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45 lines
2.8 KiB
45 lines
2.8 KiB
\newcommand{\dpllDescription}[1]{Use the DPLL algorithm with conflict-driven clause learning to determine whether or not the set of clauses given is satisfiable. Decide variables in alphabetical order starting with the \textit{#1} phase. For conflicts, draw conflict graphs after the end of the table, and add the learned clause to the table.\\
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If the set of clauses resulted in \texttt{SAT}, give a satisfying model. If the set of clauses resulted in \texttt{UNSAT}, give a resolution proof that shows that the conjunction of the clauses from the table is unsatisfiable.}
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\newcommand{\dpllDescriptionAssignmentSheet}{Use the DPLL algorithm with the rules as described above to check whether the following formula in CNF is satisfiable}
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\newcommand{\ndDescription}{ For each of the following sequents, either provide a natural deduction proof, or a counter-example that proves the sequent invalid. \\
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\noindent For proofs, clearly indicate which rule, and what assumptions/premises/\\intermediate results you are using in each step. Also clearly indicate the scope of any boxes you use. \\
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\noindent For counterexamples, give a complete model. Show that the model satisfies
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the premise(s) of the sequent in question, but does not satisfy the respective conclusion.}
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\newcommand{\tseitinRules}{%
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We list the \emph{Tseitin-rewriting rules} to be applied for the following examples.
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\begin{align*}
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\chi \leftrightarrow (\phi \lor \psi) \quad &\Leftrightarrow \tseitinOr{\chi}{\phi}{\psi} \\
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\chi \leftrightarrow (\phi \land \psi) \quad &\Leftrightarrow \tseitinAnd{\chi}{\phi}{\psi} \\
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\chi \leftrightarrow \lnot \phi \quad &\Leftrightarrow \tseitinNot{\chi}{\phi}
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\end{align*}
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}
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\newcommand{\constructROBDD}[2]{%
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Construct the reduced and ordered BDD for the formula%
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$$#1$$%
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using #2.
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Compute the needed cofactors.
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You may add function nodes representing all cofactors to the final BDD.
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Use complemented edges and one terminal node representing the truth value $\T$. To simplify drawing, you may assume that dangling edges point to the constant node.}
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\newcommand{\BDDToROBDD}{%
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Transform the given Binary Decision Diagram into a reduced and ordered BDD.}
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\newcommand{\BDDToDNF}{%
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Given the following Binary Decision Diagram that represents the formula $f$. Compute its disjunctive normal form \DNF{f}.}
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\newcommand{\computeCNFDNF}[1][\varphi]{%
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Compute \DNF{#1} and \CNF{#1} using a truth table.
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}
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\newcommand{\CNFfromCircuit}{%
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Compute the propositional formula $\varphi$ represented by the following circuit. Furthermore, compute an equisatisfiable formula $\varphi'$ using the Tseitin transformation.
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}
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\newcommand{\applyTseitin}[1]{%
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Apply the Tseitin transformation to #1. For each variable you introduce, clearly indicate which subformula it represents.
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}
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