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Equisatisfiable formulae may disagree, however, for a particular choice of variables. As a result, equisatisfiability is different from logical equivalence, as two equivalent formulae always have the same models. Whereas within equisatisfiable formulae, only the primitive proposition the formula imposes is valued.
An important set of problems in computational complexity involves finding assignments to the variables of a Boolean formula expressed in conjunctive normal form, such that the formula is true. The k -SAT problem is the problem of finding a satisfying assignment to a Boolean formula expressed in CNF in which each disjunction contains at most k ...
The Tseytin transformation, alternatively written Tseitin transformation, takes as input an arbitrary combinatorial logic circuit and produces an equisatisfiable boolean formula in conjunctive normal form (CNF). The length of the formula is linear in the size of the circuit. Input vectors that make the circuit output "true" are in 1-to-1 ...
For some versions of the SAT problem, it is useful to define the notion of a generalized conjunctive normal form formula, viz. as a conjunction of arbitrarily many generalized clauses, the latter being of the form R(l 1,...,l n) for some Boolean function R and (ordinary) literals l i. Different sets of allowed Boolean functions lead to ...
More generally, one can define a weighted version of MAX-SAT as follows: given a conjunctive normal form formula with non-negative weights assigned to each clause, find truth values for its variables that maximize the combined weight of the satisfied clauses. The MAX-SAT problem is an instance of Weighted MAX-SAT where all weights are 1. [5] [6 ...
The De Morgan dual is the canonical conjunctive normal form , maxterm canonical form, or Product of Sums (PoS or POS) which is a conjunction (AND) of maxterms. These forms can be useful for the simplification of Boolean functions, which is of great importance in the optimization of Boolean formulas in general and digital circuits in particular.
The function introduced in this process is called a Skolem function (or Skolem constant if it is of zero arity) and the term is called a Skolem term. As an example, the formula ∀ x ∃ y ∀ z P ( x , y , z ) {\displaystyle \forall x\exists y\forall zP(x,y,z)} is not in Skolem normal form because it contains the existential quantifier ∃ y ...
Instances of the 2-satisfiability problem are typically expressed as Boolean formulas of a special type, called conjunctive normal form (2-CNF) or Krom formulas. Alternatively, they may be expressed as a special type of directed graph , the implication graph , which expresses the variables of an instance and their negations as vertices in a ...