented edges and a node for \texttt{true} as the only constant node. To simplify drawing, you may assume that \textit{dangling edges} point to the constant node. Write down all cofactors that you compute to obtain the final result and mark them in the graph.

bdds/0011_sol.tex

@@ -0,0 +1,207 @@
+\begin{cofactors}
+  $f$ \= $= (((p \land q) \lor (r \land \lnot s)) \lor ((\lnot p \land t) \land u))$\\
+    \>$f_{p}$ \= $= (q \lor (r \land \lnot s))$\\
+    \>\>$f_{pq}$ \= $= \true$\\
+    \>\>$f_{p\lnot q}$ \= $= (r \land \lnot s)$\\
+    \>\>\>$f_{p\lnot qr}$ \= $= \lnot s$\\
+    \>\>\>\>$f_{p\lnot qrs}$ \= $= \false$\\
+    \>\>\>\>$f_{p\lnot qr\lnot s}$ \= $= \true$\\
+    \>\>\>$f_{p\lnot q\lnot r}$ \= $= \false$\\
+    \>$f_{\lnot p}$ \= $= ((r \land \lnot s) \lor (t \land u))$\\
+    \>\>$f_{\lnot pr}$ \= $= (\lnot s \lor (t \land u))$\\
+    \>\>\>$f_{\lnot prs}$ \= $= (t \land u)$\\
+    \>\>\>\>$f_{\lnot prst}$ \= $= u$\\
+    \>\>\>\>\>$f_{\lnot prstu}$ \= $= \true$\\
+    \>\>\>\>\>$f_{\lnot prst\lnot u}$ \= $= \false$\\
+    \>\>\>\>$f_{\lnot prs\lnot t}$ \= $= \false$\\
+    \>\>\>$f_{\lnot pr\lnot s}$ \= $= \true$\\
+    \>\>$f_{\lnot p\lnot r}$ \= $= f_{\lnot prs}$\\
+  \end{cofactors}
+  The final ROBDD: 
+  \begin{center}
+  \scalebox{0.75}{
+  
+  \begin{tikzpicture}[>=latex,line join=bevel,]
+    \pgfsetlinewidth{1bp}
+  %%
+  \pgfsetcolor{black}
+    % Edge: f_E -> p0
+    \draw [] (128.5bp,326.99bp) .. controls (128.5bp,316.92bp) and (128.5bp,301.47bp)  .. (128.5bp,291.3bp);
+    % Edge: f_p -> q1
+    \draw [] (174.49bp,268.99bp) .. controls (169.27bp,258.54bp) and (161.15bp,242.31bp)  .. (156.09bp,232.18bp);
+    % Edge: f_p!q -> r2
+    \draw [] (198.75bp,210.99bp) .. controls (194.88bp,200.67bp) and (188.89bp,184.7bp)  .. (185.08bp,174.55bp);
+    \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
+    \pgfsetstrokecolor{strokecol}
+    \draw (192.1bp,193.28bp) node {\Large$\bullet$};
+    % Edge: f_p!qr -> s3
+    \draw [] (231.61bp,152.99bp) .. controls (230.71bp,142.92bp) and (229.33bp,127.47bp)  .. (228.42bp,117.3bp);
+    \draw (230.2bp,137.22bp) node {\Large$\bullet$};
+    % Edge: f_!p -> r4
+    \draw [] (82.685bp,268.99bp) .. controls (88.094bp,258.54bp) and (96.501bp,242.31bp)  .. (101.74bp,232.18bp);
+    % Edge: f_!pr -> s5
+    \draw [] (21.5bp,210.99bp) .. controls (21.5bp,200.92bp) and (21.5bp,185.47bp)  .. (21.5bp,175.3bp);
+    % Edge: f_!prs -> t6
+    \draw [] (73.394bp,152.99bp) .. controls (74.293bp,142.92bp) and (75.672bp,127.47bp)  .. (76.58bp,117.3bp);
+    % Edge: f_!prst -> u7
+    \draw [] (161.32bp,94.988bp) .. controls (155.91bp,84.544bp) and (147.5bp,68.308bp)  .. (142.26bp,58.183bp);
+    % Edge: p0 -> r4
+    \draw [] (124.77bp,269.5bp) .. controls (120.75bp,259.27bp) and (114.38bp,243.06bp)  .. (110.32bp,232.73bp);
+    \draw (117.64bp,251.35bp) node {\Large$\circ$};
+    % Edge: p0 -> q1
+    \draw [] (132.4bp,269.5bp) .. controls (136.66bp,259.14bp) and (143.42bp,242.66bp)  .. (147.66bp,232.36bp);
+    % Edge: r4 -> s5
+    \draw [] (97.724bp,215.22bp) .. controls (81.436bp,204.49bp) and (46.704bp,181.6bp)  .. (30.356bp,170.83bp);
+    % Edge: r4 -> t6
+    \draw [] (107.27bp,210.63bp) .. controls (107.97bp,197.09bp) and (108.15bp,172.79bp)  .. (102.5bp,153.0bp) .. controls (98.455bp,138.84bp) and (89.518bp,124.18bp)  .. (83.501bp,115.32bp);
+    \draw (94.602bp,131.96bp) node {\Large$\circ$};
+    % Edge: s5 -> t6
+    \draw [] (28.821bp,155.68bp) .. controls (39.494bp,145.01bp) and (59.224bp,125.28bp)  .. (69.998bp,114.5bp);
+    % Edge: s5 -> 7
+    \draw [] (21.5bp,152.99bp) .. controls (21.5bp,137.38bp) and (21.5bp,108.85bp)  .. (21.5bp,107.09bp);
+    \draw (21.5bp,127.09bp) node {\Large$\circ$};
+    % Edge: t6 -> u7
+    \draw [] (85.344bp,97.679bp) .. controls (96.906bp,86.887bp) and (118.39bp,66.836bp)  .. (129.84bp,56.146bp);
+    % Edge: t6 -> 6
+    \draw [] (77.5bp,94.988bp) .. controls (77.5bp,79.384bp) and (77.5bp,50.854bp)  .. (77.5bp,49.086bp);
+    \draw (77.5bp,69.086bp) node {\Large$\bullet$};
+    % Edge: u7 -> 4
+    \draw [] (135.28bp,36.791bp) .. controls (132.43bp,23.663bp) and (127.8bp,2.3815bp)  .. (127.51bp,1.0642bp);
+    % Edge: u7 -> 5
+    \draw [] (139.5bp,36.791bp) .. controls (142.07bp,23.663bp) and (146.23bp,2.3815bp)  .. (146.49bp,1.0642bp);
+    \draw (142.65bp,20.692bp) node {\Large$\bullet$};
+    % Edge: q1 -> r2
+    \draw [] (156.32bp,212.0bp) .. controls (161.89bp,201.61bp) and (170.99bp,184.62bp)  .. (176.59bp,174.16bp);
+    \draw (167.15bp,191.79bp) node {\Large$\bullet$};
+    % Edge: q1 -> 0
+    \draw [] (151.5bp,210.99bp) .. controls (151.5bp,195.38bp) and (151.5bp,166.85bp)  .. (151.5bp,165.09bp);
+    % Edge: r2 -> s3
+    \draw [] (188.29bp,154.74bp) .. controls (197.08bp,144.03bp) and (212.32bp,125.48bp)  .. (220.98bp,114.93bp);
+    % Edge: r2 -> 3
+    \draw [] (173.89bp,155.63bp) .. controls (164.66bp,146.64bp) and (148.71bp,130.93bp)  .. (135.5bp,117.0bp) .. controls (131.64bp,112.93bp) and (127.11bp,107.7bp)  .. (126.56bp,107.06bp);
+    \draw (139.66bp,122.18bp) node {\Large$\circ$};
+    % Edge: s3 -> 1
+    \draw [] (225.71bp,94.988bp) .. controls (222.93bp,79.384bp) and (217.83bp,50.854bp)  .. (217.52bp,49.086bp);
+    % Edge: s3 -> 2
+    \draw [] (229.11bp,94.988bp) .. controls (231.62bp,79.384bp) and (236.2bp,50.854bp)  .. (236.49bp,49.086bp);
+    \draw (233.31bp,68.833bp) node {\Large$\bullet$};
+    % Node: f_E
+  \begin{scope}
+    \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
+    \pgfsetstrokecolor{strokecol}
+    \draw (150.0bp,349.0bp) -- (107.0bp,349.0bp) -- (107.0bp,327.0bp) -- (150.0bp,327.0bp) -- cycle;
+    \draw (128.5bp,338.0bp) node {$f$};
+  \end{scope}
+    % Node: p0
+  \begin{scope}
+    \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
+    \pgfsetstrokecolor{strokecol}
+    \draw (128.5bp,280.0bp) ellipse (11.0bp and 11.0bp);
+    \draw (128.5bp,280.0bp) node {$p$};
+  \end{scope}
+    % Node: f_p
+  \begin{scope}
+    \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
+    \pgfsetstrokecolor{strokecol}
+    \draw (201.0bp,291.0bp) -- (158.0bp,291.0bp) -- (158.0bp,269.0bp) -- (201.0bp,269.0bp) -- cycle;
+    \draw (179.5bp,280.0bp) node {$f_{p}$};
+  \end{scope}
+    % Node: q1
+  \begin{scope}
+    \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
+    \pgfsetstrokecolor{strokecol}
+    \draw (151.5bp,222.0bp) ellipse (11.0bp and 11.0bp);
+    \draw (151.5bp,222.0bp) node {$q$};
+  \end{scope}
+    % Node: f_p!q
+  \begin{scope}
+    \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
+    \pgfsetstrokecolor{strokecol}
+    \draw (224.0bp,233.0bp) -- (181.0bp,233.0bp) -- (181.0bp,211.0bp) -- (224.0bp,211.0bp) -- cycle;
+    \draw (202.5bp,222.0bp) node {$f_{p\lnot q}$};
+  \end{scope}
+    % Node: r2
+  \begin{scope}
+    \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
+    \pgfsetstrokecolor{strokecol}
+    \draw (181.5bp,164.0bp) ellipse (11.0bp and 11.0bp);
+    \draw (181.5bp,164.0bp) node {$r$};
+  \end{scope}
+    % Node: f_p!qr
+  \begin{scope}
+    \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
+    \pgfsetstrokecolor{strokecol}
+    \draw (254.0bp,175.0bp) -- (211.0bp,175.0bp) -- (211.0bp,153.0bp) -- (254.0bp,153.0bp) -- cycle;
+    \draw (232.5bp,164.0bp) node {$f_{p\lnot qr}$};
+  \end{scope}
+    % Node: s3
+  \begin{scope}
+    \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
+    \pgfsetstrokecolor{strokecol}
+    \draw (227.5bp,106.0bp) ellipse (11.0bp and 11.0bp);
+    \draw (227.5bp,106.0bp) node {$s$};
+  \end{scope}
+    % Node: f_!p
+  \begin{scope}
+    \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
+    \pgfsetstrokecolor{strokecol}
+    \draw (99.0bp,291.0bp) -- (56.0bp,291.0bp) -- (56.0bp,269.0bp) -- (99.0bp,269.0bp) -- cycle;
+    \draw (77.5bp,280.0bp) node {$f_{\lnot p}$};
+  \end{scope}
+    % Node: r4
+  \begin{scope}
+    \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
+    \pgfsetstrokecolor{strokecol}
+    \draw (106.5bp,222.0bp) ellipse (11.0bp and 11.0bp);
+    \draw (106.5bp,222.0bp) node {$r$};
+  \end{scope}
+    % Node: f_!pr
+  \begin{scope}
+    \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
+    \pgfsetstrokecolor{strokecol}
+    \draw (43.0bp,233.0bp) -- (0.0bp,233.0bp) -- (0.0bp,211.0bp) -- (43.0bp,211.0bp) -- cycle;
+    \draw (21.5bp,222.0bp) node {$f_{\lnot pr}$};
+  \end{scope}
+    % Node: s5
+  \begin{scope}
+    \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
+    \pgfsetstrokecolor{strokecol}
+    \draw (21.5bp,164.0bp) ellipse (11.0bp and 11.0bp);
+    \draw (21.5bp,164.0bp) node {$s$};
+  \end{scope}
+    % Node: f_!prs
+  \begin{scope}
+    \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
+    \pgfsetstrokecolor{strokecol}
+    \draw (94.0bp,175.0bp) -- (51.0bp,175.0bp) -- (51.0bp,153.0bp) -- (94.0bp,153.0bp) -- cycle;
+    \draw (72.5bp,164.0bp) node {$f_{\lnot prs}$};
+  \end{scope}
+    % Node: t6
+  \begin{scope}
+    \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
+    \pgfsetstrokecolor{strokecol}
+    \draw (77.5bp,106.0bp) ellipse (11.0bp and 11.0bp);
+    \draw (77.5bp,106.0bp) node {$t$};
+  \end{scope}
+    % Node: f_!prst
+  \begin{scope}
+    \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
+    \pgfsetstrokecolor{strokecol}
+    \draw (188.0bp,117.0bp) -- (145.0bp,117.0bp) -- (145.0bp,95.0bp) -- (188.0bp,95.0bp) -- cycle;
+    \draw (166.5bp,106.0bp) node {$f_{\lnot prst}$};
+  \end{scope}
+    % Node: u7
+  \begin{scope}
+    \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
+    \pgfsetstrokecolor{strokecol}
+    \draw (137.5bp,48.0bp) ellipse (11.0bp and 11.0bp);
+    \draw (137.5bp,48.0bp) node {$u$};
+  \end{scope}
+  %
+  \end{tikzpicture}
+  
+  
+  }
+  \end{center}
+  
+  

bdds/0012.tex

@@ -0,0 +1,5 @@
+\item Construct a Reduced Ordered Binary Decision Diagram (ROBDD) for the formula 
+
+$$f = (p \land q) \lor (r \land s) \lor (\lnot p \land \lnot r)$$
+
+using \textit{alphabetic variable order}. Use complemented edges and a node for \texttt{true} as the only constant node. To simplify drawing, you may assume that \textit{dangling edges} point to the constant node. Write down all cofactors that you compute to obtain the final result and mark them in the graph.

bdds/0012_sol.tex

@@ -0,0 +1,136 @@
+\begin{cofactors}
+  $f$ \= $= (((p \land q) \lor (r \land s)) \lor (\lnot p \land \lnot r))$\\
+    \>$f_{p}$ \= $= (q \lor (r \land s))$\\
+    \>\>$f_{pq}$ \= $= \true$\\
+    \>\>$f_{p\lnot q}$ \= $= (r \land s)$\\
+    \>\>\>$f_{p\lnot qr}$ \= $= s$\\
+    \>\>\>\>$f_{p\lnot qrs}$ \= $= \true$\\
+    \>\>\>\>$f_{p\lnot qr\lnot s}$ \= $= \false$\\
+    \>\>\>$f_{p\lnot q\lnot r}$ \= $= \false$\\
+    \>$f_{\lnot p}$ \= $= ((r \land s) \lor \lnot r)$\\
+    \>\>$f_{\lnot pr}$ \= $= f_{p\lnot qr}$\\
+    \>\>$f_{\lnot p\lnot r}$ \= $= \true$\\
+  \end{cofactors}
+  The final ROBDD: 
+  \begin{center}
+  \scalebox{0.75}{
+  
+  \begin{tikzpicture}[>=latex,line join=bevel,]
+    \pgfsetlinewidth{1bp}
+  %%
+  \pgfsetcolor{black}
+    % Edge: f_E -> p0
+    \draw [] (76.036bp,268.99bp) .. controls (76.036bp,258.92bp) and (76.036bp,243.47bp)  .. (76.036bp,233.3bp);
+    % Edge: f_p -> q1
+    \draw [] (126.5bp,210.99bp) .. controls (125.96bp,200.92bp) and (125.13bp,185.47bp)  .. (124.59bp,175.3bp);
+    % Edge: f_p!q -> r2
+    \draw [] (171.28bp,152.99bp) .. controls (167.41bp,142.67bp) and (161.42bp,126.7bp)  .. (157.62bp,116.55bp);
+    % Edge: f_p!qr -> s3
+    \draw [] (84.036bp,94.988bp) .. controls (84.036bp,84.921bp) and (84.036bp,69.474bp)  .. (84.036bp,59.304bp);
+    % Edge: f_!p -> r4
+    \draw [] (26.109bp,210.99bp) .. controls (27.188bp,200.92bp) and (28.843bp,185.47bp)  .. (29.932bp,175.3bp);
+    % Edge: p0 -> r4
+    \draw [] (69.397bp,212.74bp) .. controls (60.889bp,202.15bp) and (46.225bp,183.9bp)  .. (37.704bp,173.3bp);
+    \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
+    \pgfsetstrokecolor{strokecol}
+    \draw (50.231bp,188.89bp) node {\Large$\circ$};
+    % Edge: p0 -> q1
+    \draw [] (82.711bp,213.21bp) .. controls (91.778bp,202.63bp) and (107.91bp,183.81bp)  .. (117.11bp,173.08bp);
+    % Edge: r4 -> s3
+    \draw [] (33.339bp,152.98bp) .. controls (36.665bp,139.41bp) and (43.48bp,114.64bp)  .. (53.036bp,95.0bp) .. controls (59.948bp,80.799bp) and (70.727bp,65.915bp)  .. (77.581bp,57.05bp);
+    % Edge: r4 -> 4
+    \draw [] (26.743bp,153.75bp) .. controls (19.589bp,138.34bp) and (5.9248bp,108.91bp)  .. (5.0778bp,107.09bp);
+    \draw (13.5bp,125.23bp) node {\Large$\circ$};
+    % Edge: q1 -> r2
+    \draw [] (128.86bp,154.0bp) .. controls (134.42bp,143.61bp) and (143.53bp,126.62bp)  .. (149.13bp,116.16bp);
+    \draw (139.68bp,133.79bp) node {\Large$\circ$};
+    % Edge: q1 -> 0
+    \draw [] (124.04bp,152.99bp) .. controls (124.04bp,137.38bp) and (124.04bp,108.85bp)  .. (124.04bp,107.09bp);
+    % Edge: r2 -> s3
+    \draw [] (145.73bp,98.356bp) .. controls (132.26bp,87.578bp) and (105.8bp,66.407bp)  .. (92.33bp,55.635bp);
+    % Edge: r2 -> 3
+    \draw [] (154.04bp,94.988bp) .. controls (154.04bp,79.384bp) and (154.04bp,50.854bp)  .. (154.04bp,49.086bp);
+    \draw (154.04bp,69.086bp) node {\Large$\bullet$};
+    % Edge: s3 -> 1
+    \draw [] (82.039bp,36.791bp) .. controls (79.47bp,23.663bp) and (75.307bp,2.3815bp)  .. (75.049bp,1.0642bp);
+    % Edge: s3 -> 2
+    \draw [] (86.256bp,36.791bp) .. controls (89.11bp,23.663bp) and (93.736bp,2.3815bp)  .. (94.023bp,1.0642bp);
+    \draw (89.774bp,20.608bp) node {\Large$\bullet$};
+    % Node: f_E
+  \begin{scope}
+    \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
+    \pgfsetstrokecolor{strokecol}
+    \draw (97.54bp,291.0bp) -- (54.54bp,291.0bp) -- (54.54bp,269.0bp) -- (97.54bp,269.0bp) -- cycle;
+    \draw (76.036bp,280.0bp) node {$f$};
+  \end{scope}
+    % Node: p0
+  \begin{scope}
+    \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
+    \pgfsetstrokecolor{strokecol}
+    \draw (76.04bp,222.0bp) ellipse (11.0bp and 11.0bp);
+    \draw (76.036bp,222.0bp) node {$p$};
+  \end{scope}
+    % Node: f_p
+  \begin{scope}
+    \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
+    \pgfsetstrokecolor{strokecol}
+    \draw (148.54bp,233.0bp) -- (105.54bp,233.0bp) -- (105.54bp,211.0bp) -- (148.54bp,211.0bp) -- cycle;
+    \draw (127.04bp,222.0bp) node {$f_{p}$};
+  \end{scope}
+    % Node: q1
+  \begin{scope}
+    \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
+    \pgfsetstrokecolor{strokecol}
+    \draw (124.04bp,164.0bp) ellipse (11.0bp and 11.0bp);
+    \draw (124.04bp,164.0bp) node {$q$};
+  \end{scope}
+    % Node: f_p!q
+  \begin{scope}
+    \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
+    \pgfsetstrokecolor{strokecol}
+    \draw (196.54bp,175.0bp) -- (153.54bp,175.0bp) -- (153.54bp,153.0bp) -- (196.54bp,153.0bp) -- cycle;
+    \draw (175.04bp,164.0bp) node {$f_{p\lnot q}$};
+  \end{scope}
+    % Node: r2
+  \begin{scope}
+    \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
+    \pgfsetstrokecolor{strokecol}
+    \draw (154.04bp,106.0bp) ellipse (11.0bp and 11.0bp);
+    \draw (154.04bp,106.0bp) node {$r$};
+  \end{scope}
+    % Node: f_p!qr
+  \begin{scope}
+    \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
+    \pgfsetstrokecolor{strokecol}
+    \draw (105.54bp,117.0bp) -- (62.54bp,117.0bp) -- (62.54bp,95.0bp) -- (105.54bp,95.0bp) -- cycle;
+    \draw (84.036bp,106.0bp) node {$f_{p\lnot qr}$};
+  \end{scope}
+    % Node: s3
+  \begin{scope}
+    \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
+    \pgfsetstrokecolor{strokecol}
+    \draw (84.04bp,48.0bp) ellipse (11.0bp and 11.0bp);
+    \draw (84.036bp,48.0bp) node {$s$};
+  \end{scope}
+    % Node: f_!p
+  \begin{scope}
+    \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
+    \pgfsetstrokecolor{strokecol}
+    \draw (46.54bp,233.0bp) -- (3.54bp,233.0bp) -- (3.54bp,211.0bp) -- (46.54bp,211.0bp) -- cycle;
+    \draw (25.036bp,222.0bp) node {$f_{\lnot p}$};
+  \end{scope}
+    % Node: r4
+  \begin{scope}
+    \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
+    \pgfsetstrokecolor{strokecol}
+    \draw (31.04bp,164.0bp) ellipse (11.0bp and 11.0bp);
+    \draw (31.036bp,164.0bp) node {$r$};
+  \end{scope}
+  %
+  \end{tikzpicture}
+  
+  
+  }
+  \end{center}
+  
+  

bdds/0013.tex

@@ -0,0 +1,5 @@
+\item Construct a Reduced Ordered Binary Decision Diagram (ROBDD) for the formula 
+
+$$f = (\lnot p \land q \land r) \lor (p \land \lnot s)$$
+
+using \textit{alphabetic variable order}. Use complemented edges and a node for \texttt{true} as the only constant node. To simplify drawing, you may assume that \textit{dangling edges} point to the constant node. Write down all cofactors that you compute to obtain the final result and mark them in the graph.

bdds/0013_sol.tex

@@ -0,0 +1,115 @@
+\begin{cofactors}
+  $f$ \= $= (((\lnot p \land q) \land r) \lor (p \land \lnot s))$\\
+    \>$f_{p}$ \= $= \lnot s$\\
+    \>\>$f_{ps}$ \= $= \false$\\
+    \>\>$f_{p\lnot s}$ \= $= \true$\\
+    \>$f_{\lnot p}$ \= $= (q \land r)$\\
+    \>\>$f_{\lnot pq}$ \= $= r$\\
+    \>\>\>$f_{\lnot pqr}$ \= $= \true$\\
+    \>\>\>$f_{\lnot pq\lnot r}$ \= $= \false$\\
+    \>\>$f_{\lnot p\lnot q}$ \= $= \false$\\
+  \end{cofactors}
+  The final ROBDD: 
+  \begin{center}
+  \scalebox{0.75}{
+  
+  \begin{tikzpicture}[>=latex,line join=bevel,]
+    \pgfsetlinewidth{1bp}
+  %%
+  \pgfsetcolor{black}
+    % Edge: f_E -> p0
+    \draw [] (72.5bp,210.99bp) .. controls (72.5bp,200.92bp) and (72.5bp,185.47bp)  .. (72.5bp,175.3bp);
+    \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
+    \pgfsetstrokecolor{strokecol}
+    \draw (72.5bp,195.3bp) node {\Large$\bullet$};
+    % Edge: f_p -> s1
+    \draw [] (123.5bp,152.99bp) .. controls (123.5bp,142.92bp) and (123.5bp,127.47bp)  .. (123.5bp,117.3bp);
+    \draw (123.5bp,137.3bp) node {\Large$\bullet$};
+    % Edge: f_!p -> q2
+    \draw [] (21.5bp,152.99bp) .. controls (21.5bp,142.92bp) and (21.5bp,127.47bp)  .. (21.5bp,117.3bp);
+    % Edge: f_!pq -> r3
+    \draw [] (70.712bp,94.988bp) .. controls (68.892bp,84.795bp) and (66.087bp,69.087bp)  .. (64.272bp,58.925bp);
+    % Edge: p0 -> q2
+    \draw [] (65.409bp,155.21bp) .. controls (55.665bp,144.51bp) and (38.243bp,125.38bp)  .. (28.531bp,114.72bp);
+    \draw (41.998bp,129.51bp) node {\Large$\bullet$};
+    % Edge: p0 -> s1
+    \draw [] (79.591bp,155.21bp) .. controls (89.335bp,144.51bp) and (106.76bp,125.38bp)  .. (116.47bp,114.72bp);
+    % Edge: q2 -> r3
+    \draw [] (27.55bp,96.737bp) .. controls (35.301bp,86.149bp) and (48.662bp,67.901bp)  .. (56.425bp,57.297bp);
+    % Edge: q2 -> 4
+    \draw [] (21.5bp,94.988bp) .. controls (21.5bp,79.384bp) and (21.5bp,50.854bp)  .. (21.5bp,49.086bp);
+    \draw (21.5bp,69.086bp) node {\Large$\bullet$};
+    % Edge: r3 -> 2
+    \draw [] (60.281bp,36.791bp) .. controls (57.427bp,23.663bp) and (52.8bp,2.3815bp)  .. (52.514bp,1.0642bp);
+    % Edge: r3 -> 3
+    \draw [] (64.497bp,36.791bp) .. controls (67.066bp,23.663bp) and (71.23bp,2.3815bp)  .. (71.487bp,1.0642bp);
+    \draw (67.647bp,20.692bp) node {\Large$\bullet$};
+    % Edge: s1 -> 0
+    \draw [] (121.71bp,94.988bp) .. controls (118.93bp,79.384bp) and (113.83bp,50.854bp)  .. (113.52bp,49.086bp);
+    % Edge: s1 -> 1
+    \draw [] (125.11bp,94.988bp) .. controls (127.62bp,79.384bp) and (132.2bp,50.854bp)  .. (132.49bp,49.086bp);
+    \draw (129.31bp,68.833bp) node {\Large$\bullet$};
+    % Node: f_E
+  \begin{scope}
+    \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
+    \pgfsetstrokecolor{strokecol}
+    \draw (94.0bp,233.0bp) -- (51.0bp,233.0bp) -- (51.0bp,211.0bp) -- (94.0bp,211.0bp) -- cycle;
+    \draw (72.5bp,222.0bp) node {$f$};
+  \end{scope}
+    % Node: p0
+  \begin{scope}
+    \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
+    \pgfsetstrokecolor{strokecol}
+    \draw (72.5bp,164.0bp) ellipse (11.0bp and 11.0bp);
+    \draw (72.5bp,164.0bp) node {$p$};
+  \end{scope}
+    % Node: f_p
+  \begin{scope}
+    \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
+    \pgfsetstrokecolor{strokecol}
+    \draw (145.0bp,175.0bp) -- (102.0bp,175.0bp) -- (102.0bp,153.0bp) -- (145.0bp,153.0bp) -- cycle;
+    \draw (123.5bp,164.0bp) node {$f_{p}$};
+  \end{scope}
+    % Node: s1
+  \begin{scope}
+    \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
+    \pgfsetstrokecolor{strokecol}
+    \draw (123.5bp,106.0bp) ellipse (11.0bp and 11.0bp);
+    \draw (123.5bp,106.0bp) node {$s$};
+  \end{scope}
+    % Node: f_!p
+  \begin{scope}
+    \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
+    \pgfsetstrokecolor{strokecol}
+    \draw (43.0bp,175.0bp) -- (0.0bp,175.0bp) -- (0.0bp,153.0bp) -- (43.0bp,153.0bp) -- cycle;
+    \draw (21.5bp,164.0bp) node {$f_{\lnot p}$};
+  \end{scope}
+    % Node: q2
+  \begin{scope}
+    \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
+    \pgfsetstrokecolor{strokecol}
+    \draw (21.5bp,106.0bp) ellipse (11.0bp and 11.0bp);
+    \draw (21.5bp,106.0bp) node {$q$};
+  \end{scope}
+    % Node: f_!pq
+  \begin{scope}
+    \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
+    \pgfsetstrokecolor{strokecol}
+    \draw (94.0bp,117.0bp) -- (51.0bp,117.0bp) -- (51.0bp,95.0bp) -- (94.0bp,95.0bp) -- cycle;
+    \draw (72.5bp,106.0bp) node {$f_{\lnot pq}$};
+  \end{scope}
+    % Node: r3
+  \begin{scope}
+    \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
+    \pgfsetstrokecolor{strokecol}
+    \draw (62.5bp,48.0bp) ellipse (11.0bp and 11.0bp);
+    \draw (62.5bp,48.0bp) node {$r$};
+  \end{scope}
+  %
+  \end{tikzpicture}
+  
+  
+  }
+  \end{center}
+  
+  

bdds/0014.tex

@@ -0,0 +1,5 @@
+\item Construct a Reduced Ordered Binary Decision Diagram (ROBDD) for the formula 
+
+$$f = (p \land q \land \lnot r) \lor (\lnot q \land s) \lor (\lnot p \land \lnot s)$$
+
+using \textit{alphabetic variable order}. Use complemented edges and a node for \texttt{true} as the only constant node. To simplify drawing, you may assume that \textit{dangling edges} point to the constant node. Write down all cofactors that you compute to obtain the final result and mark them in the graph.

bdds/0014_sol.tex

@@ -0,0 +1,140 @@
+\begin{cofactors}
+  $f$ \= $= ((((p \land q) \land \lnot r) \lor (\lnot q \land s)) \lor (\lnot p \land \lnot s))$\\
+    \>$f_{p}$ \= $= ((q \land \lnot r) \lor (\lnot q \land s))$\\
+    \>\>$f_{pq}$ \= $= \lnot r$\\
+    \>\>\>$f_{pqr}$ \= $= \false$\\
+    \>\>\>$f_{pq\lnot r}$ \= $= \true$\\
+    \>\>$f_{p\lnot q}$ \= $= s$\\
+    \>\>\>$f_{p\lnot qs}$ \= $= \true$\\
+    \>\>\>$f_{p\lnot q\lnot s}$ \= $= \false$\\
+    \>$f_{\lnot p}$ \= $= ((\lnot q \land s) \lor \lnot s)$\\
+    \>\>$f_{\lnot pq}$ \= $= \lnot f_{p\lnot q}$\\
+    \>\>$f_{\lnot p\lnot q}$ \= $= \true$\\
+  \end{cofactors}
+  The final ROBDD: 
+  \begin{center}
+  \scalebox{0.75}{
+  
+  \begin{tikzpicture}[>=latex,line join=bevel,]
+    \pgfsetlinewidth{1bp}
+  %%
+  \pgfsetcolor{black}
+    % Edge: f_E -> p0
+    \draw [] (72.5bp,210.99bp) .. controls (72.5bp,200.92bp) and (72.5bp,185.47bp)  .. (72.5bp,175.3bp);
+    \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
+    \pgfsetstrokecolor{strokecol}
+    \draw (72.5bp,195.3bp) node {\Large$\bullet$};
+    % Edge: f_p -> q1
+    \draw [] (123.5bp,152.99bp) .. controls (123.5bp,142.92bp) and (123.5bp,127.47bp)  .. (123.5bp,117.3bp);
+    \draw (123.5bp,137.3bp) node {\Large$\bullet$};
+    % Edge: f_pq -> r2
+    \draw [] (165.92bp,94.988bp) .. controls (156.53bp,84.04bp) and (141.7bp,66.729bp)  .. (133.15bp,56.763bp);
+    \draw (146.17bp,71.948bp) node {\Large$\bullet$};
+    % Edge: f_p!q -> s3
+    \draw [] (72.5bp,94.988bp) .. controls (72.5bp,84.921bp) and (72.5bp,69.474bp)  .. (72.5bp,59.304bp);
+    % Edge: f_!p -> q4
+    \draw [] (21.5bp,152.99bp) .. controls (21.5bp,142.92bp) and (21.5bp,127.47bp)  .. (21.5bp,117.3bp);
+    \draw (21.5bp,137.3bp) node {\Large$\bullet$};
+    % Edge: p0 -> q4
+    \draw [] (65.409bp,155.21bp) .. controls (55.665bp,144.51bp) and (38.243bp,125.38bp)  .. (28.531bp,114.72bp);
+    \draw (41.998bp,129.51bp) node {\Large$\circ$};
+    % Edge: p0 -> q1
+    \draw [] (79.591bp,155.21bp) .. controls (89.335bp,144.51bp) and (106.76bp,125.38bp)  .. (116.47bp,114.72bp);
+    % Edge: q4 -> s3
+    \draw [] (28.591bp,97.213bp) .. controls (38.335bp,86.514bp) and (55.757bp,67.385bp)  .. (65.469bp,56.72bp);
+    % Edge: q4 -> 4
+    \draw [] (21.5bp,94.988bp) .. controls (21.5bp,79.384bp) and (21.5bp,50.854bp)  .. (21.5bp,49.086bp);
+    \draw (21.5bp,69.086bp) node {\Large$\bullet$};
+    % Edge: q1 -> s3
+    \draw [] (116.41bp,97.213bp) .. controls (106.66bp,86.514bp) and (89.243bp,67.385bp)  .. (79.531bp,56.72bp);
+    \draw (92.998bp,71.507bp) node {\Large$\bullet$};
+    % Edge: q1 -> r2
+    \draw [] (124.04bp,94.988bp) .. controls (124.58bp,84.921bp) and (125.4bp,69.474bp)  .. (125.95bp,59.304bp);
+    % Edge: s3 -> 2
+    \draw [] (71.168bp,36.791bp) .. controls (69.456bp,23.663bp) and (66.68bp,2.3815bp)  .. (66.508bp,1.0642bp);
+    % Edge: s3 -> 3
+    \draw [] (75.257bp,37.245bp) .. controls (78.954bp,24.164bp) and (85.101bp,2.4123bp)  .. (85.481bp,1.0657bp);
+    \draw (80.042bp,20.312bp) node {\Large$\bullet$};
+    % Edge: r2 -> 0
+    \draw [] (123.11bp,37.245bp) .. controls (118.56bp,24.164bp) and (110.99bp,2.4123bp)  .. (110.52bp,1.0657bp);
+    % Edge: r2 -> 1
+    \draw [] (127.17bp,36.791bp) .. controls (128.02bp,23.663bp) and (129.41bp,2.3815bp)  .. (129.5bp,1.0642bp);
+    \draw (128.19bp,21.022bp) node {\Large$\bullet$};
+    % Node: f_E
+  \begin{scope}
+    \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
+    \pgfsetstrokecolor{strokecol}
+    \draw (94.0bp,233.0bp) -- (51.0bp,233.0bp) -- (51.0bp,211.0bp) -- (94.0bp,211.0bp) -- cycle;
+    \draw (72.5bp,222.0bp) node {$f$};
+  \end{scope}
+    % Node: p0
+  \begin{scope}
+    \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
+    \pgfsetstrokecolor{strokecol}
+    \draw (72.5bp,164.0bp) ellipse (11.0bp and 11.0bp);
+    \draw (72.5bp,164.0bp) node {$p$};
+  \end{scope}
+    % Node: f_p
+  \begin{scope}
+    \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
+    \pgfsetstrokecolor{strokecol}
+    \draw (145.0bp,175.0bp) -- (102.0bp,175.0bp) -- (102.0bp,153.0bp) -- (145.0bp,153.0bp) -- cycle;
+    \draw (123.5bp,164.0bp) node {$f_{p}$};
+  \end{scope}
+    % Node: q1
+  \begin{scope}
+    \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
+    \pgfsetstrokecolor{strokecol}
+    \draw (123.5bp,106.0bp) ellipse (11.0bp and 11.0bp);
+    \draw (123.5bp,106.0bp) node {$q$};
+  \end{scope}
+    % Node: f_pq
+  \begin{scope}
+    \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
+    \pgfsetstrokecolor{strokecol}
+    \draw (196.0bp,117.0bp) -- (153.0bp,117.0bp) -- (153.0bp,95.0bp) -- (196.0bp,95.0bp) -- cycle;
+    \draw (174.5bp,106.0bp) node {$f_{pq}$};
+  \end{scope}
+    % Node: r2
+  \begin{scope}
+    \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
+    \pgfsetstrokecolor{strokecol}
+    \draw (126.5bp,48.0bp) ellipse (11.0bp and 11.0bp);
+    \draw (126.5bp,48.0bp) node {$r$};
+  \end{scope}
+    % Node: f_p!q
+  \begin{scope}
+    \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
+    \pgfsetstrokecolor{strokecol}
+    \draw (94.0bp,117.0bp) -- (51.0bp,117.0bp) -- (51.0bp,95.0bp) -- (94.0bp,95.0bp) -- cycle;
+    \draw (72.5bp,106.0bp) node {$f_{p\lnot q}$};
+  \end{scope}
+    % Node: s3
+  \begin{scope}
+    \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
+    \pgfsetstrokecolor{strokecol}
+    \draw (72.5bp,48.0bp) ellipse (11.0bp and 11.0bp);
+    \draw (72.5bp,48.0bp) node {$s$};
+  \end{scope}
+    % Node: f_!p
+  \begin{scope}
+    \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
+    \pgfsetstrokecolor{strokecol}
+    \draw (43.0bp,175.0bp) -- (0.0bp,175.0bp) -- (0.0bp,153.0bp) -- (43.0bp,153.0bp) -- cycle;
+    \draw (21.5bp,164.0bp) node {$f_{\lnot p}$};
+  \end{scope}
+    % Node: q4
+  \begin{scope}
+    \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
+    \pgfsetstrokecolor{strokecol}
+    \draw (21.5bp,106.0bp) ellipse (11.0bp and 11.0bp);
+    \draw (21.5bp,106.0bp) node {$q$};
+  \end{scope}
+  %
+  \end{tikzpicture}
+  
+  
+  }
+  \end{center}
+  
+  

bdds/0015.tex

@@ -0,0 +1,5 @@
+\item Construct a Reduced Ordered Binary Decision Diagram (ROBDD) for the formula 
+
+$$f = (p \lor q) \land (r \lor \lnot s) \land (\lnot p \lor t)$$
+
+using \textit{alphabetic variable order}. Use complemented edges and a node for \texttt{true} as the only constant node. To simplify drawing, you may assume that \textit{dangling edges} point to the constant node. Write down all cofactors that you compute to obtain the final result and mark them in the graph.

bdds/0015_sol.tex

@@ -0,0 +1,184 @@
+\begin{cofactors}
+  $f$ \= $= (((p \lor q) \land (r \lor \lnot s)) \land (\lnot p \lor t))$\\
+    \>$f_{p}$ \= $= ((r \lor \lnot s) \land t)$\\
+    \>\>$f_{pr}$ \= $= t$\\
+    \>\>\>$f_{prt}$ \= $= \true$\\
+    \>\>\>$f_{pr\lnot t}$ \= $= \false$\\
+    \>\>$f_{p\lnot r}$ \= $= (\lnot s \land t)$\\
+    \>\>\>$f_{p\lnot rs}$ \= $= \false$\\
+    \>\>\>$f_{p\lnot r\lnot s}$ \= $= f_{pr}$\\
+    \>$f_{\lnot p}$ \= $= (q \land (r \lor \lnot s))$\\
+    \>\>$f_{\lnot pq}$ \= $= (r \lor \lnot s)$\\
+    \>\>\>$f_{\lnot pqr}$ \= $= \true$\\
+    \>\>\>$f_{\lnot pq\lnot r}$ \= $= \lnot s$\\
+    \>\>\>\>$f_{\lnot pq\lnot rs}$ \= $= \false$\\
+    \>\>\>\>$f_{\lnot pq\lnot r\lnot s}$ \= $= \true$\\
+    \>\>$f_{\lnot p\lnot q}$ \= $= \false$\\
+  \end{cofactors}
+  The final ROBDD: 
+  \begin{center}
+  \scalebox{0.75}{
+  
+  \begin{tikzpicture}[>=latex,line join=bevel,]
+    \pgfsetlinewidth{1bp}
+  %%
+  \pgfsetcolor{black}
+    % Edge: f_E -> p0
+    \draw [] (107.57bp,268.99bp) .. controls (107.57bp,258.92bp) and (107.57bp,243.47bp)  .. (107.57bp,233.3bp);
+    % Edge: f_p -> r1
+    \draw [] (188.57bp,210.99bp) .. controls (188.57bp,200.92bp) and (188.57bp,185.47bp)  .. (188.57bp,175.3bp);
+    % Edge: f_pr -> t2
+    \draw [] (231.63bp,94.988bp) .. controls (221.85bp,84.04bp) and (206.4bp,66.729bp)  .. (197.5bp,56.763bp);
+    % Edge: f_p!r -> s3
+    \draw [] (140.07bp,152.99bp) .. controls (142.62bp,142.8bp) and (146.55bp,127.09bp)  .. (149.09bp,116.93bp);
+    \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
+    \pgfsetstrokecolor{strokecol}
+    \draw (144.24bp,136.33bp) node {\Large$\bullet$};
+    % Edge: f_!p -> q4
+    \draw [] (23.569bp,210.99bp) .. controls (23.569bp,200.92bp) and (23.569bp,185.47bp)  .. (23.569bp,175.3bp);
+    % Edge: f_!pq -> r5
+    \draw [] (70.099bp,152.99bp) .. controls (65.493bp,142.67bp) and (58.363bp,126.7bp)  .. (53.833bp,116.55bp);
+    % Edge: f_!pq!r -> s6
+    \draw [] (98.781bp,94.988bp) .. controls (96.961bp,84.795bp) and (94.156bp,69.087bp)  .. (92.341bp,58.925bp);
+    \draw (95.857bp,78.614bp) node {\Large$\bullet$};
+    % Edge: p0 -> q4
+    \draw [] (98.896bp,215.22bp) .. controls (82.799bp,204.49bp) and (48.476bp,181.6bp)  .. (32.321bp,170.83bp);
+    \draw (48.962bp,181.93bp) node {\Large$\circ$};
+    % Edge: p0 -> r1
+    \draw [] (116.24bp,215.01bp) .. controls (131.75bp,204.29bp) and (164.08bp,181.93bp)  .. (179.72bp,171.12bp);
+    % Edge: q4 -> r5
+    \draw [] (27.862bp,153.75bp) .. controls (32.651bp,143.44bp) and (40.356bp,126.84bp)  .. (45.182bp,116.45bp);
+    % Edge: q4 -> 6
+    \draw [] (22.854bp,152.99bp) .. controls (21.739bp,137.38bp) and (19.701bp,108.85bp)  .. (19.575bp,107.09bp);
+    \draw (21.0bp,127.04bp) node {\Large$\bullet$};
+    % Edge: r5 -> s6
+    \draw [] (55.619bp,96.737bp) .. controls (63.37bp,86.149bp) and (76.731bp,67.901bp)  .. (84.494bp,57.297bp);
+    \draw (72.68bp,73.434bp) node {\Large$\bullet$};
+    % Edge: r5 -> 3
+    \draw [] (49.569bp,94.988bp) .. controls (49.569bp,79.384bp) and (49.569bp,50.854bp)  .. (49.569bp,49.086bp);
+    % Edge: s6 -> 4
+    \draw [] (88.35bp,36.791bp) .. controls (85.496bp,23.663bp) and (80.869bp,2.3815bp)  .. (80.583bp,1.0642bp);
+    % Edge: s6 -> 5
+    \draw [] (92.566bp,36.791bp) .. controls (95.135bp,23.663bp) and (99.299bp,2.3815bp)  .. (99.556bp,1.0642bp);
+    \draw (95.716bp,20.692bp) node {\Large$\bullet$};
+    % Edge: r1 -> s3
+    \draw [] (182.79bp,154.25bp) .. controls (175.8bp,143.67bp) and (164.11bp,125.98bp)  .. (157.19bp,115.51bp);
+    \draw (168.21bp,132.19bp) node {\Large$\bullet$};
+    % Edge: r1 -> t2
+    \draw [] (188.75bp,152.88bp) .. controls (189.12bp,131.3bp) and (190.01bp,80.962bp)  .. (190.39bp,59.258bp);
+    % Edge: s3 -> t2
+    \draw [] (157.66bp,96.25bp) .. controls (165.03bp,85.668bp) and (177.35bp,67.976bp)  .. (184.65bp,57.505bp);
+    \draw (173.22bp,73.917bp) node {\Large$\bullet$};
+    % Edge: s3 -> 2
+    \draw [] (151.57bp,94.988bp) .. controls (151.57bp,79.384bp) and (151.57bp,50.854bp)  .. (151.57bp,49.086bp);
+    % Edge: t2 -> 0
+    \draw [] (188.35bp,36.791bp) .. controls (185.5bp,23.663bp) and (180.87bp,2.3815bp)  .. (180.58bp,1.0642bp);
+    % Edge: t2 -> 1
+    \draw [] (192.57bp,36.791bp) .. controls (195.13bp,23.663bp) and (199.3bp,2.3815bp)  .. (199.56bp,1.0642bp);
+    \draw (195.72bp,20.692bp) node {\Large$\bullet$};
+    % Node: f_E
+  \begin{scope}
+    \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
+    \pgfsetstrokecolor{strokecol}
+    \draw (129.07bp,291.0bp) -- (86.07bp,291.0bp) -- (86.07bp,269.0bp) -- (129.07bp,269.0bp) -- cycle;
+    \draw (107.57bp,280.0bp) node {$f$};
+  \end{scope}
+    % Node: p0
+  \begin{scope}
+    \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
+    \pgfsetstrokecolor{strokecol}
+    \draw (107.57bp,222.0bp) ellipse (11.0bp and 11.0bp);
+    \draw (107.57bp,222.0bp) node {$p$};
+  \end{scope}
+    % Node: f_p
+  \begin{scope}
+    \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
+    \pgfsetstrokecolor{strokecol}
+    \draw (210.07bp,233.0bp) -- (167.07bp,233.0bp) -- (167.07bp,211.0bp) -- (210.07bp,211.0bp) -- cycle;
+    \draw (188.57bp,222.0bp) node {$f_{p}$};
+  \end{scope}
+    % Node: r1
+  \begin{scope}
+    \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
+    \pgfsetstrokecolor{strokecol}
+    \draw (188.57bp,164.0bp) ellipse (11.0bp and 11.0bp);
+    \draw (188.57bp,164.0bp) node {$r$};
+  \end{scope}
+    % Node: f_pr
+  \begin{scope}
+    \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
+    \pgfsetstrokecolor{strokecol}
+    \draw (262.07bp,117.0bp) -- (219.07bp,117.0bp) -- (219.07bp,95.0bp) -- (262.07bp,95.0bp) -- cycle;
+    \draw (240.57bp,106.0bp) node {$f_{pr}$};
+  \end{scope}
+    % Node: t2
+  \begin{scope}
+    \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
+    \pgfsetstrokecolor{strokecol}
+    \draw (190.57bp,48.0bp) ellipse (11.0bp and 11.0bp);
+    \draw (190.57bp,48.0bp) node {$t$};
+  \end{scope}
+    % Node: f_p!r
+  \begin{scope}
+    \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
+    \pgfsetstrokecolor{strokecol}
+    \draw (159.07bp,175.0bp) -- (116.07bp,175.0bp) -- (116.07bp,153.0bp) -- (159.07bp,153.0bp) -- cycle;
+    \draw (137.57bp,164.0bp) node {$f_{p\lnot r}$};
+  \end{scope}
+    % Node: s3
+  \begin{scope}
+    \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
+    \pgfsetstrokecolor{strokecol}
+    \draw (151.57bp,106.0bp) ellipse (11.0bp and 11.0bp);
+    \draw (151.57bp,106.0bp) node {$s$};
+  \end{scope}
+    % Node: f_!p
+  \begin{scope}
+    \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
+    \pgfsetstrokecolor{strokecol}
+    \draw (45.07bp,233.0bp) -- (2.07bp,233.0bp) -- (2.07bp,211.0bp) -- (45.07bp,211.0bp) -- cycle;
+    \draw (23.569bp,222.0bp) node {$f_{\lnot p}$};
+  \end{scope}
+    % Node: q4
+  \begin{scope}
+    \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
+    \pgfsetstrokecolor{strokecol}
+    \draw (23.57bp,164.0bp) ellipse (11.0bp and 11.0bp);
+    \draw (23.569bp,164.0bp) node {$q$};
+  \end{scope}
+    % Node: f_!pq
+  \begin{scope}
+    \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
+    \pgfsetstrokecolor{strokecol}
+    \draw (96.07bp,175.0bp) -- (53.07bp,175.0bp) -- (53.07bp,153.0bp) -- (96.07bp,153.0bp) -- cycle;
+    \draw (74.569bp,164.0bp) node {$f_{\lnot pq}$};
+  \end{scope}
+    % Node: r5
+  \begin{scope}
+    \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
+    \pgfsetstrokecolor{strokecol}
+    \draw (49.57bp,106.0bp) ellipse (11.0bp and 11.0bp);
+    \draw (49.569bp,106.0bp) node {$r$};
+  \end{scope}
+    % Node: f_!pq!r
+  \begin{scope}
+    \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
+    \pgfsetstrokecolor{strokecol}
+    \draw (122.07bp,117.0bp) -- (79.07bp,117.0bp) -- (79.07bp,95.0bp) -- (122.07bp,95.0bp) -- cycle;
+    \draw (100.57bp,106.0bp) node {$f_{\lnot pq\lnot r}$};
+  \end{scope}
+    % Node: s6
+  \begin{scope}
+    \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
+    \pgfsetstrokecolor{strokecol}
+    \draw (90.57bp,48.0bp) ellipse (11.0bp and 11.0bp);
+    \draw (90.569bp,48.0bp) node {$s$};
+  \end{scope}
+  %
+  \end{tikzpicture}
+  
+  
+  }
+  \end{center}
+  
+  

bdds/0016.tex

@@ -0,0 +1,5 @@
+\item Construct a Reduced Ordered Binary Decision Diagram (ROBDD) for the formula 
+
+$$f = (a \land b) \lor (\lnot c \land d) \lor (e \land \lnot a)$$
+
+using \textit{alphabetic variable order}. Use complemented edges and a node for \texttt{true} as the only constant node. To simplify drawing, you may assume that \textit{dangling edges} point to the constant node. Write down all cofactors that you compute to obtain the final result and mark them in the graph.

bdds/0016_sol.tex

@@ -0,0 +1,183 @@
+\begin{cofactors}
+  $f$ \= $= (((a \land b) \lor (\lnot c \land d)) \lor (e \land \lnot a))$\\
+    \>$f_{a}$ \= $= (b \lor (\lnot c \land d))$\\
+    \>\>$f_{ab}$ \= $= \true$\\
+    \>\>$f_{a\lnot b}$ \= $= (\lnot c \land d)$\\
+    \>\>\>$f_{a\lnot bc}$ \= $= \false$\\
+    \>\>\>$f_{a\lnot b\lnot c}$ \= $= d$\\
+    \>\>\>\>$f_{a\lnot b\lnot cd}$ \= $= \true$\\
+    \>\>\>\>$f_{a\lnot b\lnot c\lnot d}$ \= $= \false$\\
+    \>$f_{\lnot a}$ \= $= ((\lnot c \land d) \lor e)$\\
+    \>\>$f_{\lnot ac}$ \= $= e$\\
+    \>\>\>$f_{\lnot ace}$ \= $= \true$\\
+    \>\>\>$f_{\lnot ac\lnot e}$ \= $= \false$\\
+    \>\>$f_{\lnot a\lnot c}$ \= $= (d \lor e)$\\
+    \>\>\>$f_{\lnot a\lnot cd}$ \= $= \true$\\
+    \>\>\>$f_{\lnot a\lnot c\lnot d}$ \= $= f_{\lnot ac}$\\
+  \end{cofactors}
+  The final ROBDD: 
+  \begin{center}
+  \scalebox{0.75}{
+  
+  \begin{tikzpicture}[>=latex,line join=bevel,]
+    \pgfsetlinewidth{1bp}
+  %%
+  \pgfsetcolor{black}
+    % Edge: f_E -> a0
+    \draw [] (84.143bp,268.99bp) .. controls (84.143bp,258.92bp) and (84.143bp,243.47bp)  .. (84.143bp,233.3bp);
+    % Edge: f_a -> b1
+    \draw [] (135.14bp,210.99bp) .. controls (135.14bp,200.92bp) and (135.14bp,185.47bp)  .. (135.14bp,175.3bp);
+    % Edge: f_a!b -> c2
+    \draw [] (184.35bp,152.99bp) .. controls (182.53bp,142.8bp) and (179.73bp,127.09bp)  .. (177.91bp,116.93bp);
+    \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
+    \pgfsetstrokecolor{strokecol}
+    \draw (181.43bp,136.61bp) node {\Large$\bullet$};
+    % Edge: f_a!b!c -> d3
+    \draw [] (225.18bp,94.988bp) .. controls (223.17bp,84.795bp) and (220.09bp,69.087bp)  .. (218.09bp,58.925bp);
+    % Edge: f_!a -> c4
+    \draw [] (33.143bp,210.99bp) .. controls (33.143bp,200.92bp) and (33.143bp,185.47bp)  .. (33.143bp,175.3bp);
+    % Edge: f_!ac -> e5
+    \draw [] (33.143bp,94.988bp) .. controls (33.143bp,84.921bp) and (33.143bp,69.474bp)  .. (33.143bp,59.304bp);
+    % Edge: f_!a!c -> d6
+    \draw [] (84.143bp,152.99bp) .. controls (84.143bp,142.92bp) and (84.143bp,127.47bp)  .. (84.143bp,117.3bp);
+    % Edge: a0 -> c4
+    \draw [] (77.051bp,213.21bp) .. controls (67.308bp,202.51bp) and (49.886bp,183.38bp)  .. (40.174bp,172.72bp);
+    \draw (53.64bp,187.51bp) node {\Large$\circ$};
+    % Edge: a0 -> b1
+    \draw [] (91.234bp,213.21bp) .. controls (100.98bp,202.51bp) and (118.4bp,183.38bp)  .. (128.11bp,172.72bp);
+    % Edge: c4 -> d6
+    \draw [] (40.234bp,155.21bp) .. controls (49.978bp,144.51bp) and (67.4bp,125.38bp)  .. (77.112bp,114.72bp);
+    \draw (63.645bp,129.51bp) node {\Large$\circ$};
+    % Edge: c4 -> e5
+    \draw [] (25.797bp,155.3bp) .. controls (18.182bp,146.73bp) and (6.7572bp,132.1bp)  .. (2.1426bp,117.0bp) .. controls (-0.71421bp,107.65bp) and (-0.71421bp,104.35bp)  .. (2.1426bp,95.0bp) .. controls (6.7572bp,79.895bp) and (18.182bp,65.265bp)  .. (25.797bp,56.7bp);
+    % Edge: d6 -> e5
+    \draw [] (77.051bp,97.213bp) .. controls (67.308bp,86.514bp) and (49.886bp,67.385bp)  .. (40.174bp,56.72bp);
+    \draw (53.64bp,71.507bp) node {\Large$\circ$};
+    % Edge: d6 -> 6
+    \draw [] (84.143bp,94.988bp) .. controls (84.143bp,79.384bp) and (84.143bp,50.854bp)  .. (84.143bp,49.086bp);
+    % Edge: e5 -> 4
+    \draw [] (31.145bp,36.791bp) .. controls (28.577bp,23.663bp) and (24.413bp,2.3815bp)  .. (24.155bp,1.0642bp);
+    % Edge: e5 -> 5
+    \draw [] (35.362bp,36.791bp) .. controls (38.216bp,23.663bp) and (42.842bp,2.3815bp)  .. (43.129bp,1.0642bp);
+    \draw (38.88bp,20.608bp) node {\Large$\bullet$};
+    % Edge: b1 -> c2
+    \draw [] (141.19bp,154.74bp) .. controls (148.94bp,144.15bp) and (162.3bp,125.9bp)  .. (170.07bp,115.3bp);
+    \draw (158.25bp,131.43bp) node {\Large$\bullet$};
+    % Edge: b1 -> 0
+    \draw [] (135.14bp,152.99bp) .. controls (135.14bp,137.38bp) and (135.14bp,108.85bp)  .. (135.14bp,107.09bp);
+    % Edge: c2 -> d3
+    \draw [] (182.39bp,96.25bp) .. controls (189.95bp,85.668bp) and (202.59bp,67.976bp)  .. (210.07bp,57.505bp);
+    \draw (198.44bp,73.78bp) node {\Large$\bullet$};
+    % Edge: c2 -> 1
+    \draw [] (176.14bp,94.988bp) .. controls (176.14bp,79.384bp) and (176.14bp,50.854bp)  .. (176.14bp,49.086bp);
+    % Edge: d3 -> 2
+    \draw [] (213.92bp,36.791bp) .. controls (211.07bp,23.663bp) and (206.44bp,2.3815bp)  .. (206.16bp,1.0642bp);
+    % Edge: d3 -> 3
+    \draw [] (218.14bp,36.791bp) .. controls (220.71bp,23.663bp) and (224.87bp,2.3815bp)  .. (225.13bp,1.0642bp);
+    \draw (221.29bp,20.692bp) node {\Large$\bullet$};
+    % Node: f_E
+  \begin{scope}
+    \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
+    \pgfsetstrokecolor{strokecol}
+    \draw (105.64bp,291.0bp) -- (62.64bp,291.0bp) -- (62.64bp,269.0bp) -- (105.64bp,269.0bp) -- cycle;
+    \draw (84.143bp,280.0bp) node {$f$};
+  \end{scope}
+    % Node: a0
+  \begin{scope}
+    \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
+    \pgfsetstrokecolor{strokecol}
+    \draw (84.14bp,222.0bp) ellipse (11.0bp and 11.0bp);
+    \draw (84.143bp,222.0bp) node {$a$};
+  \end{scope}
+    % Node: f_a
+  \begin{scope}
+    \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
+    \pgfsetstrokecolor{strokecol}
+    \draw (156.64bp,233.0bp) -- (113.64bp,233.0bp) -- (113.64bp,211.0bp) -- (156.64bp,211.0bp) -- cycle;
+    \draw (135.14bp,222.0bp) node {$f_{a}$};
+  \end{scope}
+    % Node: b1
+  \begin{scope}
+    \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
+    \pgfsetstrokecolor{strokecol}
+    \draw (135.14bp,164.0bp) ellipse (11.0bp and 11.0bp);
+    \draw (135.14bp,164.0bp) node {$b$};
+  \end{scope}
+    % Node: f_a!b
+  \begin{scope}
+    \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
+    \pgfsetstrokecolor{strokecol}
+    \draw (207.64bp,175.0bp) -- (164.64bp,175.0bp) -- (164.64bp,153.0bp) -- (207.64bp,153.0bp) -- cycle;
+    \draw (186.14bp,164.0bp) node {$f_{a\lnot b}$};
+  \end{scope}
+    % Node: c2
+  \begin{scope}
+    \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
+    \pgfsetstrokecolor{strokecol}
+    \draw (176.14bp,106.0bp) ellipse (11.0bp and 11.0bp);
+    \draw (176.14bp,106.0bp) node {$c$};
+  \end{scope}
+    % Node: f_a!b!c
+  \begin{scope}
+    \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
+    \pgfsetstrokecolor{strokecol}
+    \draw (248.64bp,117.0bp) -- (205.64bp,117.0bp) -- (205.64bp,95.0bp) -- (248.64bp,95.0bp) -- cycle;
+    \draw (227.14bp,106.0bp) node {$f_{a\lnot b\lnot c}$};
+  \end{scope}
+    % Node: d3
+  \begin{scope}
+    \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
+    \pgfsetstrokecolor{strokecol}
+    \draw (216.14bp,48.0bp) ellipse (11.0bp and 11.0bp);
+    \draw (216.14bp,48.0bp) node {$d$};
+  \end{scope}
+    % Node: f_!a
+  \begin{scope}
+    \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
+    \pgfsetstrokecolor{strokecol}
+    \draw (54.64bp,233.0bp) -- (11.64bp,233.0bp) -- (11.64bp,211.0bp) -- (54.64bp,211.0bp) -- cycle;
+    \draw (33.143bp,222.0bp) node {$f_{\lnot a}$};
+  \end{scope}
+    % Node: c4
+  \begin{scope}
+    \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
+    \pgfsetstrokecolor{strokecol}
+    \draw (33.14bp,164.0bp) ellipse (11.0bp and 11.0bp);
+    \draw (33.143bp,164.0bp) node {$c$};
+  \end{scope}
+    % Node: f_!ac
+  \begin{scope}
+    \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
+    \pgfsetstrokecolor{strokecol}
+    \draw (54.64bp,117.0bp) -- (11.64bp,117.0bp) -- (11.64bp,95.0bp) -- (54.64bp,95.0bp) -- cycle;
+    \draw (33.143bp,106.0bp) node {$f_{\lnot ac}$};
+  \end{scope}
+    % Node: e5
+  \begin{scope}
+    \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
+    \pgfsetstrokecolor{strokecol}
+    \draw (33.14bp,48.0bp) ellipse (11.0bp and 11.0bp);
+    \draw (33.143bp,48.0bp) node {$e$};
+  \end{scope}
+    % Node: f_!a!c
+  \begin{scope}
+    \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
+    \pgfsetstrokecolor{strokecol}
+    \draw (105.64bp,175.0bp) -- (62.64bp,175.0bp) -- (62.64bp,153.0bp) -- (105.64bp,153.0bp) -- cycle;
+    \draw (84.143bp,164.0bp) node {$f_{\lnot a\lnot c}$};
+  \end{scope}
+    % Node: d6
+  \begin{scope}
+    \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
+    \pgfsetstrokecolor{strokecol}
+    \draw (84.14bp,106.0bp) ellipse (11.0bp and 11.0bp);
+    \draw (84.143bp,106.0bp) node {$d$};
+  \end{scope}
+  %
+  \end{tikzpicture}
+  
+  
+  }
+  \end{center}
+  
+  

bdds/0017.tex

@@ -0,0 +1,5 @@
+\item Construct a Reduced Ordered Binary Decision Diagram (ROBDD) for the formula 
+
+$$f = (a \land b) \lor (\lnot c \land d) \lor (e \land \lnot a)$$
+
+using \textit{reverse alphabetic variable order}. Use complemented edges and a node for \texttt{true} as the only constant node. To simplify drawing, you may assume that \textit{dangling edges} point to the constant node. Write down all cofactors that you compute to obtain the final result and mark them in the graph.

bdds/0017_sol.tex

@@ -0,0 +1,206 @@
+\begin{cofactors}
+  $f$ \= $= (((a \land b) \lor (\lnot c \land d)) \lor (e \land \lnot a))$\\
+    \>$f_{e}$ \= $= (((a \land b) \lor (\lnot c \land d)) \lor \lnot a)$\\
+    \>\>$f_{ed}$ \= $= (((a \land b) \lor \lnot c) \lor \lnot a)$\\
+    \>\>\>$f_{edc}$ \= $= ((a \land b) \lor \lnot a)$\\
+    \>\>\>\>$f_{edcb}$ \= $= \true$\\
+    \>\>\>\>$f_{edc\lnot b}$ \= $= \lnot a$\\
+    \>\>\>\>\>$f_{edc\lnot ba}$ \= $= \false$\\
+    \>\>\>\>\>$f_{edc\lnot b\lnot a}$ \= $= \true$\\
+    \>\>\>$f_{ed\lnot c}$ \= $= \true$\\
+    \>\>$f_{e\lnot d}$ \= $= f_{edc}$\\
+    \>$f_{\lnot e}$ \= $= ((a \land b) \lor (\lnot c \land d))$\\
+    \>\>$f_{\lnot ed}$ \= $= ((a \land b) \lor \lnot c)$\\
+    \>\>\>$f_{\lnot edc}$ \= $= (a \land b)$\\
+    \>\>\>\>$f_{\lnot edcb}$ \= $= \lnot f_{edc\lnot b}$\\
+    \>\>\>\>$f_{\lnot edc\lnot b}$ \= $= \false$\\
+    \>\>\>$f_{\lnot ed\lnot c}$ \= $= \true$\\
+    \>\>$f_{\lnot e\lnot d}$ \= $= f_{\lnot edc}$\\
+  \end{cofactors}
+  The final ROBDD: 
+  \begin{center}
+  \scalebox{0.75}{
+  
+  \begin{tikzpicture}[>=latex,line join=bevel,]
+    \pgfsetlinewidth{1bp}
+  %%
+  \pgfsetcolor{black}
+    % Edge: f_E -> e0
+    \draw [] (114.56bp,326.99bp) .. controls (114.56bp,316.92bp) and (114.56bp,301.47bp)  .. (114.56bp,291.3bp);
+    % Edge: f_e -> d1
+    \draw [] (195.56bp,268.99bp) .. controls (195.56bp,258.92bp) and (195.56bp,243.47bp)  .. (195.56bp,233.3bp);
+    % Edge: f_ed -> c2
+    \draw [] (145.63bp,210.99bp) .. controls (146.71bp,200.92bp) and (148.37bp,185.47bp)  .. (149.46bp,175.3bp);
+    % Edge: f_edc -> b3
+    \draw [] (236.62bp,152.99bp) .. controls (226.85bp,142.04bp) and (211.39bp,124.73bp)  .. (202.49bp,114.76bp);
+    % Edge: f_edc!b -> a4
+    \draw [] (144.56bp,94.988bp) .. controls (144.56bp,84.921bp) and (144.56bp,69.474bp)  .. (144.56bp,59.304bp);
+    \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
+    \pgfsetstrokecolor{strokecol}
+    \draw (144.56bp,79.304bp) node {\Large$\bullet$};
+    % Edge: f_!e -> d5
+    \draw [] (32.561bp,268.99bp) .. controls (32.561bp,258.92bp) and (32.561bp,243.47bp)  .. (32.561bp,233.3bp);
+    % Edge: f_!ed -> c6
+    \draw [] (83.74bp,210.99bp) .. controls (83.92bp,200.92bp) and (84.196bp,185.47bp)  .. (84.377bp,175.3bp);
+    % Edge: f_!edc -> b7
+    \draw [] (35.528bp,152.99bp) .. controls (37.53bp,142.8bp) and (40.616bp,127.09bp)  .. (42.612bp,116.93bp);
+    % Edge: e0 -> d5
+    \draw [] (105.79bp,273.01bp) .. controls (90.003bp,262.23bp) and (57.01bp,239.7bp)  .. (41.274bp,228.95bp);
+    \draw (57.79bp,240.23bp) node {\Large$\circ$};
+    % Edge: e0 -> d1
+    \draw [] (123.23bp,273.01bp) .. controls (138.74bp,262.29bp) and (171.07bp,239.93bp)  .. (186.71bp,229.12bp);
+    % Edge: d5 -> c6
+    \draw [] (39.792bp,213.21bp) .. controls (49.726bp,202.51bp) and (67.49bp,183.38bp)  .. (77.392bp,172.72bp);
+    % Edge: d5 -> b7
+    \draw [] (25.112bp,213.55bp) .. controls (13.652bp,201.13bp) and (-6.1173bp,175.38bp)  .. (2.5613bp,153.0bp) .. controls (9.2091bp,135.86bp) and (25.549bp,121.2bp)  .. (35.777bp,113.28bp);
+    \draw (20.122bp,125.73bp) node {\Large$\circ$};
+    % Edge: c6 -> b7
+    \draw [] (78.311bp,154.25bp) .. controls (70.753bp,143.67bp) and (58.116bp,125.98bp)  .. (50.636bp,115.51bp);
+    % Edge: c6 -> 5
+    \draw [] (84.561bp,152.99bp) .. controls (84.561bp,137.38bp) and (84.561bp,108.85bp)  .. (84.561bp,107.09bp);
+    \draw (84.561bp,127.09bp) node {\Large$\circ$};
+    % Edge: b7 -> a4
+    \draw [] (53.79bp,99.832bp) .. controls (72.84bp,89.164bp) and (116.32bp,64.813bp)  .. (135.36bp,54.155bp);
+    % Edge: b7 -> 4
+    \draw [] (44.561bp,94.988bp) .. controls (44.561bp,79.384bp) and (44.561bp,50.854bp)  .. (44.561bp,49.086bp);
+    \draw (44.561bp,69.086bp) node {\Large$\bullet$};
+    % Edge: d1 -> c2
+    \draw [] (188.92bp,212.74bp) .. controls (180.41bp,202.15bp) and (165.75bp,183.9bp)  .. (157.23bp,173.3bp);
+    % Edge: d1 -> b3
+    \draw [] (195.56bp,210.88bp) .. controls (195.56bp,189.3bp) and (195.56bp,138.96bp)  .. (195.56bp,117.26bp);
+    \draw (195.56bp,137.26bp) node {\Large$\circ$};
+    % Edge: c2 -> b3
+    \draw [] (157.2bp,154.74bp) .. controls (165.71bp,144.15bp) and (180.37bp,125.9bp)  .. (188.89bp,115.3bp);
+    % Edge: c2 -> 3
+    \draw [] (143.77bp,154.74bp) .. controls (131.38bp,139.65bp) and (106.2bp,108.99bp)  .. (104.64bp,107.09bp);
+    \draw (117.33bp,122.55bp) node {\Large$\circ$};
+    % Edge: b3 -> a4
+    \draw [] (188.47bp,97.213bp) .. controls (178.73bp,86.514bp) and (161.3bp,67.385bp)  .. (151.59bp,56.72bp);
+    \draw (165.06bp,71.507bp) node {\Large$\bullet$};
+    % Edge: b3 -> 0
+    \draw [] (195.56bp,94.988bp) .. controls (195.56bp,79.384bp) and (195.56bp,50.854bp)  .. (195.56bp,49.086bp);
+    % Edge: a4 -> 1
+    \draw [] (142.56bp,36.791bp) .. controls (140.0bp,23.663bp) and (135.83bp,2.3815bp)  .. (135.57bp,1.0642bp);
+    % Edge: a4 -> 2
+    \draw [] (146.78bp,36.791bp) .. controls (149.63bp,23.663bp) and (154.26bp,2.3815bp)  .. (154.55bp,1.0642bp);
+    \draw (150.3bp,20.608bp) node {\Large$\bullet$};
+    % Node: f_E
+  \begin{scope}
+    \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
+    \pgfsetstrokecolor{strokecol}
+    \draw (136.06bp,349.0bp) -- (93.06bp,349.0bp) -- (93.06bp,327.0bp) -- (136.06bp,327.0bp) -- cycle;
+    \draw (114.56bp,338.0bp) node {$f$};
+  \end{scope}
+    % Node: e0
+  \begin{scope}
+    \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
+    \pgfsetstrokecolor{strokecol}
+    \draw (114.56bp,280.0bp) ellipse (11.0bp and 11.0bp);
+    \draw (114.56bp,280.0bp) node {$e$};
+  \end{scope}
+    % Node: f_e
+  \begin{scope}
+    \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
+    \pgfsetstrokecolor{strokecol}
+    \draw (217.06bp,291.0bp) -- (174.06bp,291.0bp) -- (174.06bp,269.0bp) -- (217.06bp,269.0bp) -- cycle;
+    \draw (195.56bp,280.0bp) node {$f_{e}$};
+  \end{scope}
+    % Node: d1
+  \begin{scope}
+    \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
+    \pgfsetstrokecolor{strokecol}
+    \draw (195.56bp,222.0bp) ellipse (11.0bp and 11.0bp);
+    \draw (195.56bp,222.0bp) node {$d$};
+  \end{scope}
+    % Node: f_ed
+  \begin{scope}
+    \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
+    \pgfsetstrokecolor{strokecol}
+    \draw (166.06bp,233.0bp) -- (123.06bp,233.0bp) -- (123.06bp,211.0bp) -- (166.06bp,211.0bp) -- cycle;
+    \draw (144.56bp,222.0bp) node {$f_{ed}$};
+  \end{scope}
+    % Node: c2
+  \begin{scope}
+    \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
+    \pgfsetstrokecolor{strokecol}
+    \draw (150.56bp,164.0bp) ellipse (11.0bp and 11.0bp);
+    \draw (150.56bp,164.0bp) node {$c$};
+  \end{scope}
+    % Node: f_edc
+  \begin{scope}
+    \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
+    \pgfsetstrokecolor{strokecol}
+    \draw (267.06bp,175.0bp) -- (224.06bp,175.0bp) -- (224.06bp,153.0bp) -- (267.06bp,153.0bp) -- cycle;
+    \draw (245.56bp,164.0bp) node {$f_{edc}$};
+  \end{scope}
+    % Node: b3
+  \begin{scope}
+    \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
+    \pgfsetstrokecolor{strokecol}
+    \draw (195.56bp,106.0bp) ellipse (11.0bp and 11.0bp);
+    \draw (195.56bp,106.0bp) node {$b$};
+  \end{scope}
+    % Node: f_edc!b
+  \begin{scope}
+    \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
+    \pgfsetstrokecolor{strokecol}
+    \draw (166.06bp,117.0bp) -- (123.06bp,117.0bp) -- (123.06bp,95.0bp) -- (166.06bp,95.0bp) -- cycle;
+    \draw (144.56bp,106.0bp) node {$f_{edc\lnot b}$};
+  \end{scope}
+    % Node: a4
+  \begin{scope}
+    \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
+    \pgfsetstrokecolor{strokecol}
+    \draw (144.56bp,48.0bp) ellipse (11.0bp and 11.0bp);
+    \draw (144.56bp,48.0bp) node {$a$};
+  \end{scope}
+    % Node: f_!e
+  \begin{scope}
+    \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
+    \pgfsetstrokecolor{strokecol}
+    \draw (54.06bp,291.0bp) -- (11.06bp,291.0bp) -- (11.06bp,269.0bp) -- (54.06bp,269.0bp) -- cycle;
+    \draw (32.561bp,280.0bp) node {$f_{\lnot e}$};
+  \end{scope}
+    % Node: d5
+  \begin{scope}
+    \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
+    \pgfsetstrokecolor{strokecol}
+    \draw (32.56bp,222.0bp) ellipse (11.0bp and 11.0bp);
+    \draw (32.561bp,222.0bp) node {$d$};
+  \end{scope}
+    % Node: f_!ed
+  \begin{scope}
+    \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
+    \pgfsetstrokecolor{strokecol}
+    \draw (105.06bp,233.0bp) -- (62.06bp,233.0bp) -- (62.06bp,211.0bp) -- (105.06bp,211.0bp) -- cycle;
+    \draw (83.561bp,222.0bp) node {$f_{\lnot ed}$};
+  \end{scope}
+    % Node: c6
+  \begin{scope}
+    \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
+    \pgfsetstrokecolor{strokecol}
+    \draw (84.56bp,164.0bp) ellipse (11.0bp and 11.0bp);
+    \draw (84.561bp,164.0bp) node {$c$};
+  \end{scope}
+    % Node: f_!edc
+  \begin{scope}
+    \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
+    \pgfsetstrokecolor{strokecol}
+    \draw (55.06bp,175.0bp) -- (12.06bp,175.0bp) -- (12.06bp,153.0bp) -- (55.06bp,153.0bp) -- cycle;
+    \draw (33.561bp,164.0bp) node {$f_{\lnot edc}$};
+  \end{scope}
+    % Node: b7
+  \begin{scope}
+    \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
+    \pgfsetstrokecolor{strokecol}
+    \draw (44.56bp,106.0bp) ellipse (11.0bp and 11.0bp);
+    \draw (44.561bp,106.0bp) node {$b$};
+  \end{scope}
+  %
+  \end{tikzpicture}
+  
+  
+  }
+  \end{center}
+  
+  

bdds/0018.tex

@@ -0,0 +1,5 @@
+\item Construct a Reduced Ordered Binary Decision Diagram (ROBDD) for the formula 
+
+$$f = (p \land q) \lor (r \land s) \lor (\lnot p \land \lnot r)$$
+
+using \textit{reverse alphabetic variable order}. Use complemented edges and a node for \texttt{true} as the only constant node. To simplify drawing, you may assume that \textit{dangling edges} point to the constant node. Write down all cofactors that you compute to obtain the final result and mark them in the graph.

bdds/0018_sol.tex

@@ -0,0 +1,160 @@
+\begin{cofactors}
+  $f$ \= $= (((p \land q) \lor (r \land s)) \lor (\lnot p \land \lnot r))$\\
+    \>$f_{s}$ \= $= (((p \land q) \lor r) \lor (\lnot p \land \lnot r))$\\
+    \>\>$f_{sr}$ \= $= \true$\\
+    \>\>$f_{s\lnot r}$ \= $= ((p \land q) \lor \lnot p)$\\
+    \>\>\>$f_{s\lnot rq}$ \= $= \true$\\
+    \>\>\>$f_{s\lnot r\lnot q}$ \= $= \lnot p$\\
+    \>\>\>\>$f_{s\lnot r\lnot qp}$ \= $= \false$\\
+    \>\>\>\>$f_{s\lnot r\lnot q\lnot p}$ \= $= \true$\\
+    \>$f_{\lnot s}$ \= $= ((p \land q) \lor (\lnot p \land \lnot r))$\\
+    \>\>$f_{\lnot sr}$ \= $= (p \land q)$\\
+    \>\>\>$f_{\lnot srq}$ \= $= \lnot f_{s\lnot r\lnot q}$\\
+    \>\>\>$f_{\lnot sr\lnot q}$ \= $= \false$\\
+    \>\>$f_{\lnot s\lnot r}$ \= $= f_{s\lnot r}$\\
+  \end{cofactors}
+  The final ROBDD: 
+  \begin{center}
+  \scalebox{0.75}{
+  
+  \begin{tikzpicture}[>=latex,line join=bevel,]
+    \pgfsetlinewidth{1bp}
+  %%
+  \pgfsetcolor{black}
+    % Edge: f_E -> s0
+    \draw [] (123.5bp,268.99bp) .. controls (123.5bp,258.92bp) and (123.5bp,243.47bp)  .. (123.5bp,233.3bp);
+    % Edge: f_s -> r1
+    \draw [] (174.5bp,210.99bp) .. controls (174.5bp,200.92bp) and (174.5bp,185.47bp)  .. (174.5bp,175.3bp);
+    % Edge: f_s!r -> q2
+    \draw [] (123.5bp,152.99bp) .. controls (123.5bp,142.92bp) and (123.5bp,127.47bp)  .. (123.5bp,117.3bp);
+    % Edge: f_s!r!q -> p3
+    \draw [] (72.5bp,94.988bp) .. controls (72.5bp,84.921bp) and (72.5bp,69.474bp)  .. (72.5bp,59.304bp);
+    \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
+    \pgfsetstrokecolor{strokecol}
+    \draw (72.5bp,79.304bp) node {\Large$\bullet$};
+    % Edge: f_!s -> r4
+    \draw [] (72.5bp,210.99bp) .. controls (72.5bp,200.92bp) and (72.5bp,185.47bp)  .. (72.5bp,175.3bp);
+    % Edge: f_!sr -> q5
+    \draw [] (21.5bp,152.99bp) .. controls (21.5bp,142.92bp) and (21.5bp,127.47bp)  .. (21.5bp,117.3bp);
+    % Edge: s0 -> r4
+    \draw [] (116.41bp,213.21bp) .. controls (106.66bp,202.51bp) and (89.243bp,183.38bp)  .. (79.531bp,172.72bp);
+    \draw (92.998bp,187.51bp) node {\Large$\circ$};
+    % Edge: s0 -> r1
+    \draw [] (130.59bp,213.21bp) .. controls (140.34bp,202.51bp) and (157.76bp,183.38bp)  .. (167.47bp,172.72bp);
+    % Edge: r4 -> q5
+    \draw [] (65.409bp,155.21bp) .. controls (55.665bp,144.51bp) and (38.243bp,125.38bp)  .. (28.531bp,114.72bp);
+    % Edge: r4 -> q2
+    \draw [] (79.591bp,155.21bp) .. controls (89.335bp,144.51bp) and (106.76bp,125.38bp)  .. (116.47bp,114.72bp);
+    \draw (103.0bp,129.51bp) node {\Large$\circ$};
+    % Edge: q5 -> p3
+    \draw [] (28.591bp,97.213bp) .. controls (38.335bp,86.514bp) and (55.757bp,67.385bp)  .. (65.469bp,56.72bp);
+    % Edge: q5 -> 4
+    \draw [] (21.5bp,94.988bp) .. controls (21.5bp,79.384bp) and (21.5bp,50.854bp)  .. (21.5bp,49.086bp);
+    \draw (21.5bp,69.086bp) node {\Large$\bullet$};
+    % Edge: r1 -> q2
+    \draw [] (167.41bp,155.21bp) .. controls (157.66bp,144.51bp) and (140.24bp,125.38bp)  .. (130.53bp,114.72bp);
+    \draw (144.0bp,129.51bp) node {\Large$\circ$};
+    % Edge: r1 -> 0
+    \draw [] (174.5bp,152.99bp) .. controls (174.5bp,137.38bp) and (174.5bp,108.85bp)  .. (174.5bp,107.09bp);
+    % Edge: q2 -> p3
+    \draw [] (116.41bp,97.213bp) .. controls (106.66bp,86.514bp) and (89.243bp,67.385bp)  .. (79.531bp,56.72bp);
+    \draw (92.998bp,71.507bp) node {\Large$\bullet$};
+    % Edge: q2 -> 1
+    \draw [] (123.5bp,94.988bp) .. controls (123.5bp,79.384bp) and (123.5bp,50.854bp)  .. (123.5bp,49.086bp);
+    % Edge: p3 -> 2
+    \draw [] (70.281bp,36.791bp) .. controls (67.427bp,23.663bp) and (62.8bp,2.3815bp)  .. (62.514bp,1.0642bp);
+    % Edge: p3 -> 3
+    \draw [] (74.497bp,36.791bp) .. controls (77.066bp,23.663bp) and (81.23bp,2.3815bp)  .. (81.487bp,1.0642bp);
+    \draw (77.647bp,20.692bp) node {\Large$\bullet$};
+    % Node: f_E
+  \begin{scope}
+    \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
+    \pgfsetstrokecolor{strokecol}
+    \draw (145.0bp,291.0bp) -- (102.0bp,291.0bp) -- (102.0bp,269.0bp) -- (145.0bp,269.0bp) -- cycle;
+    \draw (123.5bp,280.0bp) node {$f$};
+  \end{scope}
+    % Node: s0
+  \begin{scope}
+    \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
+    \pgfsetstrokecolor{strokecol}
+    \draw (123.5bp,222.0bp) ellipse (11.0bp and 11.0bp);
+    \draw (123.5bp,222.0bp) node {$s$};
+  \end{scope}
+    % Node: f_s
+  \begin{scope}
+    \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
+    \pgfsetstrokecolor{strokecol}
+    \draw (196.0bp,233.0bp) -- (153.0bp,233.0bp) -- (153.0bp,211.0bp) -- (196.0bp,211.0bp) -- cycle;
+    \draw (174.5bp,222.0bp) node {$f_{s}$};
+  \end{scope}
+    % Node: r1
+  \begin{scope}
+    \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
+    \pgfsetstrokecolor{strokecol}
+    \draw (174.5bp,164.0bp) ellipse (11.0bp and 11.0bp);
+    \draw (174.5bp,164.0bp) node {$r$};
+  \end{scope}
+    % Node: f_s!r
+  \begin{scope}
+    \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
+    \pgfsetstrokecolor{strokecol}
+    \draw (145.0bp,175.0bp) -- (102.0bp,175.0bp) -- (102.0bp,153.0bp) -- (145.0bp,153.0bp) -- cycle;
+    \draw (123.5bp,164.0bp) node {$f_{s\lnot r}$};
+  \end{scope}
+    % Node: q2
+  \begin{scope}
+    \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
+    \pgfsetstrokecolor{strokecol}
+    \draw (123.5bp,106.0bp) ellipse (11.0bp and 11.0bp);
+    \draw (123.5bp,106.0bp) node {$q$};
+  \end{scope}
+    % Node: f_s!r!q
+  \begin{scope}
+    \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
+    \pgfsetstrokecolor{strokecol}
+    \draw (94.0bp,117.0bp) -- (51.0bp,117.0bp) -- (51.0bp,95.0bp) -- (94.0bp,95.0bp) -- cycle;
+    \draw (72.5bp,106.0bp) node {$f_{s\lnot r\lnot q}$};
+  \end{scope}
+    % Node: p3
+  \begin{scope}
+    \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
+    \pgfsetstrokecolor{strokecol}
+    \draw (72.5bp,48.0bp) ellipse (11.0bp and 11.0bp);
+    \draw (72.5bp,48.0bp) node {$p$};
+  \end{scope}
+    % Node: f_!s
+  \begin{scope}
+    \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
+    \pgfsetstrokecolor{strokecol}
+    \draw (94.0bp,233.0bp) -- (51.0bp,233.0bp) -- (51.0bp,211.0bp) -- (94.0bp,211.0bp) -- cycle;
+    \draw (72.5bp,222.0bp) node {$f_{\lnot s}$};
+  \end{scope}
+    % Node: r4
+  \begin{scope}
+    \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
+    \pgfsetstrokecolor{strokecol}
+    \draw (72.5bp,164.0bp) ellipse (11.0bp and 11.0bp);
+    \draw (72.5bp,164.0bp) node {$r$};
+  \end{scope}
+    % Node: f_!sr
+  \begin{scope}
+    \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
+    \pgfsetstrokecolor{strokecol}
+    \draw (43.0bp,175.0bp) -- (0.0bp,175.0bp) -- (0.0bp,153.0bp) -- (43.0bp,153.0bp) -- cycle;
+    \draw (21.5bp,164.0bp) node {$f_{\lnot sr}$};
+  \end{scope}
+    % Node: q5
+  \begin{scope}
+    \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
+    \pgfsetstrokecolor{strokecol}
+    \draw (21.5bp,106.0bp) ellipse (11.0bp and 11.0bp);
+    \draw (21.5bp,106.0bp) node {$q$};
+  \end{scope}
+  %
+  \end{tikzpicture}
+  
+  
+  }
+  \end{center}
+  
+  

bdds/1005.tex

@@ -0,0 +1,4 @@
+\item
+What is the worst-case size of a binary decision diagram?
+What is the advantage of computing a reduced and ordered BDD to represent a formula compared to using a truth table?
+

bdds/1006.tex

@@ -0,0 +1,10 @@
+\item How many nodes does a \textit{Reduced Ordered Binary Decision Diagrams (ROBDDs)} for a Boolean formula with $n$ variables have, in worst-case?
+
+\begin{itemize}
+  \item[$\square$] $2n$
+  \item[$\square$] $\mathcal{O}(n^2)$
+  \item[$\square$] $\mathcal{O}(2^n)$
+  \item[$\square$] $2^{n+1}-1$
+  \item[$\square$] $n^2$
+  \item[$\square$] infinitely many
+\end{itemize}

bdds/1008.tex

@@ -0,0 +1,9 @@
+\item \self Tick all properties that apply to a reduced and ordered binary decision diagram.
+\begin{itemize}
+  \item[$\square$] A reduced and ordered BDD is a canonical representation of the formula it represents, for any fixed variable order.
+  \item[$\square$] Since it is reduced, the number of nodes in the reduced and ordered BDD does not exceed $2n^2$, where $n$ is the number of variables.
+  \item[$\square$] The graph of an BDD may contain cycles.
+  \item[$\square$] A BDD represents a propositional formula as directed acyclic graph (DAG).
+  \item[$\square$] Every node with two non-complemented outgoing edges has two distinct child nodes.
+  \item[$\square$] No two nodes in an reduced and ordered BDD represent the same cofactor.
+\end{itemize}

bdds/1009.tex

@@ -0,0 +1 @@
+\item Given you have computed the reduced and ordered BDD for a formula $f$. How can you compute the BDD representation for $\neg f$ in \textit{constant} time?

bdds/1010.tex

@@ -0,0 +1 @@
+\item How can you compute the propositional formula $f$ represented by a given BDD?

bdds/1011.tex

@@ -0,0 +1 @@
+\item Give the definition of redundant nodes in a BDD. Give an example for a BDD that contains at least one redundant node.

bdds/1012.tex

@@ -0,0 +1,29 @@
+\item \BDDToROBDD
+
+\begin{center}
+\begin{forest}
+for tree={circle, draw, no edge,
+            minimum size=2em,
+            inner sep=0pt,
+            s sep=2mm,
+            l sep=2mm}
+[$f$, rectangle, draw,  tikz={\draw [line] () to (a.north);}
+    [$a$, name=a, tikz={\draw [line] () to (b1.north east); \draw [dot=.5] () to (b2.north west);}
+        [$b$, name=b1, tikz={\draw [line] () to (c1.north); \draw [dot=.5] () to (c2.north);}
+            [$c$, name=c1, tikz={\draw [line, bend right] () to (one.west); \draw [dot=.5, bend right] () to (zero.west);}]
+            [$c$, l sep=15mm, name=c2, tikz={\draw [line, bend right] () to (one.north); \draw [dot=.5, bend left] () to (one.north east);}
+                [$1$, rectangle, draw, name=one]
+                [,phantom]
+            ]
+        ]
+        [$b$, name=b2, tikz={\draw [line] () to (c3.north); \draw [dot=.5] () to (c4.north);}
+            [$c$, l sep=15mm, name=c3, tikz={\draw [line, bend left] (.south) to (one.east); \draw [dot=.3, bend left] (.south east) to (one.south east);}
+                [,phantom]
+                [$0$, rectangle, draw, name=zero]
+            ]
+            [$c$, name=c4, tikz={\draw [line, bend left] (.south) to (zero.north); \draw [dot=.5, bend left] () to (zero.north east);}]
+        ]
+    ]
+]
+\end{forest}
+\end{center}

bdds/1013.tex

@@ -0,0 +1,29 @@
+\item \BDDToROBDD
+
+\begin{center}
+\begin{forest}
+for tree={circle, draw, no edge,
+            minimum size=2em,
+            inner sep=0pt,
+            s sep=5mm,
+            l sep=5mm}
+[$f$, rectangle, draw,  tikz={\draw [fulldot=.5] () to (a.north);}
+    [$a$, name=a, tikz={\draw [line] () to (b1.north); \draw [dot=.5] () to (b2.north);}
+        [$b$, name=b1, tikz={\draw [line] () to (d1.north); \draw [dot=.5] () to (260:3.3);}
+            [$d$, name=d1, tikz={\draw [line] () to (c1.north); \draw [fulldot=.5] () to (255:4.6);}
+                [$c$, name=c1, tikz={\draw [line] () to (243:6.5); \draw [dot=.5] () to (253:6.05);}]
+                [,phantom]
+            ]
+            [,phantom]
+        ]
+        [$b$, name=b2, tikz={\draw [line] () to (d2.north); \draw [dot=.5] () to (300:3.7);}
+            [$d$, name=d2, tikz={\draw [line] () to (c2.north); \draw [fulldot=.5] () to (286:4.7);}
+                [$c$, name=c2, tikz={\draw [line] () to (265:5.8); \draw [dot=.5] () to (275:5.8);}]
+                [,phantom]
+            ]
+            [,phantom]
+        ]
+    ]
+]
+\end{forest}
+\end{center}

bdds/1014.tex

@@ -0,0 +1,27 @@
+\item \BDDToROBDD
+
+\begin{center}
+\begin{forest}
+for tree={circle, draw, no edge,
+            minimum size=2em,
+            inner sep=0pt,
+            s sep=5mm,
+            l sep=5mm}
+[$f$, rectangle, draw,  tikz={\draw [line] () to (a.north);}
+    [$a$, name=a, tikz={\draw [line] () to (b1.north); \draw [fulldot=.5] () to (b2.north);}
+        [$b$, name=b1, tikz={\draw [line] () to (c1.north); \draw [dot=.5] () to (c2.north);}
+            [$c$, name=c1, tikz={\draw [line, bend right] () to (d1.north west); \draw [fulldot=.5, bend left] () to (d1.north east);}
+                [$d$, name=d1, tikz={\draw [line] () to (249:6.37); \draw [fulldot=.5] () to (256:6.13);}]
+            ]
+            [$c$, name=c2, tikz={\draw [line] () to (258:4.7); \draw [fulldot=.5] () to (267:4.6);}]
+        ]
+        [$b$, name=b2, tikz={\draw [line] () to (c3.north); \draw [dot=.5] () to (c4.north);}
+            [$c$, name=c3, tikz={\draw [line, bend right] () to (d2.north west); \draw [dot=.5, bend left] () to (d2.north east);}
+                [$d$, name=d2, tikz={\draw [line] () to (272:5.9); \draw [fulldot=.5] () to (280:5.98);}]
+            ]
+            [$c$, name=c4, tikz={\draw [line] () to (288:4.83); \draw [fulldot=.5] () to (295:5.05);}]
+        ]
+    ]
+]
+\end{forest}
+\end{center}

bdds/1015.tex

@@ -0,0 +1 @@
+\item \self In the context of \textit{Binary Decision Diagrams (BDDs)}, how does the variable order impact the BDD?

bdds/1016.tex

@@ -0,0 +1,8 @@
+\item \self Tick all properties that apply to a reduced and ordered BDD.
+\begin{itemize}
+  \item[$\square$] If the \textit{else}-edge of a node is complemented, it may point to the same child node as the \textit{then}-edge.
+  \item[$\square$] Using the reduced and ordered BDD representation of formula $f$, it is possible to whether $f$ is valid in constant time.
+  \item[$\square$] The size of a BDD is independent on the variable order.
+  \item[$\square$] Using complemented edges, negation can be performed in constant time.
+  \item[$\square$] The size of a BDD is independent of the variable order.
+\end{itemize}

bdds/1017.tex

@@ -0,0 +1,6 @@
+\item \self Tick all properties that apply to a \textit{reduced} and \textit{ordered} BDD (ROBDD).
+\begin{itemize}
+  \item[$\square$] Checks for entailment can be done in constant time.
+  \item[$\square$] Some formulas that can be expressed by truth tables cannot be expressed by BDDs.
+  \item[$\square$] Equivalence checks can be performed in constant time (assuming that the BDDs for the formula to check are already available).
+\end{itemize}

bdds/1018.tex

@@ -0,0 +1 @@
+\item When do we consider a BDD to be reduced? Explain the types of redudancies that are not allowed to appear in a reduced and ordered BDD.

bdds/1019.tex

@@ -0,0 +1,22 @@
+\item \BDDToDNF
+
+\begin{center}
+\begin{forest}
+for tree={circle, draw, no edge,
+            minimum size=2em,
+            inner sep=0pt,
+            s sep=2mm,
+            l sep=1mm}
+[$f$, rectangle, draw,  tikz={\draw [line] () to (a.north);}
+    [$a$, name=a, tikz={\draw [line] () to (b.north east); \draw [fulldot=.5] () to (d.north);}
+        [$b$, name=b, tikz={\draw [fulldot=.5, bend left] () to (c.north east); \draw [line, bend right] () to (c.north west);}
+            [$c$, name=c, s sep=20mm, tikz={\draw [line] () to (250:4); \draw [fulldot=.5] () to (d.north west);}
+                [,phantom]
+                [$d$, name=d, s sep=10mm, tikz={\draw [fulldot=.5] () to (287:5); \draw [line] () to (275:4.8);}]
+            ]
+        ]
+        [,phantom]
+    ]
+]
+\end{forest}
+\end{center}

bdds/1020.tex

@@ -0,0 +1,21 @@
+\item \BDDToDNF
+
+\begin{center}
+\begin{forest}
+for tree={circle, draw, no edge,
+          minimum size=2em,
+          inner sep=0pt,
+          s sep=10mm,
+          l sep=4mm}
+[$f$, rectangle, draw,  tikz={\draw [fulldot=.5] () to (c.north);}
+   [$c$, name=c, tikz={\draw [line] () to (d1.north); \draw [dot=.5] () to (d2.north);}
+      [$d$, name=d1, tikz={\draw [line] () to (b.north); \draw [dot=.5] () to (266:2.7);}
+         [$b$, name=b, tikz={\draw [line] () to (a.north); \draw [fulldot=.5] () to (266:3.8);}
+            [$a$, name=a, tikz={\draw [line] () to (256:5.2); \draw [fulldot=.5] () to (266:5);}]
+         ]
+      ]
+      [$d$, name=d2, tikz={\draw [line] () to (b.east); \draw [dot=.5] () to (a.east);}]
+   ]
+]
+\end{forest}
+\end{center}

bdds/1021.tex

@@ -0,0 +1,26 @@
+\item For the following binary decision diagram:
+
+\begin{center}
+\begin{forest}
+for tree={circle, draw, no edge,
+          minimum size=2em,
+          inner sep=0pt,
+          s sep=10mm,
+          l sep=5mm}
+[$f$, rectangle, draw,  tikz={\draw [fulldot=.5] () to (a.north);}
+   [$a$, name=a, tikz={\draw [line] () to (b1.north); \draw [fulldot=.5] () to (c.north);}
+      [$b$, name=b1, tikz={\draw [line, bend right] () to (d.north west); \draw [dot=.5, bend left] () to (d.north east);}
+         [$d$, name=d, tikz={\draw [line] () to (254:4.6); \draw [fulldot=.5] () to (265:4.42);}]
+      ]
+      [$c$, name=c, tikz={\draw [line] () to (d.east); \draw [fulldot=.5] () to (291:3.53);}]
+   ]
+]
+\end{forest}
+\end{center}
+
+Check if the following models are satisfying:
+\begin{align*}
+  \Model_1 =~& \{a = \top, b = \top, c = \bot, d = \bot\},\\
+  \Model_2 =~& \{a = \bot, b = \bot, c = \top, d = \top\},\text{~and~}
+\end{align*}
+compute \DNF{f}.

bdds/1022.tex

@@ -0,0 +1,28 @@
+\item For the following binary decision diagram:
+
+
+\begin{center}
+\begin{forest}
+for tree={circle, draw, no edge,
+          minimum size=2em,
+          inner sep=0pt,
+          s sep=10mm,
+          l sep=5mm}
+[$f$, rectangle, draw,  tikz={\draw [line] () to (a.north);}
+   [$a$, name=a, tikz={\draw [line] () to (b1.north); \draw [fulldot=.5] () to (c2.north);}
+      [$b$, name=b1, tikz={\draw [line, bend right] () to (c1.north west); \draw [dot=.5, bend left] () to (c1.north east);}
+         [$c$, name=c1, tikz={\draw [line] () to (254:4.6); \draw [fulldot=.5] () to (265:4.42);}]
+      ]
+      [$c$, name=c2, tikz={\draw [line] () to (277:3.35); \draw [fulldot=.5] () to (291:3.53);}]
+   ]
+]
+\end{forest}
+\end{center}
+
+
+Check if the following models are satisfying:
+\begin{align*}
+  \Model_1 =~&\{a = \top, b = \top, c = \bot\},\\
+  \Model_2 =~&\{a = \bot, b = \bot, c = \bot\},\text{~and~}
+\end{align*}
+compute \DNF{f}.

bdds/1023.tex

@@ -0,0 +1,29 @@
+\item For the following binary decision diagram:
+
+
+Note: Else-edges are marked with circles. Filled circles represent the \emph{complemented} attribute. Dangling edges are assumed to point to the constant node \texttt{true}.
+
+\begin{center}
+\begin{forest}
+for tree={circle, draw, no edge,
+          minimum size=2em,
+          inner sep=0pt,
+          s sep=10mm,
+          l sep=5mm}
+[$f$, rectangle, draw,  tikz={\draw [fulldot=.5] () to (a.north);}
+   [$a$, name=a, tikz={\draw [line] () to (b1.north); \draw [fulldot=.5] () to (c.north);}
+      [$b$, name=b1, tikz={\draw [line, bend right] () to (d.north west); \draw [dot=.5, bend left] () to (d.north east);}
+         [$d$, name=d, tikz={\draw [line] () to (254:4.6); \draw [fulldot=.5] () to (265:4.42);}]
+      ]
+      [$c$, name=c, tikz={\draw [line] () to (d.east); \draw [fulldot=.5] () to (291:3.53);}]
+   ]
+]
+\end{forest}
+\end{center}
+
+Check if the following models are satisfying:
+\begin{align*}
+  \Model_1 =~& \{a = \top, b = \top, c = \bot, d = \bot\},\\
+  \Model_2 =~& \{a = \bot, b = \bot, c = \top, d = \top\},\text{~and~}
+\end{align*}
+compute \DNF{f}.

bdds/1024.tex

@@ -0,0 +1 @@
+\item Give the definition of a cofactor of a formula $f$ with respect to an assignment $A$.

bdds/1025.tex

@@ -0,0 +1,7 @@
+\item \self Construct a Reduced Ordered Binary Decision Diagram (ROBDD) for the formula 
+
+$$f = (\neg a \vee b) \wedge (a \vee b),$$
+
+using \textit{alphabetic variable order}. Use complemented edges and a node for \texttt{true} as the only constant node. To simplify drawing, you may assume that \textit{dangling edges} point to the constant node. Write down all cofactors that you compute to obtain the final result and mark them in the graph.
+
+% (not a or b) and (a or b)

bdds/1025_sol.tex

@@ -0,0 +1,44 @@
+\begin{cofactors}
+  $f$ \= $= ((\lnot a \lor b) \land (a \lor b))$\\
+    \>$f_{b}$ \= $= \true$\\
+    \>$f_{\lnot b}$ \= $= \false$\\
+  \end{cofactors}
+  The final ROBDD: 
+  \begin{center}
+  \scalebox{0.75}{
+  
+  \begin{tikzpicture}[>=latex,line join=bevel,]
+    \pgfsetlinewidth{1bp}
+  %%
+  \pgfsetcolor{black}
+    % Edge: f_E -> b0
+    \draw [] (21.5bp,94.988bp) .. controls (21.5bp,84.921bp) and (21.5bp,69.474bp)  .. (21.5bp,59.304bp);
+    % Edge: b0 -> 0
+    \draw [] (19.503bp,36.791bp) .. controls (16.934bp,23.663bp) and (12.77bp,2.3815bp)  .. (12.513bp,1.0642bp);
+    % Edge: b0 -> 1
+    \draw [] (23.719bp,36.791bp) .. controls (26.573bp,23.663bp) and (31.2bp,2.3815bp)  .. (31.486bp,1.0642bp);
+    \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
+    \pgfsetstrokecolor{strokecol}
+    \draw (27.237bp,20.608bp) node {\Large$\bullet$};
+    % Node: f_E
+  \begin{scope}
+    \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
+    \pgfsetstrokecolor{strokecol}
+    \draw (43.0bp,117.0bp) -- (0.0bp,117.0bp) -- (0.0bp,95.0bp) -- (43.0bp,95.0bp) -- cycle;
+    \draw (21.5bp,106.0bp) node {$f$};
+  \end{scope}
+    % Node: b0
+  \begin{scope}
+    \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
+    \pgfsetstrokecolor{strokecol}
+    \draw (21.5bp,48.0bp) ellipse (11.0bp and 11.0bp);
+    \draw (21.5bp,48.0bp) node {$b$};
+  \end{scope}
+  %
+  \end{tikzpicture}
+  
+  
+  }
+  \end{center}
+  
+  

bdds/1026.tex

@@ -0,0 +1,6 @@
+\item \self  Construct a Reduced Ordered Binary Decision Diagram (ROBDD) for the formula 
+$$f = (\lnot x \lor \lnot y) \land (x \land (y \lor z)),$$
+
+using \textit{variable order} $y < z < x$. Use complemented edges and a node for \texttt{true} as the only constant node. To simplify drawing, you may assume that \textit{dangling edges} point to the constant node. Write down all cofactors that you compute to obtain the final result and mark them in the graph.
+
+% (not x or not y) and (x and (y or z))

bdds/1026_sol.tex

@@ -0,0 +1,91 @@
+\begin{cofactors}
+  $f$ \= $= ((\lnot x \lor \lnot y) \land (x \land (y \lor z)))$\\
+    \>$f_{y}$ \= $= \false$\\
+    \>$f_{\lnot y}$ \= $= (x \land z)$\\
+    \>\>$f_{\lnot yz}$ \= $= x$\\
+    \>\>\>$f_{\lnot yzx}$ \= $= \true$\\
+    \>\>\>$f_{\lnot yz\lnot x}$ \= $= \false$\\
+    \>\>$f_{\lnot y\lnot z}$ \= $= \false$\\
+  \end{cofactors}
+  The final ROBDD: 
+  \begin{center}
+  \scalebox{0.75}{
+  
+  \begin{tikzpicture}[>=latex,line join=bevel,]
+    \pgfsetlinewidth{1bp}
+  %%
+  \pgfsetcolor{black}
+    % Edge: f_E -> y0
+    \draw [] (72.5bp,210.99bp) .. controls (72.5bp,200.92bp) and (72.5bp,185.47bp)  .. (72.5bp,175.3bp);
+    \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
+    \pgfsetstrokecolor{strokecol}
+    \draw (72.5bp,195.3bp) node {\Large$\bullet$};
+    % Edge: f_!y -> z1
+    \draw [] (25.254bp,152.99bp) .. controls (29.124bp,142.67bp) and (35.113bp,126.7bp)  .. (38.918bp,116.55bp);
+    % Edge: f_!yz -> x2
+    \draw [] (108.92bp,94.988bp) .. controls (105.24bp,84.669bp) and (99.535bp,68.698bp)  .. (95.911bp,58.551bp);
+    % Edge: y0 -> z1
+    \draw [] (67.68bp,154.0bp) .. controls (62.115bp,143.61bp) and (53.01bp,126.62bp)  .. (47.408bp,116.16bp);
+    \draw (56.852bp,133.79bp) node {\Large$\bullet$};
+    % Edge: y0 -> 0
+    \draw [] (72.5bp,152.99bp) .. controls (72.5bp,137.38bp) and (72.5bp,108.85bp)  .. (72.5bp,107.09bp);
+    % Edge: z1 -> x2
+    \draw [] (49.452bp,97.213bp) .. controls (59.005bp,86.514bp) and (76.085bp,67.385bp)  .. (85.607bp,56.72bp);
+    % Edge: z1 -> 3
+    \draw [] (42.5bp,94.988bp) .. controls (42.5bp,79.384bp) and (42.5bp,50.854bp)  .. (42.5bp,49.086bp);
+    \draw (42.5bp,69.086bp) node {\Large$\bullet$};
+    % Edge: x2 -> 1
+    \draw [] (90.503bp,36.791bp) .. controls (87.934bp,23.663bp) and (83.77bp,2.3815bp)  .. (83.513bp,1.0642bp);
+    % Edge: x2 -> 2
+    \draw [] (94.719bp,36.791bp) .. controls (97.573bp,23.663bp) and (102.2bp,2.3815bp)  .. (102.49bp,1.0642bp);
+    \draw (98.237bp,20.608bp) node {\Large$\bullet$};
+    % Node: f_E
+  \begin{scope}
+    \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
+    \pgfsetstrokecolor{strokecol}
+    \draw (94.0bp,233.0bp) -- (51.0bp,233.0bp) -- (51.0bp,211.0bp) -- (94.0bp,211.0bp) -- cycle;
+    \draw (72.5bp,222.0bp) node {$f$};
+  \end{scope}
+    % Node: y0
+  \begin{scope}
+    \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
+    \pgfsetstrokecolor{strokecol}
+    \draw (72.5bp,164.0bp) ellipse (11.0bp and 11.0bp);
+    \draw (72.5bp,164.0bp) node {$y$};
+  \end{scope}
+    % Node: f_!y
+  \begin{scope}
+    \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
+    \pgfsetstrokecolor{strokecol}
+    \draw (43.0bp,175.0bp) -- (0.0bp,175.0bp) -- (0.0bp,153.0bp) -- (43.0bp,153.0bp) -- cycle;
+    \draw (21.5bp,164.0bp) node {$f_{\lnot y}$};
+  \end{scope}
+    % Node: z1
+  \begin{scope}
+    \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
+    \pgfsetstrokecolor{strokecol}
+    \draw (42.5bp,106.0bp) ellipse (11.0bp and 11.0bp);
+    \draw (42.5bp,106.0bp) node {$z$};
+  \end{scope}
+    % Node: f_!yz
+  \begin{scope}
+    \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
+    \pgfsetstrokecolor{strokecol}
+    \draw (134.0bp,117.0bp) -- (91.0bp,117.0bp) -- (91.0bp,95.0bp) -- (134.0bp,95.0bp) -- cycle;
+    \draw (112.5bp,106.0bp) node {$f_{\lnot yz}$};
+  \end{scope}
+    % Node: x2
+  \begin{scope}
+    \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
+    \pgfsetstrokecolor{strokecol}
+    \draw (92.5bp,48.0bp) ellipse (11.0bp and 11.0bp);
+    \draw (92.5bp,48.0bp) node {$x$};
+  \end{scope}
+  %
+  \end{tikzpicture}
+  
+  
+  }
+  \end{center}
+  
+  

bdds/1027.tex

@@ -0,0 +1,7 @@
+\item \self  Construct a Reduced Ordered Binary Decision Diagram (ROBDD) for the formula  
+
+$$f = (\lnot x \land \lnot y)\lor (x \land y),$$
+
+using \textit{variable order} $z < x < y$. Use complemented edges and a node for \texttt{true} as the only constant node. To simplify drawing, you may assume that \textit{dangling edges} point to the constant node. Write down all cofactors that you compute to obtain the final result and mark them in the graph.
+
+% (not x and not y) or (x and y)

bdds/1027_sol.tex

@@ -0,0 +1,67 @@
+\begin{cofactors}
+  $f$ \= $= ((\lnot x \land \lnot y) \lor (x \land y))$\\
+    \>$f_{y}$ \= $= x$\\
+    \>\>$f_{yx}$ \= $= \true$\\
+    \>\>$f_{y\lnot x}$ \= $= \false$\\
+    \>$f_{\lnot y}$ \= $= \lnot f_y$\\
+  \end{cofactors}
+  The final ROBDD: 
+  \begin{center}
+  \scalebox{0.75}{
+  
+  \begin{tikzpicture}[>=latex,line join=bevel,]
+    \pgfsetlinewidth{1bp}
+  %%
+  \pgfsetcolor{black}
+    % Edge: f_E -> y0
+    \draw [] (21.5bp,152.99bp) .. controls (21.5bp,142.92bp) and (21.5bp,127.47bp)  .. (21.5bp,117.3bp);
+    % Edge: f_y -> x1
+    \draw [] (63.382bp,94.988bp) .. controls (53.412bp,84.04bp) and (37.646bp,66.729bp)  .. (28.57bp,56.763bp);
+    % Edge: y0 -> x1
+    \draw [] (16.385bp,96.003bp) .. controls (14.19bp,85.491bp) and (14.205bp,68.213bp)  .. (16.429bp,57.79bp);
+    % Edge: y0 -> x1
+    \draw [] (26.615bp,96.003bp) .. controls (28.81bp,85.491bp) and (28.795bp,68.213bp)  .. (26.571bp,57.79bp);
+    \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
+    \pgfsetstrokecolor{strokecol}
+    \draw (30.134bp,77.47bp) node {\Large$\bullet$};
+    % Edge: x1 -> 0
+    \draw [] (19.503bp,36.791bp) .. controls (16.934bp,23.663bp) and (12.77bp,2.3815bp)  .. (12.513bp,1.0642bp);
+    % Edge: x1 -> 1
+    \draw [] (23.719bp,36.791bp) .. controls (26.573bp,23.663bp) and (31.2bp,2.3815bp)  .. (31.486bp,1.0642bp);
+    \draw (27.237bp,20.608bp) node {\Large$\bullet$};
+    % Node: f_E
+  \begin{scope}
+    \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
+    \pgfsetstrokecolor{strokecol}
+    \draw (43.0bp,175.0bp) -- (0.0bp,175.0bp) -- (0.0bp,153.0bp) -- (43.0bp,153.0bp) -- cycle;
+    \draw (21.5bp,164.0bp) node {$f$};
+  \end{scope}
+    % Node: y0
+  \begin{scope}
+    \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
+    \pgfsetstrokecolor{strokecol}
+    \draw (21.5bp,106.0bp) ellipse (11.0bp and 11.0bp);
+    \draw (21.5bp,106.0bp) node {$y$};
+  \end{scope}
+    % Node: f_y
+  \begin{scope}
+    \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
+    \pgfsetstrokecolor{strokecol}
+    \draw (94.0bp,117.0bp) -- (51.0bp,117.0bp) -- (51.0bp,95.0bp) -- (94.0bp,95.0bp) -- cycle;
+    \draw (72.5bp,106.0bp) node {$f_{y}$};
+  \end{scope}
+    % Node: x1
+  \begin{scope}
+    \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
+    \pgfsetstrokecolor{strokecol}
+    \draw (21.5bp,48.0bp) ellipse (11.0bp and 11.0bp);
+    \draw (21.5bp,48.0bp) node {$x$};
+  \end{scope}
+  %
+  \end{tikzpicture}
+  
+  
+  }
+  \end{center}
+  
+  

bdds/1028.tex

@@ -0,0 +1,7 @@
+\item \self Construct a Reduced Ordered Binary Decision Diagram (ROBDD) for the formula 
+
+$$f = (\neg p \vee r) \land (q \vee \neg p) \land (\neg q \vee  p) $$
+
+using \textit{variable order} $r < q < p$. Use complemented edges and a node for \texttt{true} as the only constant node. To simplify drawing, you may assume that \textit{dangling edges} point to the constant node. Write down all cofactors that you compute to obtain the final result and mark them in the graph.
+
+% (not p or r) and (q or not p) and (not q or p)

bdds/1028_sol.tex

@@ -0,0 +1,138 @@
+\begin{cofactors}
+  $f$ \= $= (((\lnot p \lor r) \land (q \lor \lnot p)) \land (\lnot q \lor p))$\\
+    \>$f_{r}$ \= $= ((q \lor \lnot p) \land (\lnot q \lor p))$\\
+    \>\>$f_{rq}$ \= $= p$\\
+    \>\>\>$f_{rqp}$ \= $= \true$\\
+    \>\>\>$f_{rq\lnot p}$ \= $= \false$\\
+    \>\>$f_{r\lnot q}$ \= $= \lnot f_{rq}$\\
+    \>$f_{\lnot r}$ \= $= ((\lnot p \land (q \lor \lnot p)) \land (\lnot q \lor p))$\\
+    \>\>$f_{\lnot rq}$ \= $= \false$\\
+    \>\>$f_{\lnot r\lnot q}$ \= $= (\lnot p \land \lnot p)$\\
+    \>\>\>$f_{\lnot r\lnot qp}$ \= $= \false$\\
+    \>\>\>$f_{\lnot r\lnot q\lnot p}$ \= $= \true$\\
+  \end{cofactors}
+  The final ROBDD: 
+  \begin{center}
+  \scalebox{0.75}{
+  
+  \begin{tikzpicture}[>=latex,line join=bevel,]
+    \pgfsetlinewidth{1bp}
+  %%
+  \pgfsetcolor{black}
+    % Edge: f_E -> r0
+    \draw [] (72.5bp,210.99bp) .. controls (72.5bp,200.92bp) and (72.5bp,185.47bp)  .. (72.5bp,175.3bp);
+    % Edge: f_r -> q1
+    \draw [] (123.5bp,152.99bp) .. controls (123.5bp,142.92bp) and (123.5bp,127.47bp)  .. (123.5bp,117.3bp);
+    % Edge: f_rq -> p2
+    \draw [] (165.38bp,94.988bp) .. controls (155.41bp,84.04bp) and (139.65bp,66.729bp)  .. (130.57bp,56.763bp);
+    % Edge: f_!r -> q3
+    \draw [] (21.5bp,152.99bp) .. controls (21.5bp,142.92bp) and (21.5bp,127.47bp)  .. (21.5bp,117.3bp);
+    \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
+    \pgfsetstrokecolor{strokecol}
+    \draw (21.5bp,137.3bp) node {\Large$\bullet$};
+    % Edge: f_!r!q -> p4
+    \draw [] (70.712bp,94.988bp) .. controls (68.892bp,84.795bp) and (66.087bp,69.087bp)  .. (64.272bp,58.925bp);
+    \draw (67.788bp,78.614bp) node {\Large$\bullet$};
+    % Edge: r0 -> q3
+    \draw [] (65.409bp,155.21bp) .. controls (55.665bp,144.51bp) and (38.243bp,125.38bp)  .. (28.531bp,114.72bp);
+    \draw (41.998bp,129.51bp) node {\Large$\bullet$};
+    % Edge: r0 -> q1
+    \draw [] (79.591bp,155.21bp) .. controls (89.335bp,144.51bp) and (106.76bp,125.38bp)  .. (116.47bp,114.72bp);
+    % Edge: q3 -> p4
+    \draw [] (27.55bp,96.737bp) .. controls (35.301bp,86.149bp) and (48.662bp,67.901bp)  .. (56.425bp,57.297bp);
+    \draw (44.611bp,73.434bp) node {\Large$\circ$};
+    % Edge: q3 -> 2
+    \draw [] (21.5bp,94.988bp) .. controls (21.5bp,79.384bp) and (21.5bp,50.854bp)  .. (21.5bp,49.086bp);
+    % Edge: p4 -> 3
+    \draw [] (60.281bp,36.791bp) .. controls (57.427bp,23.663bp) and (52.8bp,2.3815bp)  .. (52.514bp,1.0642bp);
+    % Edge: p4 -> 4
+    \draw [] (64.497bp,36.791bp) .. controls (67.066bp,23.663bp) and (71.23bp,2.3815bp)  .. (71.487bp,1.0642bp);
+    \draw (67.647bp,20.692bp) node {\Large$\bullet$};
+    % Edge: q1 -> p2
+    \draw [] (118.39bp,96.003bp) .. controls (116.19bp,85.491bp) and (116.21bp,68.213bp)  .. (118.43bp,57.79bp);
+    % Edge: q1 -> p2
+    \draw [] (128.61bp,96.003bp) .. controls (130.81bp,85.491bp) and (130.79bp,68.213bp)  .. (128.57bp,57.79bp);
+    \draw (132.13bp,77.47bp) node {\Large$\bullet$};
+    % Edge: p2 -> 0
+    \draw [] (121.28bp,36.791bp) .. controls (118.43bp,23.663bp) and (113.8bp,2.3815bp)  .. (113.51bp,1.0642bp);
+    % Edge: p2 -> 1
+    \draw [] (125.5bp,36.791bp) .. controls (128.07bp,23.663bp) and (132.23bp,2.3815bp)  .. (132.49bp,1.0642bp);
+    \draw (128.65bp,20.692bp) node {\Large$\bullet$};
+    % Node: f_E
+  \begin{scope}
+    \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
+    \pgfsetstrokecolor{strokecol}
+    \draw (94.0bp,233.0bp) -- (51.0bp,233.0bp) -- (51.0bp,211.0bp) -- (94.0bp,211.0bp) -- cycle;
+    \draw (72.5bp,222.0bp) node {$f$};
+  \end{scope}
+    % Node: r0
+  \begin{scope}
+    \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
+    \pgfsetstrokecolor{strokecol}
+    \draw (72.5bp,164.0bp) ellipse (11.0bp and 11.0bp);
+    \draw (72.5bp,164.0bp) node {$r$};
+  \end{scope}
+    % Node: f_r
+  \begin{scope}
+    \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
+    \pgfsetstrokecolor{strokecol}
+    \draw (145.0bp,175.0bp) -- (102.0bp,175.0bp) -- (102.0bp,153.0bp) -- (145.0bp,153.0bp) -- cycle;
+    \draw (123.5bp,164.0bp) node {$f_{r}$};
+  \end{scope}
+    % Node: q1
+  \begin{scope}
+    \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
+    \pgfsetstrokecolor{strokecol}
+    \draw (123.5bp,106.0bp) ellipse (11.0bp and 11.0bp);
+    \draw (123.5bp,106.0bp) node {$q$};
+  \end{scope}
+    % Node: f_rq
+  \begin{scope}
+    \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
+    \pgfsetstrokecolor{strokecol}
+    \draw (196.0bp,117.0bp) -- (153.0bp,117.0bp) -- (153.0bp,95.0bp) -- (196.0bp,95.0bp) -- cycle;
+    \draw (174.5bp,106.0bp) node {$f_{rq}$};
+  \end{scope}
+    % Node: p2
+  \begin{scope}
+    \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
+    \pgfsetstrokecolor{strokecol}
+    \draw (123.5bp,48.0bp) ellipse (11.0bp and 11.0bp);
+    \draw (123.5bp,48.0bp) node {$p$};
+  \end{scope}
+    % Node: f_!r
+  \begin{scope}
+    \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
+    \pgfsetstrokecolor{strokecol}
+    \draw (43.0bp,175.0bp) -- (0.0bp,175.0bp) -- (0.0bp,153.0bp) -- (43.0bp,153.0bp) -- cycle;
+    \draw (21.5bp,164.0bp) node {$f_{\lnot r}$};
+  \end{scope}
+    % Node: q3
+  \begin{scope}
+    \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
+    \pgfsetstrokecolor{strokecol}
+    \draw (21.5bp,106.0bp) ellipse (11.0bp and 11.0bp);
+    \draw (21.5bp,106.0bp) node {$q$};
+  \end{scope}
+    % Node: f_!r!q
+  \begin{scope}
+    \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
+    \pgfsetstrokecolor{strokecol}
+    \draw (94.0bp,117.0bp) -- (51.0bp,117.0bp) -- (51.0bp,95.0bp) -- (94.0bp,95.0bp) -- cycle;
+    \draw (72.5bp,106.0bp) node {$f_{\lnot r\lnot q}$};
+  \end{scope}
+    % Node: p4
+  \begin{scope}
+    \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
+    \pgfsetstrokecolor{strokecol}
+    \draw (62.5bp,48.0bp) ellipse (11.0bp and 11.0bp);
+    \draw (62.5bp,48.0bp) node {$p$};
+  \end{scope}
+  %
+  \end{tikzpicture}
+  
+  
+  }
+  \end{center}
+  
+  

bdds/1029.tex

@@ -0,0 +1,7 @@
+\item \self  Construct a Reduced Ordered Binary Decision Diagram (ROBDD) for the formula
+
+$$f = (q \land \lnot s) \lor (s \land (\neg r \vee p)) \lor (p \land q \land r)$$
+
+using \textit{variable order} $p < q < r < s$. Use complemented edges and a node for \texttt{true} as the only constant node. To simplify drawing, you may assume that \textit{dangling edges} point to the constant node. Write down all cofactors that you compute to obtain the final result and mark them in the graph.
+
+% (q and not s) or (s and (not r or p)) or (p and q and r)

bdds/1029_sol.tex

@@ -0,0 +1,162 @@
+\begin{cofactors}
+  $f$ \= $= (((q \land \lnot s) \lor (s \land (\lnot r \lor p))) \lor ((p \land q) \land r))$\\
+    \>$f_{p}$ \= $= (((q \land \lnot s) \lor s) \lor (q \land r))$\\
+    \>\>$f_{pq}$ \= $= \true$\\
+    \>\>$f_{p\lnot q}$ \= $= s$\\
+    \>\>\>$f_{p\lnot qs}$ \= $= \true$\\
+    \>\>\>$f_{p\lnot q\lnot s}$ \= $= \false$\\
+    \>$f_{\lnot p}$ \= $= ((q \land \lnot s) \lor (s \land \lnot r))$\\
+    \>\>$f_{\lnot pq}$ \= $= (\lnot s \lor (s \land \lnot r))$\\
+    \>\>\>$f_{\lnot pqr}$ \= $= \lnot f_{p\lnot q}$\\
+    \>\>\>$f_{\lnot pq\lnot r}$ \= $= \true$\\
+    \>\>$f_{\lnot p\lnot q}$ \= $= (s \land \lnot r)$\\
+    \>\>\>$f_{\lnot p\lnot qr}$ \= $= \false$\\
+    \>\>\>$f_{\lnot p\lnot q\lnot r}$ \= $= f_{p\lnot q}$\\
+  \end{cofactors}
+  The final ROBDD: 
+  \begin{center}
+  \scalebox{0.75}{
+  
+  \begin{tikzpicture}[>=latex,line join=bevel,]
+    \pgfsetlinewidth{1bp}
+  %%
+  \pgfsetcolor{black}
+    % Edge: f_E -> p0
+    \draw [] (123.5bp,268.99bp) .. controls (123.5bp,258.92bp) and (123.5bp,243.47bp)  .. (123.5bp,233.3bp);
+    % Edge: f_p -> q1
+    \draw [] (174.5bp,210.99bp) .. controls (174.5bp,200.92bp) and (174.5bp,185.47bp)  .. (174.5bp,175.3bp);
+    % Edge: f_p!q -> s2
+    \draw [] (74.288bp,94.988bp) .. controls (76.108bp,84.795bp) and (78.913bp,69.087bp)  .. (80.728bp,58.925bp);
+    % Edge: f_!p -> q3
+    \draw [] (72.5bp,210.99bp) .. controls (72.5bp,200.92bp) and (72.5bp,185.47bp)  .. (72.5bp,175.3bp);
+    \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
+    \pgfsetstrokecolor{strokecol}
+    \draw (72.5bp,195.3bp) node {\Large$\bullet$};
+    % Edge: f_!pq -> r4
+    \draw [] (123.5bp,152.99bp) .. controls (123.5bp,142.92bp) and (123.5bp,127.47bp)  .. (123.5bp,117.3bp);
+    \draw (123.5bp,137.3bp) node {\Large$\bullet$};
+    % Edge: f_!p!q -> r5
+    \draw [] (21.5bp,152.99bp) .. controls (21.5bp,142.92bp) and (21.5bp,127.47bp)  .. (21.5bp,117.3bp);
+    \draw (21.5bp,137.3bp) node {\Large$\bullet$};
+    % Edge: p0 -> q3
+    \draw [] (116.41bp,213.21bp) .. controls (106.66bp,202.51bp) and (89.243bp,183.38bp)  .. (79.531bp,172.72bp);
+    \draw (92.998bp,187.51bp) node {\Large$\bullet$};
+    % Edge: p0 -> q1
+    \draw [] (130.59bp,213.21bp) .. controls (140.34bp,202.51bp) and (157.76bp,183.38bp)  .. (167.47bp,172.72bp);
+    % Edge: q3 -> r5
+    \draw [] (65.409bp,155.21bp) .. controls (55.665bp,144.51bp) and (38.243bp,125.38bp)  .. (28.531bp,114.72bp);
+    \draw (41.998bp,129.51bp) node {\Large$\circ$};
+    % Edge: q3 -> r4
+    \draw [] (79.591bp,155.21bp) .. controls (89.335bp,144.51bp) and (106.76bp,125.38bp)  .. (116.47bp,114.72bp);
+    % Edge: r5 -> s2
+    \draw [] (29.226bp,97.908bp) .. controls (40.895bp,87.195bp) and (62.899bp,66.995bp)  .. (74.644bp,56.212bp);
+    \draw (59.911bp,69.737bp) node {\Large$\bullet$};
+    % Edge: r5 -> 4
+    \draw [] (21.5bp,94.988bp) .. controls (21.5bp,79.384bp) and (21.5bp,50.854bp)  .. (21.5bp,49.086bp);
+    % Edge: r4 -> s2
+    \draw [] (117.45bp,96.737bp) .. controls (109.7bp,86.149bp) and (96.338bp,67.901bp)  .. (88.575bp,57.297bp);
+    % Edge: r4 -> 3
+    \draw [] (123.5bp,94.988bp) .. controls (123.5bp,79.384bp) and (123.5bp,50.854bp)  .. (123.5bp,49.086bp);
+    \draw (123.5bp,69.086bp) node {\Large$\bullet$};
+    % Edge: q1 -> s2
+    \draw [] (171.65bp,152.93bp) .. controls (167.29bp,138.85bp) and (157.95bp,113.01bp)  .. (143.5bp,95.0bp) .. controls (128.72bp,76.585bp) and (105.0bp,61.54bp)  .. (91.923bp,54.08bp);
+    \draw (109.2bp,64.149bp) node {\Large$\circ$};
+    % Edge: q1 -> 0
+    \draw [] (177.46bp,153.25bp) .. controls (182.18bp,137.7bp) and (190.93bp,108.87bp)  .. (191.47bp,107.09bp);
+    % Edge: s2 -> 1
+    \draw [] (80.281bp,36.791bp) .. controls (77.427bp,23.663bp) and (72.8bp,2.3815bp)  .. (72.514bp,1.0642bp);
+    % Edge: s2 -> 2
+    \draw [] (84.497bp,36.791bp) .. controls (87.066bp,23.663bp) and (91.23bp,2.3815bp)  .. (91.487bp,1.0642bp);
+    \draw (87.647bp,20.692bp) node {\Large$\bullet$};
+    % Node: f_E
+  \begin{scope}
+    \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
+    \pgfsetstrokecolor{strokecol}
+    \draw (145.0bp,291.0bp) -- (102.0bp,291.0bp) -- (102.0bp,269.0bp) -- (145.0bp,269.0bp) -- cycle;
+    \draw (123.5bp,280.0bp) node {$f$};
+  \end{scope}
+    % Node: p0
+  \begin{scope}
+    \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
+    \pgfsetstrokecolor{strokecol}
+    \draw (123.5bp,222.0bp) ellipse (11.0bp and 11.0bp);
+    \draw (123.5bp,222.0bp) node {$p$};
+  \end{scope}
+    % Node: f_p
+  \begin{scope}
+    \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
+    \pgfsetstrokecolor{strokecol}
+    \draw (196.0bp,233.0bp) -- (153.0bp,233.0bp) -- (153.0bp,211.0bp) -- (196.0bp,211.0bp) -- cycle;
+    \draw (174.5bp,222.0bp) node {$f_{p}$};
+  \end{scope}
+    % Node: q1
+  \begin{scope}
+    \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
+    \pgfsetstrokecolor{strokecol}
+    \draw (174.5bp,164.0bp) ellipse (11.0bp and 11.0bp);
+    \draw (174.5bp,164.0bp) node {$q$};
+  \end{scope}
+    % Node: f_p!q
+  \begin{scope}
+    \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
+    \pgfsetstrokecolor{strokecol}
+    \draw (94.0bp,117.0bp) -- (51.0bp,117.0bp) -- (51.0bp,95.0bp) -- (94.0bp,95.0bp) -- cycle;
+    \draw (72.5bp,106.0bp) node {$f_{p\lnot q}$};
+  \end{scope}
+    % Node: s2
+  \begin{scope}
+    \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
+    \pgfsetstrokecolor{strokecol}
+    \draw (82.5bp,48.0bp) ellipse (11.0bp and 11.0bp);
+    \draw (82.5bp,48.0bp) node {$s$};
+  \end{scope}
+    % Node: f_!p
+  \begin{scope}
+    \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
+    \pgfsetstrokecolor{strokecol}
+    \draw (94.0bp,233.0bp) -- (51.0bp,233.0bp) -- (51.0bp,211.0bp) -- (94.0bp,211.0bp) -- cycle;
+    \draw (72.5bp,222.0bp) node {$f_{\lnot p}$};
+  \end{scope}
+    % Node: q3
+  \begin{scope}
+    \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
+    \pgfsetstrokecolor{strokecol}
+    \draw (72.5bp,164.0bp) ellipse (11.0bp and 11.0bp);
+    \draw (72.5bp,164.0bp) node {$q$};
+  \end{scope}
+    % Node: f_!pq
+  \begin{scope}
+    \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
+    \pgfsetstrokecolor{strokecol}
+    \draw (145.0bp,175.0bp) -- (102.0bp,175.0bp) -- (102.0bp,153.0bp) -- (145.0bp,153.0bp) -- cycle;
+    \draw (123.5bp,164.0bp) node {$f_{\lnot pq}$};
+  \end{scope}
+    % Node: r4
+  \begin{scope}
+    \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
+    \pgfsetstrokecolor{strokecol}
+    \draw (123.5bp,106.0bp) ellipse (11.0bp and 11.0bp);
+    \draw (123.5bp,106.0bp) node {$r$};
+  \end{scope}
+    % Node: f_!p!q
+  \begin{scope}
+    \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
+    \pgfsetstrokecolor{strokecol}
+    \draw (43.0bp,175.0bp) -- (0.0bp,175.0bp) -- (0.0bp,153.0bp) -- (43.0bp,153.0bp) -- cycle;
+    \draw (21.5bp,164.0bp) node {$f_{\lnot p\lnot q}$};
+  \end{scope}
+    % Node: r5
+  \begin{scope}
+    \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
+    \pgfsetstrokecolor{strokecol}
+    \draw (21.5bp,106.0bp) ellipse (11.0bp and 11.0bp);
+    \draw (21.5bp,106.0bp) node {$r$};
+  \end{scope}
+  %
+  \end{tikzpicture}
+  
+  
+  }
+  \end{center}
+  
+  

bdds/1030.tex

@@ -0,0 +1,36 @@
+\begin{minipage}{.65\textwidth}
+\begin{enumerate}
+\item Use the BDD shown in the figure on the right to check if  the formula it represents evaluates to \texttt{true} or \texttt{false} with the
+following variable assignments.
+\begin{enumerate}
+         \item $\mathcal{M}_1: ~ p = \top, r = \bot, q = \top, s=\bot$
+         \item $\mathcal{M}_2: ~ p = \bot, r = \bot, q = \bot, s=\top$
+\end{enumerate}
+
+\item Find the formula $f$ that is represented by the BDD.
+ \end{enumerate}
+
+\end{minipage}
+\begin{minipage}{.3\textwidth}
+    \begin{center}
+    \begin{forest}
+    for tree={circle, draw, no edge,
+                minimum size=2em,
+                inner sep=0pt,
+                s sep=6mm,
+                l sep=6mm}
+    [$f$, rectangle, draw,  tikz={\draw [fulldot=.5] () to (p.north);}
+        [$p$, name=p, tikz={\draw [line] () to (r1.north); \draw [fulldot=.5] () to (r2.north);}
+            [$r$, name=r1, tikz={\draw [line] () to (254:3.8); \draw [fulldot=.5] () to (q1.north west);}]
+            [$r$, name=r2, tikz={\draw [line] () to (q1.north east); \draw [dot=.5] () to (q2.north west);}
+                [$q$, name=q1, tikz={\draw [line] () to (s.north); \draw[fulldot=.5] () to (277:5);}
+                    [$s$, name=s, tikz={\draw [line] () to (260:6.3); \draw[fulldot=.5] () to (268:6.2);}]
+                    [,phantom]
+                ]
+                [$q$, name=q2, tikz={\draw [line] () to (280:5.03); \draw[fulldot=.5] () to (290:5.23);}]
+            ]
+        ]
+    ]
+    \end{forest}
+    \end{center}
+\end{minipage}

bdds/1031.tex

@@ -0,0 +1,39 @@
+\item Use the BDD shown in the figure on the right to check if  the formula it represents evaluates to \texttt{true} or \texttt{false} with the
+following variable assignments.
+\begin{minipage}{.65\textwidth}
+\begin{enumerate}
+\begin{enumerate}
+         \item $\mathcal{M}_1: ~ a = \bot, b = \top, c = \bot, d = \top$
+         \item $\mathcal{M}_2: ~ a = \top, b = \top, c = \top, d = \top$
+\end{enumerate}
+
+\item Find the formula $f$ that is represented by the BDD.
+ \end{enumerate}
+
+\end{minipage}
+\begin{minipage}{.3\textwidth}
+    \begin{center}
+\begin{forest}
+for tree={circle, draw, no edge,
+          minimum size=2em,
+          inner sep=0pt,
+          s sep=5mm,
+          l sep=5mm}
+[$f$, rectangle, draw,  tikz={\draw [fulldot=.5] () to (c.north);}
+   [$c$, name=c, tikz={\draw [line] () to (d1.north); \draw [dot=.5] () to (d2.north);}
+      [$d$, name=d1, tikz={\draw [line] () to (b1.north); \draw [fulldot=.5] () to (261:3.25);}
+         [$b$, name=b1, tikz={\draw [line] () to (242:5); \draw [fulldot=.5] () to (256:4.55);}]
+         [,phantom]
+      ]
+      [$d$, name=d2, tikz={\draw [line] () to (b2.north); \draw [dot=.5] () to (299:3.7);}
+         [$b$, name=b2, tikz={\draw [line] () to (a.north); \draw [dot=.5] () to (286:4.6);}
+            [$a$, name=a, tikz={\draw [line] () to (264:5.7); \draw [fulldot=.5] () to (276:5.7);}]
+            [,phantom]
+         ]
+         [,phantom]
+      ]
+   ]
+]
+\end{forest}
+    \end{center}
+\end{minipage}

bdds/1031_sol.tex

@@ -0,0 +1,10 @@
+\begin{enumerate}
+\item
+\begin{enumerate}
+     \item false
+     \item false
+\end{enumerate}
+
+\item
+$f = (p \land \lnot r \land q \land s) \lor (\lnot p \land r \land q \land s) \lor (\lnot p \land \lnot r \land q)$
+\end{enumerate}

bdds/1032.tex

@@ -0,0 +1,29 @@
+\item \self Convert the following BDD into a \emph{reduced  ordered} BDD.
+
+\begin{center}
+\begin{forest}
+for tree={circle, draw, no edge,
+            minimum size=2em,
+            inner sep=0pt,
+            s sep=6mm,
+            l sep=6mm}
+[$f$, rectangle, draw,  tikz={\draw [line] () to (p.north);}
+    [$q$, name=q, tikz={\draw [line] () to (s1.north); \draw [dot=.5] () to (s2.north);}
+        [$s$, name=s1, tikz={\draw [line] () to (240:4.1); \draw [fulldot=.5] () to (r1.north);}
+            [,phantom]
+            [$r$, name=r1, tikz={\draw [line] () to (p1.north); \draw[dot=.5] () to (p2.north);}
+                [$p$, name=p1, tikz={\draw [line] () to (254:6.57); \draw[fulldot=.5] () to (263:6.37);}]
+                [$p$, name=p2, tikz={\draw [line] () to (266:6.33); \draw[dot=.5] () to (275:6.35);}]
+            ]
+        ]
+        [$s$, name=s2, tikz={\draw [line] () to (281:3.6); \draw [fulldot=.5] () to (r2.north);}
+            [,phantom]
+            [$r$, name=r2, tikz={\draw [line] () to (p3.north); \draw[dot=.5] () to (p4.north);}
+                [$p$, name=p3, tikz={\draw [line] () to (277:6.38); \draw[fulldot=.5] () to (286:6.6);}]
+                [$p$, name=p4, tikz={\draw [line] () to (288:6.65); \draw[dot=.5] () to (297:7.1);}]
+            ]
+        ]
+    ]
+]
+\end{forest}
+\end{center}

bdds/1032_sol.tex

@@ -0,0 +1,18 @@
+\begin{center}
+\begin{forest}
+for tree={circle, draw, no edge,
+            minimum size=2em,
+            inner sep=0pt,
+            s sep=6mm,
+            l sep=6mm}
+[$f$, rectangle, draw,  tikz={\draw [line] () to (s.north);}
+    [$s$, name=s, tikz={\draw [line] () to (-0.6, -2.2); \draw [fulldot=.5] () to (r.north);}
+        [,phantom]
+        [$r$, name=r, tikz={\draw [line] () to (p.north); \draw[dot=.5] () to (1.3,-3.5);}
+            [$p$, name=p, tikz={\draw [line] () to (-0.6, -4.8); \draw[fulldot=.5] () to (0.6, -4.8);}]
+            [,phantom]
+        ]
+    ]
+]
+\end{forest}
+\end{center}

bdds/1033.tex

@@ -0,0 +1,5 @@
+\item \self Construct a ROBDD for the formula
+
+$$f =(a\land d\land c)\lor(b\land\lnot d\land\lnot a)\lor(c \implies \lnot d) \lor (a \implies \lnot b)$$
+
+using \textit{variable order} $b < a < d < c$. Use complemented edges and a node for \texttt{true} as the only constant node. To simplify drawing, you may assume that \textit{dangling edges} point to the constant node. Write down all cofactors that you compute to obtain the final result and mark them in the graph.

bdds/1033_sol.tex

@@ -0,0 +1,21 @@
+\begin{minipage}{0.5\textwidth}
+\begin{tabbing}
+$f$ \= $= (a\land d\land c)\lor(b\land\lnot d\land\lnot a)\lor(c \implies \lnot d) \lor (a \implies \lnot b)$\\
+	\>$ f_b $ \= $= \top$ \\
+	\>$ f_{\lnot b} =\top$ \\
+\end{tabbing}
+\end{minipage}
+\begin{minipage}{0.5\textwidth}
+\begin{center}
+\begin{forest}
+for tree={circle, draw, no edge,
+            minimum size=2em,
+            inner sep=0pt,
+            s sep=5mm,
+            l sep=5mm}
+[$f$, rectangle, draw,  tikz={\draw [line] () to (0,-1);}
+]
+\end{forest}
+\end{center}
+
+\end{minipage}

bdds/1034.tex

@@ -0,0 +1,5 @@
+\item \self Construct a reduced ordered binary decision diagram (ROBDD) for the formula
+
+$$f = (p \oplus q) \land \lnot r$$
+
+using \textit{variable order} $p < q < r$. Use complemented edges and a node for \texttt{true} as the only constant node. To simplify drawing, you may assume that \textit{dangling edges} point to the constant node. Write down all cofactors that you compute to obtain the final result and mark them in the graph.

bdds/1034_sol.tex

@@ -0,0 +1,40 @@
+\begin{minipage}{0.5\textwidth}
+\begin{tabbing}
+$f$ \= $= (p \oplus q) \land \lnot r$\\
+	\>$ f_p $ \= $= \lnot q \land \lnot r$ \\
+	    \>\>$ f_{pq} $ \= $= \bot$ \\
+	    \>\>$ f_{p \lnot q} $ \= $= \lnot r$ \\
+	        \>\>\>$ f_{p \lnot q r} $ \= $= \bot$ \\
+	        \>\>\>$ f_{p \lnot q \lnot r} $ \= $= \top$ \\
+	\>$ f_{\lnot p} = q \land \lnot r$ \\
+	    \>\>$ f_{\lnot pq} $ \= $= \lnot r = f_{p \lnot q}$ \\
+	    \>\>$ f_{\lnot p \lnot q} $ \= $= \bot$ \\
+\end{tabbing}
+\end{minipage}
+\begin{minipage}{0.5\textwidth}
+The final ROBDD:
+\begin{center}
+\begin{forest}
+for tree={circle, draw, no edge,
+            minimum size=2em,
+            inner sep=0pt,
+            s sep=6mm,
+            l sep=6mm}
+[$f$, rectangle, draw,  tikz={\draw [fulldot=.3] () to (p.north);}
+    [$f_q$, rectangle, draw,  tikz={\draw [fulldot=.3] () to (q1.north);}]
+    [$p$, name=p, tikz={\draw [line] () to (q1.north east); \draw [dot=.3] () to (q2.north west);}
+        [$q$, name=q1, tikz={\draw [line] () to (-2,-3.5); \draw [dot=.3] () to (r.north west);}
+            [$f_{p \lnot q}$, rectangle, draw,  tikz={\draw [fulldot=.3] () to (r.west);}]
+        ]
+        [$q$, name=q2, tikz={\draw [line] () to (r.north east); \draw [dot=.3] () to (2,-3.5);}
+            [$r$, name=r, tikz={\draw [line] () to (-0.5,-4.8); \draw [fulldot=.3] () to (0.5, -4.8);}]
+            [$f_{\lnot pq}$, rectangle, draw,  tikz={\draw [fulldot=.3] () to (r.east);}]
+            [,phantom]
+        ]
+    ]
+    [$f_{\lnot q}$, rectangle, draw,  tikz={\draw [fulldot=.3] () to (q2.north);}]
+]
+\end{forest}
+\end{center}
+
+\end{minipage}

bdds/1035.tex

@@ -0,0 +1,5 @@
+\item \self Construct a ROBDD for the formula
+
+$$f = (p \leftrightarrow q) \land (r \leftrightarrow s)$$
+
+using \textit{variable order} $r < s < p < q$. Use complemented edges and a node for \texttt{true} as the only constant node. To simplify drawing, you may assume that \textit{dangling edges} point to the constant node. Write down all cofactors that you compute to obtain the final result and mark them in the graph.

bdds/1035_sol.tex

@@ -0,0 +1,49 @@
+\begin{minipage}{0.5\textwidth}
+\begin{tabbing}
+$f$ \= $= (p \leftrightarrow q) \land (r \leftrightarrow s)$\\
+	\>$ f_r $ \= $= (p \leftrightarrow q) \land s$ \\
+	    \>\>$ f_{rs} $ \= $= (p \leftrightarrow q)$ \\
+	        \>\>\>$ f_{rsp} $ \= $= q$ \\
+	            \>\>\>\>$ f_{rspq} $ \= $= \top$ \\
+	            \>\>\>\>$ f_{rsp \lnot q} $ \= $= \bot$ \\
+	        \>\>\>$ f_{rs \lnot p} $ \= $= \lnot q = \lnot f_{rsp}$ \\
+        \>\>$ f_{r \lnot s} $ \= $= \bot$ \\
+	\>$ f_{\lnot r} = (p \leftrightarrow q) \land \lnot s$ \\
+	    \>\>$ f_{\lnot rs} $ \= $= \bot$ \\
+	    \>\>$ f_{\lnot r \lnot s} $ \= $= (p \leftrightarrow q) = f_{rs}$ \\
+\end{tabbing}
+\end{minipage}
+\begin{minipage}{0.5\textwidth}
+The final ROBDD:
+\begin{center}
+\begin{forest}
+for tree={circle, draw, no edge,
+            minimum size=2em,
+            inner sep=0pt,
+            s sep=6mm,
+            l sep=6mm}
+[$f$, rectangle, draw,  tikz={\draw [line] () to (r.north);}
+    [$f_r$, rectangle, draw,  tikz={\draw [line] () to (s1.north west);}]
+    [$r$, name=r, tikz={\draw [line] () to (s1.north); \draw [fulldot=.5] () to (s2.north);}
+        [$f_{\lnot r \lnot s}$, rectangle, draw,  tikz={\draw [line] () to (p.north west);}]
+        [$f_{rs}$ , rectangle, draw,  tikz={\draw [line] () to (p.north west);}]
+        [$s$, name=s1, tikz={\draw [line] () to (p.north); \draw [fulldot=.5] () to (-0.2, -3.5);}
+            [$f_{\lnot rsp}$, rectangle, draw,  tikz={\draw [line] () to (q.north west);}]
+            [$p$, name=p, tikz={\draw [line, bend right=14] () to (q.north); \draw [fulldot=.5, bend left=14] () to (q.north east);}
+                [$q$, name=q, tikz={\draw [line] () to (-2.3, -6.2); \draw [fulldot=.5] () to (-1.5, -6.2);}
+                ]
+                [,phantom]
+            ]
+            [,phantom]
+            [,phantom]
+        ]
+        [$s$, name=s2, tikz={\draw [line] () to (0.2,-3.5); \draw [fulldot=.5, bend left=70] () to (p.east);}]
+        [,phantom]
+        [,phantom]
+    ]
+    [$f_{\lnot r}$, rectangle, draw,  tikz={\draw [fulldot=.5] () to (s2.north east);}]
+]
+\end{forest}
+\end{center}
+
+\end{minipage}

bdds/1036.tex

@@ -0,0 +1,11 @@
+\item \self
+
+  \begin{enumerate}
+    \item Construct a Reduced Ordered Binary Decision Diagram (ROBDD) for the formula
+
+$$f= (a \lor b \lor c) \land \lnot d$$
+
+using \textit{variable order} $a < b < c < d$. Use complemented edges and a node for \texttt{true} as the only constant node. To simplify drawing, you may assume that \textit{dangling edges} point to the constant node. Write down all cofactors that you compute to obtain the final result and mark them in the graph.
+
+  \item Construct a Reduced Ordered Binary Decision Diagram (ROBDD) for $f$ with a different variable order. The ROBDD should result in a \textit{smaller} ROBDD, w.r.t. the number of nodes.
+  \end{enumerate}

bdds/1036_sol.tex

@@ -0,0 +1,100 @@
+\begin{enumerate}
+\item using \textit{variable order} $c < a < d < b$:\\
+\begin{minipage}{0.5\textwidth}
+\begin{tabbing}
+$f$ \= $= (a \lor b \lor c) \land \lnot d$\\
+	\>$ f_c $ \= $= \lnot d$ \\
+	    \>\>$ f_{ca} $ \= $= \lnot d$ \\
+	    \>\>$ f_{c \lnot a} $ \= $= \lnot d$ \\
+	    \>\>\Rightarrow a does not have an influence\\
+	    \>\>on the formula.\\
+	    \>\>These cofactors can be skipped.\\
+        \>\>$ f_{cd} $ \= $= \bot$ \\
+        \>\>$ f_{c \lnot d} $ \= $= \top $ \\
+	\>$ f_{\lnot c} $ \= $= (a \lor b) \land \lnot d$ \\
+	    \>\>$ f_{\lnot ca} $ \= $= \lnot d = f_{c}$ \\
+	    \>\>$ f_{\lnot c \lnot a} $ \= $= b \land \lnot d$ \\
+	        \>\>\>$ f_{\lnot c \lnot ad} $ \= $= \bot$ \\
+	        \>\>\>$ f_{\lnot c \lnot a \lnot d} $ \= $= b$ \\
+	            \>\>\>\>$ f_{\lnot c \lnot a \lnot db} $ \= $= \top$ \\
+	            \>\>\>\>$ f_{\lnot c \lnot a \lnot db} $ \= $= \bot$ \\
+\end{tabbing}
+\end{minipage}
+\begin{minipage}{0.5\textwidth}
+\begin{center}
+\begin{forest}
+for tree={circle, draw, no edge,
+            minimum size=2em,
+            inner sep=0pt,
+            s sep=6mm,
+            l sep=6mm}
+[$f$, rectangle, draw,  tikz={\draw [fulldot=.5] () to (c.north);}
+    [,phantom]
+    [$c$, name=c, tikz={\draw [line] () to (d1.north); \draw [dot=.5] () to (a.north);}
+        [$f_{c}$, rectangle, draw,  tikz={\draw [fulldot=.5] () to (d1.north west);}]
+        [,phantom]
+        [$a$, name=a, tikz={\draw [line,bend left=30] () to (d1.north); \draw [dot=.5] () to (d2.north);}
+            [$d$, name=d1, tikz={\draw [line] () to (-2,-5); \draw [fulldot=.5] () to (-1,-5);}]
+            [,phantom]
+            [$d$, name=d2, tikz={\draw [line] () to (0.6,-5); \draw [fulldot=.5] () to (b.north);}
+                [,phantom]
+                [$b$, name=b, tikz={\draw [line] () to (1.3,-6.3); \draw [fulldot=.5] () to (2.3,-6.3);}]
+            ]
+            [$f_{\lnot c \lnot a \lnot d}$, rectangle, draw,  tikz={\draw [line] () to (b.north east);}]
+        ]
+        [$f_{\lnot c \lnot a}$, rectangle, draw,  tikz={\draw [fulldot=.5] () to (d2.north east);}]
+    ]
+    [$f_{\lnot c}$, rectangle, draw,  tikz={\draw [fulldot=.5] () to (a.north east);}]
+]
+\end{forest}
+\end{center}
+\end{minipage}
+
+\item
+using \textit{variable order} $d < a < b < c$:\\
+\begin{minipage}{0.5\textwidth}
+\begin{tabbing}
+$f$ \= $= (a \lor b \lor c) \land \lnot d$\\
+	\>$ f_d $ \= $= \bot$ \\
+	\>$ f_{\lnot d} $ \= $= a \lor b \lor c$ \\
+	    \>\>$ f_{\lnot da} $ \= $= \top$ \\
+	    \>\>$ f_{\lnot d \lnot a} $ \= $= b \lor c$ \\
+	        \>\>\>$ f_{\lnot d \lnot ab} $ \= $= \top$ \\
+	        \>\>\>$ f_{\lnot d \lnot a \lnot b} $ \= $= c$ \\
+	            \>\>\>\>$ f_{\lnot d \lnot a \lnot bc} $ \= $= \top$ \\
+	            \>\>\>\>$ f_{\lnot d \lnot a \lnot b \lnot c} $ \= $= \bot$ \\
+\end{tabbing}
+\end{minipage}
+\begin{minipage}{0.5\textwidth}
+\begin{center}
+\begin{forest}
+for tree={circle, draw, no edge,
+            minimum size=2em,
+            inner sep=0pt,
+            s sep=6mm,
+            l sep=6mm}
+[$f$, rectangle, draw,  tikz={\draw [fulldot=.5] () to (d.north);}
+    [,phantom]
+    [$d$, name=d, tikz={\draw [line] () to (-0.5,-2.3); \draw [fulldot=.5] () to (a.north);}
+        [,phantom]
+        [,phantom]
+        [$a$, name=a, tikz={\draw [line] () to (0,-3.6); \draw [dot=.5] () to (b.north);}
+            [,phantom]
+            [,phantom]
+            [$b$, name=b, tikz={\draw [line] () to (0.5,-4.9); \draw [dot=.5] () to (c.north);}
+                [,phantom]
+                [,phantom]
+                [$c$, name=c, tikz={\draw [line] () to (1.3,-6.2); \draw [fulldot=.5] () to (2.3,-6.2);}
+                ]
+                [,phantom]
+            ]
+            [$f_{\lnot d \lnot a \lnot b}$, rectangle, draw,  tikz={\draw [line] () to (c.north east);}]
+        ]
+        [$f_{\lnot d \lnot a}$, rectangle, draw,  tikz={\draw [line] () to (b.north east);}]
+    ]
+    [$f_{\lnot d}$, rectangle, draw,  tikz={\draw [line] () to (a.north east);}]
+]
+\end{forest}
+\end{center}
+\end{minipage}
+\end{enumerate}

bdds/2018.tex

@@ -0,0 +1 @@
+\item Explain how a reduced and ordered BDD can be used to determine the satisfiability of the formula $f$ it is representing.

bdds/2019.tex

@@ -0,0 +1,2 @@
+\item Explain how a reduced and ordered BDD can be used to determine whether the formula $f$ it is representing is valid.
+

bdds/bdds.tex

@@ -0,0 +1,175 @@
+\begin{questionSection}{Reduced Ordered Binary Decision Diagrams}
+
+
+\question{bdds/0001.tex}
+         {bdds/0001_sol.tex}
+         {3cm}
+\question{bdds/0007.tex}
+         {bdds/0007_sol.tex}
+         {3cm}
+
+\question{bdds/1019.tex}
+         {no_solution}
+         {3cm}
+
+\question{bdds/1020.tex}
+         {no_solution}
+         {3cm}
+
+\question{bdds/1021.tex}
+         {no_solution}
+         {3cm}
+
+\question{bdds/1022.tex}
+         {no_solution}
+         {3cm}
+
+\question{bdds/1023.tex}
+         {no_solution}
+         {3cm}
+
+
+\question{bdds/0004.tex}
+         {bdds/0004_sol.tex}
+         {3cm}
+
+\question{bdds/1013.tex}
+         {no_solution}
+         {3cm}
+
+\question{bdds/1014.tex}
+         {no_solution}
+         {3cm}
+
+\question{bdds/1012.tex}
+         {no_solution}
+         {3cm}
+
+\question{bdds/0005.tex}
+         {bdds/0005_sol.tex}
+         {3cm}
+
+
+
+% We start with theory questions here
+\question{bdds/0006.tex}
+         {bdds/0006_sol.tex}
+         {3cm}
+
+\question{bdds/1024.tex}
+         {no_solution}
+         {3cm}
+
+
+\question{bdds/1005.tex}
+         {no_solution}
+         {3cm}
+
+\mcquestion{bdds/1008.tex}
+           {bdds/1008.tex}
+
+\question{bdds/1009.tex}
+         {no_solution}
+         {3cm}
+
+\question{bdds/1010.tex}
+         {no_solution}
+         {3cm}
+
+\question{bdds/1011.tex}
+         {no_solution}
+         {3cm}
+
+
+\question{bdds/1015.tex}
+         {no_solution}
+         {3cm}
+
+\mcquestion{bdds/1016.tex}
+           {bdds/1016.tex}
+
+
+
+\question{bdds/1018.tex}
+         {no_solution}
+         {3cm}
+
+\question{bdds/2018.tex}
+         {no_solution}
+         {3cm}
+
+\question{bdds/2019.tex}
+         {no_solution}
+         {3cm}
+
+\end{questionSection}
+
+\clearpage
+\begin{questionSection}{Construction of Reduced Ordered BDDs}
+
+
+\question{bdds/1026.tex}
+         {bdds/1026_sol.tex}
+         {3cm}
+
+\question{bdds/1027.tex}
+         {bdds/1027_sol.tex}
+         {3cm}
+
+\question{bdds/1028.tex}
+         {bdds/1028_sol.tex}
+         {3cm}
+
+\question{bdds/1029.tex}
+         {bdds/1029_sol.tex}
+         {3cm}
+
+
+\question{bdds/1033.tex}
+         {bdds/1033_sol.tex}
+         {3cm}
+
+\question{bdds/1034.tex}
+         {bdds/1034_sol.tex}
+         {3cm}
+
+\question{bdds/1035.tex}
+         {bdds/1035_sol.tex}
+         {3cm}
+
+\question{bdds/1036.tex}
+         {bdds/1036_sol.tex}
+         {3cm}
+\question{bdds/0008.tex}
+         {bdds/0008_sol.tex}
+         {3cm}
+
+\question{bdds/0009.tex}
+         {bdds/0009_sol.tex}
+         {3cm}
+
+\question{bdds/0010.tex}
+         {bdds/0010_sol.tex}
+         {3cm}
+
+
+%\section{New in 2024 - Remove me later} % TODO: Remove me!
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\question{bdds/0012.tex}
+         {bdds/0012_sol.tex}
+         {3cm}
+
+\question{bdds/0013.tex}
+         {bdds/0013_sol.tex}
+         {3cm}
+
+\question{bdds/0014.tex}
+         {bdds/0014_sol.tex}
+         {3cm}
+
+\question{bdds/0018.tex}
+         {bdds/0018_sol.tex}
+         {3cm}
+
+\end{questionSection}

compile

@@ -1,4 +1,4 @@
-#!/bin/sh
+#!/bin/bash
 . util/util.sh
 
 TOGGLEFILE=util/toggle.tex
@@ -7,9 +7,10 @@ ALLCHAPTERFILE=util/allchapter.tex
 FILENAMEPREFIX=questionnaire
 SINK=/dev/null
 
-VALID_ARGS="bdh"
+VALID_ARGS="abdh"
 BURST=0
 DEBUG=0
+ANNOTATE=0
 
 printUsage() {
   printf "A shell wrapper for this questionnaire using latexmk.\n"
@@ -18,6 +19,7 @@ printUsage() {
   printf "Flags:\n"
   printf " -b     Compile in burst mode, i.e. each chapter into a individual file.\n"
   printf " -d     Print LuaLaTeX output to terminal.\n"
+  printf " -a     Annotate all questions and solutions with their base filename.\n"
   printf " -h     Print this help message and exit.\n"
 
 }
@@ -30,6 +32,8 @@ do
        ;;
     d) DEBUG=1
        ;;
+    a) ANNOTATE=1
+       ;;
     h) printUsage
        info "Exiting..."
        exit
@@ -42,6 +46,10 @@ set_chapter() {
   printf "$1\n" > $CHAPTERFILE
 }
 
+do_annotate() {
+  echo "\\annotatetrue" >> $TOGGLEFILE
+}
+
 rename() {
   filename="$FILENAMEPREFIX"
   while [ $# -gt 0 ]
@@ -58,12 +66,14 @@ compile_file() {
   if [ $mode = "solution" ]
   then
     echo "\solutiontrue" > $TOGGLEFILE
+    [ $ANNOTATE -eq 1 ] && do_annotate
     [ "$#" -eq 1 ] && info "Compiling questionnaire with solutions." || info "Compiling chapter $chapter with solutions."
     [ $DEBUG -eq 1 ] && make pdf || make pdf >> $SINK 2>&1
     rename "with_solutions" $chapter
   elif [ $mode = "spacing" ]
   then
     echo "\spacingtrue" > $TOGGLEFILE
+    [ $ANNOTATE -eq 1 ] && do_annotate
     [ "$#" -eq 1 ] && info "Compiling questionnaire with space for your solutions." || info "Compiling chapter $chapter with space for your solutions."
     [ $DEBUG -eq 1 ] && make pdf || make pdf >> $SINK 2>&1
     rename "with_spacing" $chapter
@@ -73,6 +83,7 @@ compile_file() {
 compile_simple_file() {
   [ "$#" -eq 0 ] && info "Compiling whole questionnaire." || info "Compiling chapter $1 from the questionnaire."
   echo "" > $TOGGLEFILE
+  [ $ANNOTATE -eq 1 ] && do_annotate
   [ $DEBUG -eq 1 ] && make all || make all >> $SINK 2>&1
   rename $1
 }
@@ -92,7 +103,8 @@ compile_wrapper() {
 }
 
 compile_chapter() {
-  for chapter in one two three four five six seven eight nine ten eleven
+  #for chapter in smtzthree proplogic satsolver ndpred predlogic ndpred smt bdd eqchecking symbenc temporal
+  for chapter in bdd satsolver
   do
     set_chapter "\\chapter${chapter}true"
     compile_wrapper "$chapter" "$@"

main.tex

@@ -8,6 +8,7 @@
 \input{util/styling_macros}
 \input{util/constants}
 \input{util/chapter}
+\input{util/tcolorbox}
 
 \pagestyle{fancy}
 
@@ -30,7 +31,6 @@
 \makeatother
 
 \newcounter{beforeCounter}
-
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
 \begin{document}
@@ -57,82 +57,28 @@
 \clearpage
 
 
-\ifchapterone
+\ifchapterproplogic
   \setsectionnum{0}
-  \section{SMT Solvers and Z3}
-  \input{smt_and_z3/smt_and_z3.tex}
-  \pagebreak
-\fi
-
-\ifchaptertwo
-  \setsectionnum{1}
   \section{Propositional Logic}
-  \input{propositional_logic/propositional_logic.tex}
+    \input{propositional_logic/propositional_logic.tex}
   \pagebreak
 \fi
 
-\ifchapterthree
-  \setsectionnum{2}
+\ifchaptersatsolver
+  \setsectionnum{1}
   \section{SAT Solvers}
   \input{sat_solvers_dpll/sat_solvers_dpll.tex}
   \pagebreak
 \fi
 
-\ifchapterfour
-  \setsectionnum{3}
-  \section{Natural Deduction for Propositional Logic}
-  \input{natural_deduction_propositional_logic/natural_deduction_prop_logic.tex}
-  \pagebreak
-\fi
-
-\ifchapterfive
-  \setsectionnum{4}
-  \section{Predicate Logic}
-  \input{predicate_logic/predicate_logic.tex}
-  \pagebreak
-\fi
-
-\ifchaptersix
-  \setsectionnum{5}
-  \section{Natural Deduction for Predicate Logic}
-  \input{natural_deduction_predicate_logic/natural_deduction_predicate_logic.tex}
-  \pagebreak
-\fi
-
-\ifchapterseven
-  \setsectionnum{6}
-  \section{Satisfiability Modulo Theories}
-  \input{smt/smt.tex}
-  \pagebreak
-\fi
-
-\ifchaptereight
-  \setsectionnum{7}
+\ifchapterbdd
+  \setsectionnum{2}
   \section{Binary Decision Diagrams}
   \input{bdds/bdds.tex}
   \pagebreak
 \fi
 
-\ifchapternine
-  \setsectionnum{8}
-  \section{Combinational Equivalence Checking}
-  \input{equivalence_checking/equivalence_checking.tex}
-  \pagebreak
-\fi
-
-\ifchapterten
-  \setsectionnum{9}
-  \section{Symbolic Encoding}
-  \input{symbolic_encoding/symbolic_encoding.tex}
-  \pagebreak
-\fi
 
-\ifchaptereleven
-  \setsectionnum{10}
-  \section{Temporal Logic}
-  \input{temporal_logic/temporal_logic.tex}
-  \pagebreak
-\fi
 %---------------------------------------------------------
 
 

no_solution.tex

@@ -1 +1 @@
-There is no solution available for this question yet.
+\textit{There is no solution available for this question yet.}

propositional_logic/0001.tex

@@ -0,0 +1,9 @@
+\item \lect Look at the following statements and tick them if they are true.
+
+\begin{itemize}
+\item[$\square$] "\textit{Give me the butter}." is a declarative sentence.
+\item[$\square$] Questions are always declarative sentences.
+\item[$\square$] Declarative sentences can be true and false at the same time.
+\item[$\square$] "\textit{My best friend is staying overnight.}" is a declarative sentence.
+
+\end{itemize}

propositional_logic/0001_sol.tex

@@ -0,0 +1,9 @@
+\item \lect Look at the following statements and tick them if they are true.
+
+\begin{itemize}
+\item[$\square$] "\textit{Give me the butter}." is a declarative sentence.
+\item[$\square$] Questions are always declarative sentences.
+\item[$\square$] Declarative sentences can be true and false at the same time.
+\item[$\ticked$] "\textit{My best friend is staying overnight.}" is a declarative sentence.
+
+\end{itemize}

propositional_logic/0002.tex

@@ -0,0 +1,7 @@
+\item \lect Model the following sentences as detailed as possible in propositional logic.
+
+\begin{enumerate}
+    \item Alice will either take the bike or the tram to get to the concert, not both.
+    \item Students will have to take an exam at the end of the semester.
+    \item If he is hungry and the fridge is not empty, he cooks for himself.
+\end{enumerate}

propositional_logic/0002_sol.tex

@@ -0,0 +1,14 @@
+\begin{enumerate}
+    \item $p:$ \quad Alice will take the bike to get to the concert.
+
+          $q:$ \quad Alice will take the tram to get to the concert.
+          $$ (p \land \lnot q) \lor (\lnot p \land q) $$
+    \item $p:$ \quad Students will have to take an exam at the end of the semester.
+          $$ p $$
+    \item $p:$ \quad He is hungry.
+
+          $q:$ \quad The fridge is empty.
+
+          $r:$ \quad He cooks for himself.
+          $$ p \land \lnot q \imp r $$
+\end{enumerate}

propositional_logic/0003.tex

@@ -0,0 +1,7 @@
+\item \lect Model the following sentences as detailed as possible in propositional logic.
+
+\begin{enumerate}
+    \item If the air temperature is above 30°C, then the water temperature is above 20°C and I am able to go for a swim.
+    \item Your kid will be safe if and only if it learns to swim.
+    \item What time is it?
+\end{enumerate}

propositional_logic/0003_sol.tex

@@ -0,0 +1,13 @@
+\begin{enumerate}
+    \item $p:$ \quad The air temperature is above 30°C.
+
+          $q:$ \quad The water temperature is above 20°C.
+
+          $r:$ \quad I am able to go for a swim.
+          $$ p \imp q \land r $$
+    \item $p:$ \quad Your kid will be safe.
+
+          $q:$ \quad Your kid learns to swim.
+          $$ p \leftrightarrow q $$
+    \item This is not a declarative sentence.
+\end{enumerate}

propositional_logic/0004.tex

@@ -0,0 +1 @@
+\item \lect Give the definition of well-formed formulas in propositional logic. \\

propositional_logic/0004_sol.tex

@@ -0,0 +1,5 @@
+We give the definition of well-formed formulas in propositional logic 
+using a grammar in Backus-Naur form (BNF) as:
+%
+$$\varphi \coloneqq ~ < \text{atomic proposition>} ~ | ~\varphi \wedge \varphi~ |~ \varphi \vee \varphi~ | ~\neg \varphi ~| ~\varphi \imp \varphi~ | ~\varphi \leftrightarrow \varphi~ | ~( \varphi )$$
+

propositional_logic/0004_sol.tex~

@@ -0,0 +1,5 @@
+We give the definition of well-formed formulas in propositional logic 
+using a grammar in Backus-Naur form (BNF) as:
+%
+$$\varphi \coloneqq ~ < \text{atomic proposition>} ~ | ~\varphi \wedge \varphi~ |~ \varphi \vee \varphi~ | ~\neg \varphi ~| ~\varphi \imp \varphi~ | ~\varphi \leftrightarrow \varphi~ | ~( \varphi )$$
+

propositional_logic/0005.tex

@@ -0,0 +1,10 @@
+\item \lect Let $p, q$ and $r$ be a atomic propositions. Tick all statements that are true.
+
+\begin{itemize}
+\item[$\square$] "$\lnot p \land \lor \enspace q$" is a propositional formula.
+\item[$\square$] "$(p \land q) \lor (r \imp p)$" is a propositional formula.
+\item[$\square$] "$\lnot p$" is a propositional formula.
+\item[$\square$] "$\lor$" is a propositional formula.
+\item[$\square$] "$p$" is a propositional formula.
+
+\end{itemize}

propositional_logic/0005_sol.tex

@@ -0,0 +1,9 @@
+\item \lect Let $p, q$ and $r$ be a atomic propositions. Tick all statements that are true.
+
+\begin{itemize}
+\item[$\square$] "$\lnot p \land \lor \enspace q$" is a propositional formula.
+\item[$\ticked$] "$(p \land q) \lor (r \imp p)$" is a propositional formula.
+\item[$\ticked$] "$\lnot p$" is a propositional formula.
+\item[$\square$] "$\lor$" is a propositional formula.
+\item[$\ticked$] "$p$" is a propositional formula.
+\end{itemize}

propositional_logic/0006.tex

@@ -0,0 +1,2 @@
+\item \lect  Determine whether the string $\neg (a \vee \neg \neg b)$ is a well-formed formula using the parse
+tree. Explain your answer.