Satisfying Truth Assignments Research Articles

Relating planar graph drawings to planar satisfiability problems

A SAT graph G(Φ) of a satisfiability instance Φ consists of a vertex for each clause and a vertex for each variable, where there exists an edge between a clause vertex and a variable vertex if and only if the variable or its negation appears in that clause. Many satisfiability problems, which are NP-hard, become polynomial-time solvable when the SAT graph is restricted to satisfy some graph properties. A rich body of research attempts to narrow down the boundary between the NP-hardness and polynomial-time solvability of various satisfiability problems. In this paper, we examine planar satisfiability problems and leverage planar graph drawing algorithms to improve our understanding of these problems. A rich body of graph drawing algorithms exists to check whether a planar graph admits a drawing that satisfies certain drawing aesthetics. We show how the existing graph drawing knowledge could be used to establish sufficient conditions for a SAT instance to always be satisfiable and give algorithms to efficiently find a satisfying truth assignment. In some cases, our algorithm can find a truth assignment by setting a small number of variables to true, which relates to the satisfiability variants that seek to minimize the number of ones.

Verifying the DPLL Algorithm in Dafny

Modern high-performance SAT solvers quickly solve large satisfiability instances that occur in practice. If the instance is satisfiable, then the SAT solver can provide a witness which can be checked independently in the form of a satisfying truth assignment. However, if the instance is unsatisfiable, the certificates could be exponentially large or the SAT solver might not be able to output certificates. The implementation of the SAT solver should then be trusted not to contain bugs. However, the data structures and algorithms implemented by a typical high-performance SAT solver are complex enough to allow for subtle programming errors. To counter this issue, we build a verified SAT solver using the Dafny system. We discuss its implementation in the present article.

Model counting with error-correcting codes

The idea of counting the number of satisfying truth assignments (models) of a formula by adding random parity constraints can be traced back to the seminal work of Valiant and Vazirani showing that NP is as easy as detecting unique solutions. While theoretically sound, the random parity constraints used in that construction suffer from the following drawback: each constraint, on average, involves half of all variables. As a result, the branching factor associated with searching for models that also satisfy the parity constraints quickly gets out of hand. In this work we prove that one can work with much shorter parity constraints and still get rigorous mathematical guarantees, especially when the number of models is large so that many constraints need to be added. Our work is motivated by the realization that the essential feature for a system of parity constraints to be useful in probabilistic model counting is that its set of solutions resembles an error-correcting code.

On Distributed Solution to SAT by Membrane Computing

Tissue P systems with evolutional communication rules and cell division (TPec, for short) are a class of bio-inspired parallel computational models, which can solve NP-complete problems in a feasible time. In this work, a variant of TPec, called $k$-distributed tissue P systems with evolutional communication and cell division ($k\text{-}\Delta_{TP_{ec}}$, for short) is proposed. A uniform solution to the SAT problem by $k\text{-}\Delta_{TP_{ec}}$ under balanced fixed-partition is presented. The solution provides not only the precise satisfying truth assignments for all Boolean formulas, but also a precise amount of possible such satisfying truth assignments. It is shown that the communication resource for one-way and two-way uniform $k$-P protocols are increased with respect to $k$; while a single communication is shown to be possible for bi-directional uniform $k$-P protocols for any $k$. We further show that if the number of clauses is at least equal to the square of the number of variables of the given boolean formula, then $k\text{-}\Delta_{TP_{ec}}$ for solving the SAT problem are more efficient than TPec as show in \cite{bosheng2017}; if the number of clauses is equal to the number of variables, then $k\text{-}\Delta_{TP_{ec}}$ for solving the SAT problem work no much faster than TPec.

XSAT and NAE-SAT of linear CNF classes

XSAT and NAE-SAT are important variants of the propositional satisfiability problem (SAT). XSAT contains all CNF formulas that can be satisfied by setting exactly one literal in each clause to 1, whereas NAE-SAT requires satisfying truth assignments not setting all literals equally in any clause. Both variants are studied here regarding their computational complexity of linear CNF formulas, whose clauses are allowed to have at most one variable in common. We prove that both variants remain NP-complete for (monotone) linear formulas, yielding the conclusion that also bicolorability of linear hypergraphs is NP-complete. The reduction used gives rise to the complexity investigations of several monotone linear subclasses of both variants that are parameterized by the size of clauses, or by the number of occurrences of variables. For particular values of those parameters, we are able to show the NP-completeness of XSAT and NAE-SAT, though we cannot provide a complete treatment. Finally, we focus on exact linear formulas, where clauses intersect pairwise, and for which SAT is known to be polynomial-time solvable. Here, we prove NP-completeness of XSAT for exact linear formulas. If in addition the clauses are required to be of uniform length, we show that both XSAT and NAE-SAT are decidable in polynomial time, what even holds for their counting versions.

Exploiting independent subformulas: A faster approximation scheme for #k-SAT

We present an improvement on Thurleyʼs recent randomized approximation scheme for #k-SAT where the task is to count the number of satisfying truth assignments of a Boolean function Φ given as an n-variable k-CNF. We introduce a novel way to identify independent substructures of Φ and can therefore reduce the size of the search space considerably. Our randomized algorithm works for any k. For #3-SAT, it runs in time O(ε−2⋅1.51426n), for #4-SAT, it runs in time O(ε−2⋅1.60816n), with error bound ε.

Compactly Generating All Satisfying Truth Assignments of a Horn Formula

While it was known that all models of a Horn formula can be generated in outputpolynomial time, here we present an explicit algorithm as opposed to the rather vague oracle-scheme suggested in the proof of [6, Thm.4]. It is an instance of some principle of exclusion that compactly (thus not one by one) generates all models of certain Boolean formulae given in CNF. The principle of exclusion can be adapted to generate only the models of weight k. We compare and contrast it with constraint programming, 0; 1 integer programming, and binary decision diagrams.

Absorbing random walks and the NAE2SAT problem

In this paper, we propose a simple randomized algorithm for the NAE2SAT problem. The analysis of the algorithm uses the theory of symmetric, absorbing random walks. Inasmuch as the NAE2SAT problem is in the complexity class L (Deterministic Logarithmic Space) and L ⊆ P ⊆ RP, the existence of such an algorithm is not surprising. However, the simplicity of our approach and the insights revealed by its analysis make the current study worthwhile. Not-all-equal SAT (NAESAT) is the variant of the satisfiability problem (SAT), in which we are interested in an assignment that satisfies all the clauses, but falsifies at least one literal in each clause. NAESAT is an interesting SAT variant in its own right and has been studied by SAT theoreticians, on account of its connections to the graph colouring problem. It is well-known that the variant of NAESAT with exactly three literals per clause (NAE3SAT) is NP-complete [M.R. Garey and D.S. Johnson, Computers and Intractability: A Guide to the Theory of NP-Completeness, W. H. Freeman Company, San Francisco, 1979]. Likewise, there seem to be a number of polynomial time algorithms for NAE2SAT as part of algorithm design folklore [D.S. Johnson, Personal Communication]. Here we show that the NAE2SAT problem admits an extremely simple, literal-flipping algorithm, in precisely the same way that 2SAT does. On a satisfiable instance involving n variables, the probability that our algorithm does not find a satisfying assignment is at most 1/24. The randomized algorithm takes O(1) extra space, in the presence of a verifier and provides an interesting insight into checking whether a graph is bipartite. It must be noted that space-parsimonious randomized algorithms for a problem, such as the one proposed in this paper, invariably lead to error–bounded, online algorithms for the same. As part of our analysis, we argue that a restricted variant of NAE2SAT is L-complete. We note that the bounds derived in this paper are sharper than the ones in Papadimitriou [C.H. Papadimitriou, On selecting a satisfying truth assignment, in Proceedings of the 32nd Annual Symposium on Foundations of Computer Science, San Juan, Puerto Rico, 1–4 October 1991, IEEE, ed., IEEE, Washington, DC, 1991, pp. 163–169].

Models and quantifier elimination for quantified Horn formulas

In this paper, quantified Horn formulas ( QHORN) are investigated. We prove that the behavior of the existential quantifiers depends only on the cases where at most one of the universally quantified variables is zero. Accordingly, we give a detailed characterization of QHORN satisfiability models which describe the set of satisfying truth assignments to the existential variables. We also consider quantified Horn formulas with free variables ( QHORN * ) and show that they have monotone equivalence models. The main application of these findings is that any quantified Horn formula Φ of length | Φ | with free variables, | ∀ | universal quantifiers and an arbitrary number of existential quantifiers can be transformed into an equivalent quantified Horn formula of length O ( | ∀ | · | Φ | ) which contains only existential quantifiers. We also obtain a new algorithm for solving the satisfiability problem for quantified Horn formulas with or without free variables in time O ( | ∀ | · | Φ | ) by transforming the input formula into a satisfiability-equivalent propositional formula. Moreover, we show that QHORN satisfiability models can be found with the same complexity.

The Complexity of Some Subclasses of Minimal Unsatisfiable Formulas

This paper is concerned with the complexity of some natural subclasses of minimal unsatisable formulas. We show the D P {completeness of the classes of maximal and marginal minimal unsatisable formulas. Then we consider the class Unique{MU of minimal unsatisable formulas which have after removing a clause exactly one satisfying truth assignment. We show that Unique{MU has the same complexity as the unique satisabilit y problem with respect to polynomial reduction. However, a slight modication of this class leads to the D P {completeness. Finally we show that the class of minimal unsatisable formulas which can be divided for every variable into two separate minimal unsatisable formulas is at least as hard as the unique satisabilit y problem.

The unsatisfiability threshold revisited

The problem of determining the unsatisfiability threshold for random 3-SAT formulas consists in determining the clause to variable ratio that marks the experimentally observed abrupt change from almost surely satisfiable formulas to almost surely unsatisfiable. Up to now, there have been rigorously established increasingly better lower and upper bounds to the actual threshold value. In this paper, we consider the problem of bounding the threshold value from above using methods that, we believe, are of interest on their own right. More specifically, we show how the method of local maximum satisfying truth assignments can be combined with results for the occupancy problem in schemes of random allocation of balls into bins in order to achieve an upper bound for the unsatisfiability threshold less than 4.571. In order to obtain this value, we establish a bound on the q-binomial coefficients (a generalization of the binomial coefficients). No such bound was previously known, despite the extensive literature on q-binomial coefficients. Finally, to prove our result we had to establish certain relations among the conditional probabilities of an event in various probabilistic models for random formulas. It turned out that these relations were considerably harder to prove than the corresponding ones for unconditional probabilities, which were previously known.

Hiding Satisfying Assignments: Two are Better than One

The evaluation of incomplete satisfiability solvers depends critically on the availability of hard satisfiable instances. A plausible source of such instances consists of random k-SAT formulas whose clauses are chosen uniformly from among all clauses satisfying some randomly chosen truth assignment A. Unfortunately, instances generated in this manner tend to be relatively easy and can be solved efficiently by practical heuristics. Roughly speaking, for a number of different algorithms, A acts as a stronger and stronger attractor as the formula's density increases. Motivated by recent results on the geometry of the space of satisfying truth assignments of random k-SAT and NAE-k-SAT formulas, we introduce a simple twist on this basic model, which appears to dramatically increase its hardness. Namely, in addition to forbidding the clauses violated by the hidden assignment A, we also forbid the clauses violated by its complement, so that both A and compliment of A are satisfying. It appears that under this "symmetrization" the effects of the two attractors largely cancel out, making it much harder for algorithms to find any truth assignment. We give theoretical and experimental evidence supporting this assertion.

On a simple randomized algorithm for finding a 2-factor in sparse graphs

We analyze the performance of a simple randomized algorithm for finding 2-factors in directed Hamiltonian graphs of out-degree at most two and in undirected Hamiltonian graphs of degree at most three. For the directed case, the algorithm finds a 2-factor in O ( n 2 ) expected time. The analysis of our algorithm is based on random walks on the line and interestingly resembles the analysis of a randomized algorithm for the 2-SAT problem given by Papadimitriou [On selecting a satisfying truth assignment, in: Proc. 32nd Annual IEEE Symp. on the Foundations of Computer Science (FOCS), 1991, p. 163]. For the undirected case, the algorithm finds a 2-factor in O ( n 3 ) expected time. We also analyze random versions of these graphs and show that cycles of length Ω ( n / log n ) can be found with high probability in polynomial time. This partially answers an open question of Broder et al. [Finding hidden Hamilton cycles, Random Structures Algorithms 5 (1994) 395] on finding hidden Hamiltonian cycles in sparse random graphs and improves on a result of Karger et al. [On approximating the longest path in a graph, Algorithmica 18 (1997) 82].

On Computing Belief Change Operations using Quantified Boolean Formulas

In this paper, we show how an approach to belief revision and belief contraction can be axiomatized by means of quantified Boolean formulas. Specifically, we consider the approach of belief change scenarios, a general framework that has been introduced for expressing different forms of belief change. The essential idea is that for a belief change scenario (K, R, C), the set of formulas K, representing the knowledge base, is modified so that the sets of formulas R and C are respectively true in, and consistent with the result. By restricting the form of a belief change scenario, one obtains specific belief change operators including belief revision, contraction, update, and merging. For both the general approach and for specific operators, we give a quantified Boolean formula such that satisfying truth assignments to the free variables correspond to belief change extensions in the original approach. Hence, we reduce the problem of determining the results of a belief change operation to that of satisfiability. This approach has several benefits. First, it furnishes an axiomatic specification of belief change with respect to belief change scenarios. This then leads to further insight into the belief change framework. Second, this axiomatization allows us to identify strict complexity bounds for the considered reasoning tasks. Third, we have implemented these different forms of belief change by means of existing solvers for quantified Boolean formulas. As well, it appears that this approach may be straightforwardly applied to other specific approaches to belief change.

Selecting Complementary Pairs of Literals

Abstract We present and rigorously analyze a heuristic that searches for a satisfying truth assignment of a given random instance of the 3-SAT problem. We prove that the heuristic asymptotically certainly succeeds in producing a satisfying truth assignment for formulas with clauses to variables ratio (density) of up to 3.52. Thus the experimentally observed threshold of the density where a typical formula's phase changes from asymptotically certainly satisfiable to asymptotically certainly contradictory is rigorously shown to be at least 3.52. The best previous lower bound in the long series of mathematically rigorous approximations by various research groups of the experimental threshold was 3.42. That was the first result where the probabilistic analysis was based on random formulas with a pre-specified, typical number of appearances for each literal. However, in that result, in order to simplify the analysis, the number of appearances of each literal was decoupled from the number of appearances of its negation. In this work, we assume not only that each literal has the typical number of occurrences, but that for each variable both numbers of occurrences of its positive and negated appearances are typical. By standard techniques, our algorithm can be easily modified to run in linear time. Thus not only the satisfiability threshold, but also the threshold (experimental again) where the complexity of searching for satisfying truth assignments jumps from polynomial to exponential is at least 3.52. This should be contrasted with the value 3.9 for the complexity threshold given by theoretical (but not mathematically rigorous) techniques of Statistical Physics.

An efficient local search method for random 3-satisfiability

Abstract We report on some exceptionally good results in the solution of randomly generated 3-satisfiability instances using the “record-to-record travel (RRT)” local search method. When this simple, but less-studied algorithm is applied to random one-million variable instances from the problem's satisfiable phase, it seems to find satisfying truth assignments almost always in linear time, with the coefficient of linearity depending on the ratio α of clauses to variables in the generated instances. RRT has a parameter for tuning “greediness”. By lessening greediness, the linear time phase can be extended up to very close to the satisfiability threshold α c . Such linear time complexity is typical for random-walk based local search methods for small values of α . Previously, however, it has been suspected that these methods necessarily lose their time linearity far below the satisfiability threshold. The only previously introduced algorithm reported to have nearly linear time complexity also close to the satisfiability threshold is the survey propagation (SP) algorithm. However, SP is not a local search method and is more complicated to implement than RRT. Comparative experiments with the WalkSAT local search algorithm show behavior somewhat similar to RRT, but with the linear time phase not extending quite as close to the satisfiability threshold.

The complexity of minimal satisfiability problems

A dichotomy theorem for a class of decision problems is a result asserting that certain problems in the class are solvable in polynomial time, while the rest are NP-complete. The first remarkable such dichotomy theorem was proved by Schaefer in 1978. It concerns the class of generalized satisfiability problems Sat (S) , whose input is a CNF (S) -formula, i.e., a formula constructed from elements of a fixed set S of generalized connectives using conjunctions and substitutions by variables. Here, we investigate the complexity of minimal satisfiability problems Min Sat (S) , where S is a fixed set of generalized connectives. The input to such a problem is a CNF (S) -formula and a satisfying truth assignment; the question is to decide whether there is another satisfying truth assignment that is strictly smaller than the given truth assignment with respect to the coordinate-wise partial order on truth assignments. Minimal satisfiability problems were first studied by researchers in artificial intelligence while investigating the computational complexity of propositional circumscription. The question of whether dichotomy theorems can be proved for these problems was raised at that time, but was left open. We settle this question affirmatively by establishing a dichotomy theorem for the class of all Min Sat (S) -problems, where S is a finite set of generalized connectives. We also prove a dichotomy theorem for a variant of Min Sat (S) in which the minimization is restricted to a subset of the variables, whereas the remaining variables may vary arbitrarily (this variant is related to extensions of propositional circumscription and was first studied by Cadoli). Moreover, we show that similar dichotomy theorems hold also when some of the variables are assigned constant values. Finally, we give simple criteria that tell apart the polynomial-time solvable cases of these minimal satisfiability problems from the NP-complete ones.

Recognition of Simple Enlarged Horn Formulas and Simple Extended Horn Formulas

The enlarged Horn formulas generalize the extended Horn formulas introduced by Chandru and Hooker (1991). Their satisfying truth assignments can be generated with polynomial delay. Unfortunately no polynomial algorithm is known for recognizing enlarged Horn formulas or extended Horn formulas. In this paper we define the class of simple enlarged Horn formulas, a subclass of the enlarged Horn formulas, that contains the simple extended Horn formulas introduced by Swaminathan and Wagner (1995). We present recognition algorithms for the simple enlarged Horn formulas and the simple extended Horn formulas whose complexity is bounded by the complexity of the arborescence-realization problem.

The unsatisfiability threshold revisited

Abstract Abstract The problem of determining the unsatisfiability threshold for random 3-SAT formulas consists in determining the clause to variable ratio that marks the experimentally observed abrupt change from almost surely satisfiable formulas to almost surely unsatisfiable. Up to now, there have been rigorously established increasingly better lower and upper bounds to the actual threshold value. In this paper, we consider the problem of bounding the threshold value from above using methods that, we believe, are of interest on their own right. More specifically, we show how the method of local maximum satisfying truth assignments can be combined with results for the occupancy problem in random allocation schemes of balls into bins in order to achieve an upper bound for the unsatisfiability threshold less than 4.571. Thus we improve over the best, with an available complete proof, previous upper bound, which was 4.596. In order to obtain this value, we also establish a bound on the (q-binomial coefficients (a generalization of the binomial coefficients) which, we believe, is of independent interest.

The SAT phase transition

Phase transition is an important feature of SAT problem. For randomk-SAT model, it is proved that asr (ratio of clauses to variables) increases, the structure of solutions will undergo a sudden change like satisfiability phase transition whenr reaches a threshold point (r=r σ ). This phenomenon shows that the satisfying truth assignments suddenly shift from being relatively different from each other to being very similar to each other.