\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.\\
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.}

\newcommand{\dpllDescriptionAssignmentSheet}{Use the DPLL algorithm with the rules as described above to check whether the following formula in CNF is satisfiable}

\newcommand{\ndDescription}{ For each of the following sequents, either provide a natural deduction proof, or a counter-example that proves the sequent invalid. \\

\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. \\

\noindent For counterexamples, give a complete model. Show that the model satisfies
the premise(s) of the sequent in question, but does not satisfy the respective conclusion.}

\newcommand{\tseitinRules}{%
We list the \emph{Tseitin-rewriting rules} to be applied for the following examples.
         \begin{align*}
         \chi \leftrightarrow (\phi \lor \psi) \quad &\Leftrightarrow \tseitinOr{\chi}{\phi}{\psi} \\
         \chi \leftrightarrow (\phi \land \psi) \quad &\Leftrightarrow \tseitinAnd{\chi}{\phi}{\psi} \\
         \chi \leftrightarrow \lnot \phi \quad &\Leftrightarrow \tseitinNot{\chi}{\phi}
         \end{align*}
}

\newcommand{\constructROBDD}[2]{%
Construct the reduced and ordered BDD for the formula%
$$#1$$%
using #2.
Compute the needed cofactors.
You may add function nodes representing all cofactors to the final BDD.
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.}

\newcommand{\BDDToROBDD}{%
Transform the given Binary Decision Diagram into a reduced and ordered BDD.}

\newcommand{\BDDToDNF}{%
Given the following Binary Decision Diagram that represents the formula $f$. Compute its disjunctive normal form \DNF{f}.}

