Showing papers on "Conjunctive normal form published in 1996"

Stålmarck's Algorithm as a HOL Derived Rule

Abstract: Stalmarck’s algorithm is a patented technique for tautology-checking which has been used successfully for industrial-scale problems Here we describe the algorithm and explore its implementation as a HOL derived rule

Computing prime implicants by integer programming

Abstract: An enumerative approach for selective generation of prime implicants of a theory in conjunctive normal form is presented. The method is based on 0-1 programming. Optimal solutions of the integer linear program associated with the theory correspond to prime implicants. All prime implicants can be obtained by augmenting the integer program with new constraints which discard the already obtained solutions. The method allows to implement preference criteria in the choice of the prime implicants to find.

Cutting planes, connectivity, and threshold logic

Abstract: Originating from work in operations research the cutting plane refutation systemCP is an extension of resolution, where unsatisfiable propositional logic formulas in conjunctive normal form are recognized by showing the non-existence of boolean solutions to associated families of linear inequalities. Polynomial sizeCP proofs are given for the undirecteds-t connectivity principle. The subsystemsCPq ofCP, forq≥2, are shown to be polynomially equivalent toCP, thus answering problem 19 from the list of open problems of [8]. We present a normal form theorem forCP2-proofs and thereby for arbitraryCP-proofs. As a corollary, we show that the coefficients and constant terms in arbitrary cutting plane proofs may be exponentially bounded by the number of steps in the proof, at the cost of an at most polynomial increase in the number of steps in the proof. The extensionCPLE+, introduced in [9] and there shown top-simulate Frege systems, is proved to be polynomially equivalent to Frege systems. Lastly, since linear inequalities are related to threshold gates, we introduce a new threshold logic and prove a completeness theorem.

Computation of prime implicates and prime implicants by a variant of the Davis and Putnam procedure

Abstract: The problem is the transformation of a conjunctive normal form (CNF) into a minimized (for the inclusion operator) disjunctive normal form (DNF) and vice versa This operation is called the unionist product For a CNF (resp DNF), one pass of the unionist product provides the prime implicants (resp implicates); two passes provide the prime implicates (resp implicants) An algorithm built upon the classical Davis and Putnam procedure is presented for calculating, without the explicit minimization for the inclusion, this unionist product

A linear-time transformation of linear inequalities into conjunctive normal form

Abstract: We present a technique that transforms any binary programming problem with integral coefficients to a satisfiability problem of propositional logic in linear time. Preliminary computational experience using this transformation, shows that a pure logical solver can be a valuable tool for solving binary programming problems. In a number of cases it competes favourably with well known techniques from operations research, especially for hard unsatisfiable problems.

Logic-Based Outer-Approximation and Benders Decomposition Algorithms for the Synthesis of Process Networks

Abstract: In this paper the MINLP problem for the optimal synthesis of process networks is modeled as a discrete optimization problem involving logic disjunctions with nonlinear equations and pure logic relations. The logic disjunctions allow the conditional modeling of equations. The outer approximation algorithm is used as a basis to derive a logic-based OA solution method which naturally gives rise to NLP subproblems that avoid zero flows and a disjunctive LP master problem. The NLP subproblems are selected through a set covering problem for which we consider both the cases of disjunctive and conjunctive normal form logic. The master problem, on the other hand, is converted to mixed-integer form using a convex-hull representation. Furthermore, based on some interesting relations of outer-approximation with Generalized Benders Decomposition it is also shown that it is possible to derive a logic-based method for the latter algorithm. The performance of the proposed algorithms illustrated with two process network problems.

The Inverse Satisfiability Problem

Abstract: We study the complexity of telling whether a set of bitvectors represents the set of all satisfying truth assignments of a Boolean expression of a certain type. We show that the problem is coNP-complete when the expression is required to be in conjunctive normal form with three literals per clause (3CNF). We also prove a Dichotomy Theorem analogous to the classical one by Schaefer, stating that, unless P=NP, the problem can be solved in polynomial time if and only if the clauses allowed are either all Horn, or all anti-Horn, or all 2CNF, or all equivalent to equations modulo two.

The Logic of Normative Systems

Abstract: This paper exploits the properties of adjointness to develop a theory of norms. The adjoints are a pair of mappings between two logics, one called IMP, a logic of imperatives, and another called PROP, which is ordinary logic. In particular, L is a mapping form Prop to IMP and it has a right adjoint R from IMP to PROP. Norms are defined, formally, as statements that contain occurrences of RU where U is in IMP. A sample theorem (reminiscent of Kant’s “ought implies can”): Responsibility implies power and, moreover, incapacity implies immunity.

utation of P nam Procedure

Abstract: The problem is the transformation of a conjunctive normal form (CNF) into U minimized (for the inclusion operator) disjunclive normul form (DNF) and reciprocally. This operation is called the unionist product. For a CNF (resp. DNF), one pass of the unionist product provides the prime implicants (resp. implicates); two passes provide the prime implicates (resp. implicants). An algorithm built upon the classical Davis and Putnum Procedure is presented for calculating, wirhout the explicit minimizalion for the inclusion, this unionist product.

utation of Prime Implicates and t avis a

Abstract: The problem is the (runsformation of a conjunctive normal form (CNF) into u minimized (for the inclusion operator) disjunctive normulform (DNF) und reciprocally. This operation is called the unionist product. For a CNF (resp. DNF), one pass of the unionist product provides the prime implicants (resp. implicates); two passes provide the prime implicates (resp. implicants). An ulgorithm built upon the classical Davis and Putnam Procedure is presented for calculating, without the explicir minimization for the inclusion, this unionist product. The prime implicates and implicants play an important role in propositional logic and in tools for AI. They arc used for theoretical and practical applications. If the number of these prime implicants is small, their computation time can be expensive. It is the case for most of the unsatisfiable formulas. So there is no “efficient” method for the computation of the prime implicants or implicates. Moreover, for the problem of satisfiability, lot of progress has been made thanks lo the Davis and Putnam procedure (DP). The proposed method for the computation of the prime implicants (and implicates) relies upon DP and reuses all the works on the heuristics and refinements of DP. Using DP is not sufficient to design an elficient algorithm. The set inclusion simplification reduces the efficiency of the prime implicants and implicates computation methods [3]. The notion oi ’ necessary set introduced in this article discards the explicit minimization