Solution-Graphs of Boolean Formulas and Isomorphism1

Lecture Notes in Computer Science, 2014

For Boolean satisfiability problems, the structure of the solution space is characterized by the solution graph, where the vertices are the solutions, and two solutions are connected iff they differ in exactly one variable. In 2006, Gopalan et al. studied connectivity properties of the solution graph and related complexity issues for CSPs [GKMP09], motivated mainly by research on satisfiability algorithms and the satisfiability threshold. They proved dichotomies for the diameter of connected components and for the complexity of the st-connectivity question, and conjectured a trichotomy for the connectivity question. Recently, we were able to establish the trichotomy [Sch13]. Here, we consider connectivity issues of satisfiability problems defined by Boolean circuits and propositional formulas that use gates, resp. connectives, from a fixed set of Boolean functions. We obtain dichotomies for the diameter and the two connectivity problems: On one side, the diameter is linear in the number of variables, and both problems are in P, while on the other side, the diameter can be exponential, and the problems are PSPACE-complete. For partially quantified formulas, we show an analogous dichotomy.

2012

We investigate the complexity of the syntactic isomorphism problem of two Boolean Formulas in Conjunctive Normal Form (CNF): given two CNF Boolean formulas ϕ(a1,. .. , an) and ϕ(b1,. .. , bn) decide whether there exists a permutation of clauses, a permutation of literals and a bijection between their variables such that ϕ(a1,. .. , an) and ϕ(b1,. .. , bn) become syntactically identical. We first show that the CNF Syntactic Formulas Isomorphism (CSFI) problem is polynomial time reducible to the graph isomorphism problem (GI) and then we show that GI is polynomial time reducible to a special case of the CSFI problem (MCSFI) that is CSFI-complete and also GI-complete, thus concluding that the syntactic isomorphism problem for CNF Boolean formulas is GI-complete. Finally we observe that the same results hold when considering DNF Boolean formulas (DSFI).

We examine the complexity of the formula value problem for Boolean formulas, which is the following decision problem: Given a Boolean formula without variables, does it evaluate to true? We show that the complexity of this problem is determined by certain closure properties of the connectives allowed to build the formula, and achieve a complete classification: The formula value problem is either in LOGTIME, complete for one of the classes NLOGTIME, coNLOGTIME or NC 1 , or equivalent to counting modulo 2 under very strict reductions.

European Journal of Combinatorics, 2010

a r t i c l e i n f o Article history: Available online xxxx a b s t r a c t

This short note deals with several classes of Boolean formulae which have the property, that satisfiabilty can be tested for them in polynomial time with respect to the length of the formula. The studied classes known ffom the literature build uP on the $\mathrm{w}\mathrm{e}\mathrm{U}$-known classes of quadratic on Horn formulae. We prove several interesting properties of these classes and show their mutual positions with respect to inclusion, aproblem which was not previously studied. 1Introduction The class of Horn formulae is avery important and extensively studied subclass of general Boolean formulae. The principal reason for their importance is the fact, that the $\mathrm{s}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{s}\mathrm{f}\mathrm{i}\mathrm{a}\mathrm{b}\underline{\mathrm{i}\mathrm{l}\mathrm{i}}\mathrm{t}\mathrm{y}$ problem (SAT), which is well-known to be $\mathrm{N}\mathrm{P}$-complete for general Boolean formulae, can be solved efficiently (in linear time with respect to the length of the formula) for Horn formulae [11, 15, 17]. This has significant practical implications. Many real-life problems require for their solution to solve SAT as asubproblem, and hence are in general intractable; however, they become tractable if the underlying Boolean formula in the problem is Horn. Such problems arise in several application areas, among others in artificial inteUigence [9, 13, 14] and database design $[10, 16]$. The limiting factor in using Horn formulae is their expressing power. Not every real-life problem can be formulated in such away, that the underlying formula is Horn. For the above reasons it is obvious, that finding broader classes of formulae, which preserve the property that satisfiability is decidable for them in polynomial time, is highly desirable. Several attempts in this direction were successfully made. The &st natural generalzation that was considered is the class of hidden Horn formulae, which are in literature sometimes also called renameable or disguised Horn formulae. This class consists of formulae, which can be obtained from Horn formulae by so called "variable complementing" (also known as "vaiable renaming" or "variable switching"), i.e. by replacing some Boolean variables by their complements. Aspvall showed in [1] that recognizing whether agiven Boolean formula is hidden Horn can be done in linear time. Moreover, the recognition algorithm combined with the linear time SAT algorithm for Horn formulae [11, 15, 17] directly yields alinear time SAT algorithm for the class of hidden Horn formulae. Yamasaki and Doshita [19] defined adifferent generalzation of Horn formulae, called their class $S_{0}$ , and developed acubic time SAT algorithm for formulae in So. This was later improved to quadratic time by Arvind and Biswas [3]. Moreover, recognizing whether agiven formula belongs to $S_{0}$ can be decided also in quadratic time by astraightforward algorithm which uses in asimple way the definition of the class. The class $S_{0}$ was further generalized by Gallo and Scuteu\'a [12] who came up with arecursively

Lecture Notes in Computer Science

This paper initiates the study of SAT instances of bounded diameter. The diameter of an ordered CNF formula is defined as the maximum difference between the index of the first and the last occurrence of a variable. We study the complexity of the satisfiability, the counting and the maximization problems for formulas of bounded diameter. We investigate the relation between the diameter of a formula, and the tree-width and the path-width of its corresponding incidence graph, and show that under highly parallel and efficient transformations, diameter and path-width are equal up to a constant factor. Our main technical contribution is that the computational complexity of SAT, Max-SAT, #SAT grows smoothly with the diameter (as a function of the number of variables). Our main focus is in providing space efficient and highly parallel algorithms, while the running time of our algorithms matches previously known results. Among others, we show NL-completeness of SAT and NC 2 algorithms for Max-SAT, #SAT when diameter is O(log n). Given the tree decomposition of a formula, we further improve on the space efficiency to decide SAT as asked by Alekhnovich and Razborov [1].

2005

Schaefer proved in 1978 that the Boolean constraint satisfaction problem for a given constraint language is either in P or is NP-complete, and identified all tractable cases. Schaefer’s dichotomy theorem actually shows that there are at most two constraint satisfaction problems, up to polynomial-time isomorphism (and these isomorphism types are distinct if and only if P ≠ NP). We show that if one considers AC0 isomorphisms, then there are exactly six isomorphism types (assuming that the complexity classes NP, P, ⊕L, NL, and L are all distinct).

2006

Several propositional fragments have been considered so far as target languages for knowledge compilation and used for improving computational tasks from major AI areas (like inference, diagnosis and planning); among them are the ordered binary decision diagrams, prime implicates, prime implicants, "formulae" in decomposable negation normal form. On the other hand, the validity problem val(QPROP P S ) for Quantified Boolean Formulae (QBF) has been acknowledged for the past few years as an important issue for AI, and many solvers have been designed. In this paper, the complexity of restrictions of the validity problem for QBF obtained by imposing the matrix of the input QBF to belong to propositional fragments used as target languages for compilation, is identified. It turns out that this problem remains hard (PSPACE-complete) even under severe restrictions on the matrix of the input. Nevertheless some tractable restrictions are pointed out.

Proceedings of the thirty-first annual ACM symposium on Theory of computing - STOC '99, 1999

We show that hypergraph isomorphism can be tested in time O(c"), where n is the sire of the vertex set. In general, input of a hypergraph could require n(2") space, in which case the isomorphism test is in polynomial time. As a consequence, we put into polynomial time the classic problem of testing whether two Boolean functions, given by truth tables, are related via permutations and complementations of the variables, and therefore have structurally identical network realizations. In fact, the method is parallelizable and we put the problem even into NC. We obtain similarly an NC test of equivalence of truth tables under permutation of variables alone.

Journal on Satisfiability, Boolean Modeling and Computation

For Boolean satisfiability problems, the structure of the solution space is characterized by the solution graph, where the vertices are the solutions, and two solutions are connected iff they differ in exactly one variable. Motivated by research on heuristics and the satisfiability threshold, in 2006, Gopalan et al. studied connectivity properties of the solution graph and related complexity issues for constraint satisfaction problems [11]. They found dichotomies for the diameter of connected components and for the complexity of the stconnectivity question, and conjectured a trichotomy for the connectivity question. Their results could be improved based on findings by Makino et al. [15]. Building on this work, we here prove the trichotomy for the connectivity question. Also, we correct a minor mistake in [11], which leads to a slight shift of the boundaries towards the hard side.

An XSAT 𝜙 is satisfiable iff there exists a satisfying assignment 𝛼 to 𝜙. This paper shows that 𝜙 is satisfiable iff 𝜙 is reducible to 𝛼, viz., 𝜙 → 𝛼, thus 𝜙 ∧ 𝛼 is true, where 𝛼 = 𝑟 1 ∧ 𝑟 2 ∧ • • • ∧ 𝑟 𝑛 for some 𝑟 𝑙 ∈ {𝑥 𝑙 , 𝑥 𝑙 }. That is, the formula 𝜙 is unsatisfiable if it reduces to a simple formula 𝜓 inconsistent, otherwise 𝜙 → 𝜓 ∧𝜙 ′ such that 𝜙 ′ → 𝛼 ′ , and that 𝛼 = 𝜓 ∧ 𝛼 ′. Also, 𝛼 ′ = 𝜓 0 ∧ • • • ∧ 𝜓 𝑛 ′ and 𝜓 0 = (𝑟 ℓ 01 ∧ • • • ∧ 𝑟 ℓ 0 ñ),. .. ,𝜓 𝑛 ′ = (𝑟 ℓ 𝑛 ′ 1 ∧ • • • ∧ 𝑟 ℓ 𝑛 ′ ñ) such that {ℓ 01 , .

Theoretical Computer Science, 2010

The expressive power of existentially quantified Boolean formulas ∃CNF with free variables is investigated. We introduce a hierarchy of subclasses ∃MU * (k) of ∃CNF formulas based on the maximum deficiency k of minimal unsatisfiable subformulas of the bound part of the formulas. We will establish an upper bound of the size of minimally equivalent circuits. It will be shown, that there are constants a and b, such that for every formula in ∃MU * (k) of length m of the bound part and length l of the free part of the formula there is an equivalent circuit of size less than l + a • m b(log 2 (m)+k) 2 .

Discrete Applied Mathematics, 2008

… of Computer Science, 1999. 40th Annual …, 1999

We show that non-deterministic time N T IM E(n) is not contained in deterministic time n 2−ǫ and poly-logarithmic space, for any ǫ > 0. This implies that (infinitely often) satisfiability cannot be solved in time O(n 2−ǫ ) and polylogarithmic space. A similar result is presented for uniform circuits.

17th International Conference on Electronics, Communications and Computers (CONIELECOMP'07), 2007

Subgraph isomorphism (SI) detection is an important problem for several computer science subfields. In this paper we present a study of the Subgraph Isomorphism Problem (SIP) and its relation with the Hamiltonian cycles and SAT problems. In particular, we describe how instances of those problems can be solved throughout SI detection (using problems reductions). In our experiments we use an algorithm developed by the authors, which is capable to find all valid mappings in a SI instance. We performed several experiments, including cases for which there exists a known solution in polynomial time. In our analysis, we show the advantage and disadvantage of using a SI representation to solve Hamiltonian cycles and SAT problems.