scholarly journals On Minimum Representations of Matched Formulas

2014 ◽  
Vol 51 ◽  
pp. 707-723
Author(s):  
O. Cepek ◽  
S. Gursky ◽  
P. Kucera
Keyword(s):  
Normal Form ◽  
Boolean Function ◽  
Minimization Problem ◽  
Conjunctive Normal Form ◽  
Boolean Formula ◽  
Polynomial Hierarchy ◽  
Partial Assignment ◽  
Complete Problem ◽  
Boolean Minimization ◽  
Np Complete

A Boolean formula in conjunctive normal form (CNF) is called matched if the system of sets of variables which appear in individual clauses has a system of distinct representatives. Each matched CNF is trivially satisfiable (each clause can be satisfied by its representative variable). Another property which is easy to see, is that the class of matched CNFs is not closed under partial assignment of truth values to variables. This latter property leads to a fact (proved here) that given two matched CNFs it is co-NP complete to decide whether they are logically equivalent. The construction in this proof leads to another result: a much shorter and simpler proof of the fact that the Boolean minimization problem for matched CNFs is a complete problem for the second level of the polynomial hierarchy. The main result of this paper deals with the structure of clause minimum CNFs. We prove here that if a Boolean function f admits a representation by a matched CNF then every clause minimum CNF representation of f is matched.

scholarly journals On Minimum Representations of Matched Formulas (Extended Abstract)

2017 ◽  
Author(s):  
Ondřej Čepek ◽  
Štefan Gurský ◽  
Petr Kučera
Keyword(s):  
Normal Form ◽  
Boolean Function ◽  
Conjunctive Normal Form ◽  
Boolean Formula ◽  
Polynomial Hierarchy ◽  
Boolean Minimization

A Boolean formula in conjunctive normal form (CNF) is called matched if the system of sets of variables which appear in individual clauses has a system of distinct representatives. We present here two results for matched CNFs: The first result is a shorter and simpler proof of the fact that Boolean minimization remains complete for the second level of polynomial hierarchy even if the input is restricted to matched CNFs. The second result is structural --- we show that if a Boolean function f admits a representation by a matched CNF then every clause minimum CNF representation of f is matched.

scholarly journals Applying Neural Networks to Find the Minimum Cost Coverage of a Boolean Function

VLSI Design ◽  
1995 ◽  
Vol 3 (1) ◽  
pp. 13-19 ◽  
Author(s):  
Pong P. Chu
Keyword(s):  
Neural Network ◽  
Boolean Function ◽  
Input Data ◽  
Selection Process ◽  
Optimal Solution ◽  
Minimum Cost ◽  
Hopfield Network ◽  
Neural Network Approach ◽  
Complete Problem ◽  
Np Complete

To find a minimal expression of a boolean function includes a step to select the minimum cost cover from a set of implicants. Since the selection process is an NP-complete problem, to find an optimal solution is impractical for large input data size. Neural network approach is used to solve this problem. We first formalize the problem, and then define an “energy function” and map it to a modified Hopfield network, which will automatically search for the minima. Simulation of simple examples shows the proposed neural network can obtain good solutions most of the time.

NAE-resolution: A new resolution refutation technique to prove not-all-equal unsatisfiability

2020 ◽  
Vol 30 (7) ◽  
pp. 736-751
Author(s):  
Hans Kleine Büning ◽  
P. Wojciechowski ◽  
K. Subramani
Keyword(s):  
Normal Form ◽  
Conjunctive Normal Form ◽  
Principal Result ◽  
Satisfying Assignment ◽  
Boolean Formulas ◽  
Complete Procedure ◽  
Np Complete ◽  
Resolution Refutation ◽  
Original Formula ◽  
Cnf Formulas

AbstractIn this paper, we analyze Boolean formulas in conjunctive normal form (CNF) from the perspective of read-once resolution (ROR) refutation schemes. A read-once (resolution) refutation is one in which each clause is used at most once. Derived clauses can be used as many times as they are deduced. However, clauses in the original formula can only be used as part of one derivation. It is well known that ROR is not complete; that is, there exist unsatisfiable formulas for which no ROR exists. Likewise, the problem of checking if a 3CNF formula has a read-once refutation is NP-complete. This paper is concerned with a variant of satisfiability called not-all-equal satisfiability (NAE-satisfiability). A CNF formula is NAE-satisfiable if it has a satisfying assignment in which at least one literal in each clause is set to false. It is well known that the problem of checking NAE-satisfiability is NP-complete. Clearly, the class of CNF formulas which are NAE-satisfiable is a proper subset of satisfiable CNF formulas. It follows that traditional resolution cannot always find a proof of NAE-unsatisfiability. Thus, traditional resolution is not a sound procedure for checking NAE-satisfiability. In this paper, we introduce a variant of resolution called NAE-resolution which is a sound and complete procedure for checking NAE-satisfiability in CNF formulas. The focus of this paper is on a variant of NAE-resolution called read-once NAE-resolution in which each clause (input or derived) can be part of at most one NAE-resolution step. Our principal result is that read-once NAE-resolution is a sound and complete procedure for 2CNF formulas. Furthermore, we provide an algorithm to determine the smallest such NAE-resolution in polynomial time. This is in stark contrast to the corresponding problem concerning 2CNF formulas and ROR refutations. We also show that the problem of checking whether a 3CNF formula has a read-once NAE-resolution is NP-complete.

scholarly journals Maximum Exact Satisfiability: NP-completeness Proofs and Exact Algorithms

2004 ◽  
Vol 11 (19) ◽  
Author(s):  
Bolette Ammitzbøll Madsen ◽  
Peter Rossmanith
Keyword(s):  
Time Complexity ◽  
Exact Algorithm ◽  
Conjunctive Normal Form ◽  
Exact Algorithms ◽  
Fixed Parameter Tractable ◽  
Complete Problem ◽  
Satisfiability Problems ◽  
Fixed Parameter ◽  
Np Complete ◽  
Open Question

Inspired by the Maximum Satisfiability and Exact Satisfiability problems we present two Maximum Exact Satisfiability problems. The first problem called Maximum Exact Satisfiability is: given a formula in conjunctive normal form and an integer k, is there an assignment to all variables in the formula such that at least k clauses have exactly one true literal. The second problem called Restricted Maximum Exact Satisfiability has the further restriction that no clause is allowed to have more than one true literal. Both problems are proved NP-complete restricted to the versions where each clause contains at most two literals. In fact Maximum Exact Satisfiability is a generalisation of the well-known NP-complete problem MaxCut. We present an exact algorithm for Maximum Exact Satisfiability where each clause contains at most two literals with time complexity O(poly(L) . 2^{m/4}), where m is the number of clauses and L is the length of the formula. For the second version we give an algorithm with time complexity O(poly(L) . 1.324718^n) , where n is the number of variables. We note that when restricted to the versions where each clause contains exactly two literals and there are no negations both problems are fixed parameter tractable. It is an open question if this is also the case for the general problems.

scholarly journals Quantum­Inspired Genetic Algorithm for Solving the Test Suite Minimization Problem

2020 ◽  
Vol 19 ◽  
Keyword(s):  
Genetic Algorithm ◽  
Minimization Problem ◽  
Test Suite ◽  
Test Cases ◽  
Special Importance ◽  
Quantum Superposition ◽  
Complete Problem ◽  
Minimal Subset ◽  
Np Complete ◽  
Test Requirements

Test Suite Minimization problem is a nondeterministic polynomial time (NP) complete problem insoftware engineering that has a special importance in software testing. In this problem, a subset with a minimalsize that contains a number of test cases that cover all the test requirements should be found. A brute­forceapproach to solving this problem is to assume a size for the minimal subset and then search to find if there is asubset of test cases with the assumed size that solves the problem. If not, the assumed minimal size is graduallyincremented, and the search is repeated. In this paper, a quantum­inspired genetic algorithm (QIGA) will beproposed to solve this problem. In it, quantum superposition, quantum rotation and quantum measurement willbe used in an evolutionary algorithm. The paper will show that the adopted quantum techniques can speed upthe convergence of the classical genetic algorithm. The proposed method has an advantage in that it reduces theassumed minimal number of test cases using quantum measurements, which makes it able to discover the minimalnumber of test cases without any prior assumptions.

A Polynomial Algorithm for Constructing a Large Bipartite Subgraph, with an Application to a Satisfiability Problem

1982 ◽  
Vol 34 (3) ◽  
pp. 519-524 ◽  
Author(s):  
Svatopluk Poljak ◽  
Daniel Turzík
Keyword(s):  
Connected Graph ◽  
Polynomial Algorithm ◽  
Short Proof ◽  
Satisfiability Problem ◽  
Boolean Formula ◽  
Complete Problem ◽  
Bipartite Subgraph ◽  
Np Complete ◽  
Image Position

Let G be a symmetric connected graph without loops. Denote by b(G) the maximum number of edges in a bipartite subgraph of G. Determination of b(G) is polynomial for planar graphs ([6], [8]); in general it is an NP-complete problem ([5]). Edwards in [1], [2] found some estimates of b(G) which give, in particular,for a connected graph G of n vertices and m edges, whereand ﹛x﹜ denotes the smallest integer ≧ x.We give an 0(V3) algorithm which for a given graph constructs a bipartite subgraph B with at least f(m, n) edges, yielding a short proof of Edwards’ result.Further, we consider similar methods for obtaining some estimates for a particular case of the satisfiability problem. Let Φ be a Boolean formula of variables x1, …, xn.

A FAST ALGORITHM FOR BANDWIDTH MINIMIZATION

2007 ◽  
Vol 16 (03) ◽  
pp. 537-544 ◽  
Author(s):  
ANDREW LIM ◽  
BRIAN RODRIGUES ◽  
FEI XIAO
Keyword(s):  
Simulated Annealing ◽  
Tabu Search ◽  
Minimization Problem ◽  
Fast Algorithm ◽  
Hill Climbing ◽  
Solution Quality ◽  
Complete Problem ◽  
Np Complete ◽  
Bandwidth Minimization ◽  
Matrix Bandwidth

We propose a simple and direct node shifting method with hill climbing for the well-known matrix bandwidth minimization problem. Many heuristics have been developed for this NP-complete problem including the Cuthill-McKee (CM) and the Gibbs, Poole and Stockmeyer (GPS) algorithms. Recently, heuristics such as Simulated Annealing, Tabu Search and GRASP have been used, where Tabu Search and the GRASP with Path Relinking achieved significantly better solution quality than the CM and GPS algorithms. Experimentation shows that our method achieves the best solution quality when compared with these while being much faster than newly-developed algorithms.

On Random Betweenness Constraints

2010 ◽  
Vol 19 (5-6) ◽  
pp. 775-790 ◽  
Author(s):  
ANDREAS GOERDT
Keyword(s):  
Normal Form ◽  
High Probability ◽  
Conjunctive Normal Form ◽  
Satisfiability Problem ◽  
Linear Ordering ◽  
Satisfiability Problems ◽  
Ordering Constraints ◽  
Random Instances ◽  
Np Complete

Ordering constraints are formally analogous to instances of the satisfiability problem in conjunctive normal form, but instead of a boolean assignment we consider a linear ordering of the variables in question. A clause becomes true given a linear ordering if and only if the relative ordering of its variables obeys the constraint considered.The naturally arising satisfiability problems are NP-complete for many types of constraints. We look at random ordering constraints. Previous work of the author shows that there is a sharp unsatisfiability threshold for certain types of constraints. The value of the threshold, however, is essentially undetermined. We pursue the problem of approximating the precise value of the threshold. We show that random instances of the betweenness constraint are satisfiable with high probability if the number of randomly picked clauses is ≤0.92n, where n is the number of variables considered. This improves the previous bound, which is <0.82n random clauses. The proof is based on a binary relaxation of the betweenness constraint and involves some ideas not used before in the area of random ordering constraints.

scholarly journals On the orthogonalization of arbitrary Boolean formulae

2005 ◽  
Vol 2005 (2) ◽  
pp. 61-74 ◽  
Author(s):  
Renato Bruni
Keyword(s):  
Normal Form ◽  
Boolean Function ◽  
General Procedure ◽  
Conjunctive Normal Form ◽  
Disjunctive Normal Form ◽  
Reliability Theory ◽  
Satisfiability Problem ◽  
Propositional Satisfiability ◽  
Strongly Connected ◽  
Great Relevance

The orthogonal conjunctive normal form of a Boolean function is a conjunctive normal form in which any two clauses contain at least a pair of complementary literals. Orthogonal disjunctive normal form is defined similarly. Orthogonalization is the process of transforming the normal form of a Boolean function to orthogonal normal form. The problem is of great relevance in several applications, for example, in the reliability theory. Moreover, such problem is strongly connected with the well-known propositional satisfiability problem. Therefore, important complexity issues are involved. A general procedure for transforming an arbitrary CNF or DNF to an orthogonal one is proposed. Such procedure is tested on randomly generated Boolean formulae.

METHODS OF REDUCING OF LARGE BOOLEAN FORMULAS REPRESENTED IN A CONJUNCTIVE NORMAL FORM FOR DETERMINING THEIR SATISFIABILITY

2020 ◽  
Author(s):  
N.I. Gdansky ◽  
◽  
A.A. Denisov ◽  
Keyword(s):  
Normal Form ◽  
Normal Forms ◽  
Conjunctive Normal Form ◽  
Boolean Formulas

The article explores the satisfiability of conjunctive normal forms used in modeling systems.The problems of CNF preprocessing are considered.The analysis of particular methods for reducing this formulas, which have polynomial input complexity is given.

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