Satisfiability Of Boolean Formulas Research Articles

The 2CNF Boolean formula satisfiability problem and the linear space hypothesis

We aim at investigating the solvability/insolvability of nondeterministic logarithmic-space (NL) decision, search, and optimization problems parameterized by natural size parameters using simultaneously polynomial time and sub-linear space. We are particularly focused on 2SAT3—a restricted variant of the 2CNF Boolean (propositional) formula satisfiability problem in which each variable of a given 2CNF formula appears at most 3 times in the form of literals—parameterized by the total number mvbl(ϕ) of variables of each given Boolean formula ϕ. We propose a new, practical working hypothesis, called the linear space hypothesis (LSH), which asserts that (2SAT3,mvbl) cannot be solved in polynomial time using only “sub-linear” space (i.e., mvbl(x)εpolylog(|x|) space for a constant ε∈[0,1)) on all instances x. Immediate consequences of LSH include L≠NL, LOGDCFL≠LOGCFL, and SC≠NSC. For our investigation, we fully utilize a key notion of “short reductions”, under which the class PsubLIN of all parameterized polynomial-time sub-linear-space solvable problems is indeed closed.

A Complete Diagrammatic Calculus for Boolean Satisfiability

We propose a calculus of string diagrams to reason about satisfiability of Boolean formulas, and prove it to be sound and complete. We then showcase our calculus in a few case studies. First, we consider SAT-solving. Second, we consider Horn clauses, which leads us to a new decision method for propositional logic programs equivalence under Herbrand model semantics.

Behaviour examples for synthesizing automaton models by temporal formulas

The paper deals with researching and developing the methods that make it possible to account behaviour examples when synthesizing automaton models by temporal formulas. Definitions of the terms and concepts used in work are given; the problem of synthesizing automaton systems according to the specification in the form of temporal formulas and behaviour examples is formulated; a promising algorithm for reducing the problem of synthesizing automaton systems to the Boolean formula satisfiability problem is described; an analysis of the domain and other approaches is carried out. New methods of taking into account behaviour examples in the synthesis of automaton systems according to a specification given in the form of temporal formulas are proposed. Algorithms for constructing graphs of scripts and methods for dividing graphs into clusters are described; they are designed to increase the efficiency of representing behaviour examples used for coding the behaviour examples in the form of Boolean formulas. An experimental study of the proposed methods of accounting for behaviour examples and basic approaches to the presentation of behaviour examples is carried out. The experimental results showed the superiority of the newly developed methods regarding the presentation of scripts in the form of temporal formulas. In summary, the main conclusions of the work carried out are presented.

BeBoSy: Behavior Examples Meet Bounded Synthesis

Bounded synthesis generates a state machine satisfying given temporal specification in linear temporal logic (LTL) while bounding the number of states of the target solution. Bounded synthesis methods currently do not support finite-length behavior examples, which are often helpful when formulating the specification of the desired state machine. In this work we show how to incorporate behavior examples into bounded synthesis methods and exemplify the approach by presenting BeBoSy ( Behavior Examples meet BoSy ), an extension of the bounded synthesis tool BoSy based on reductions to Boolean satisfiability (SAT) and Quantified Boolean formula satisfiability (QSAT) problems. The proposed approach is to augment the encodings used in BoSy with additional constraints that ensure the compliance of the generated state machine with the given behavior examples. Experiments with openly available data from the annual reactive synthesis competition SYNTCOMP augmented with random behavior examples show that the efficiency of the proposed approach is superior to both BoSy (with naive representation of behavior examples with LTL formulas) and a state-of-the-art counterexample-guided approach EFSM-tools.

State complexity characterizations of parameterized degree-bounded graph connectivity, sub-linear space computation, and the linear space hypothesis

The linear space hypothesis is a practical working hypothesis, which originally states the insolvability of a restricted 2CNF Boolean formula satisfiability problem parameterized by the number of Boolean variables. From this hypothesis, it naturally follows that the degree-3 directed graph connectivity problem (3DSTCON) parameterized by the number of vertices in a given graph cannot belong to PsubLIN, composed of all parameterized decision problems computable by polynomial-time, sub-linear-space deterministic Turing machines. This hypothesis immediately implies L≠NL and it was used as a solid foundation to obtain new lower bounds on the computational complexity of various NL search and NL optimization problems. The state complexity of transformation refers to the cost of converting one type of finite automata to another type, where the cost is measured in terms of the increase of the number of inner states of the converted automata from that of the original automata. We relate the linear space hypothesis to the state complexity of transforming restricted 2-way nondeterministic finite automata to computationally equivalent 2-way alternating finite automata having narrow computation graphs. For this purpose, we present state complexity characterizations of 3DSTCON and PsubLIN. We further characterize a nonuniform version of the linear space hypothesis in terms of the state complexity of transformation.

Efficient Verified (UN)SAT Certificate Checking

SAT solvers decide the satisfiability of Boolean formulas in conjunctive normal form. They are commonly used for software and hardware verification. Modern SAT solvers are highly complex and optimized programs. As a single bug in the solver may invalidate the verification of many systems, SAT solvers output certificates for their answer, which are then checked independently. However, even certificate checking requires highly optimized non-trivial programs. This paper presents the first SAT solver certificate checker that is formally verified down to the integer sequence representing the formula. Our tool supports the full DRAT standard, and is even faster than the unverified state-of-the-art tool drat-trim, on a realistic set of benchmarks drawn from the 2016 and 2017 SAT competitions. An optional multi-threaded mode further reduces the runtime, in particular for big certificates.

The Monotone Satisfiability Problem with Bounded Variable Appearances

The prominent Boolean formula satisfiability problem SAT is known to be [Formula: see text]-complete even for very restricted variants such as 3-SAT, Monotone 3-SAT, or Planar 3-SAT, or instances with bounded variable appearance. We settle the computational complexity status for two variants with bounded variable appearance: We show that Planar Monotone Sat — the variant of Monotone Sat in which the incidence graph is required to be planar — is [Formula: see text]-complete even if each clause consists of at most three distinct literals and each variable appears exactly three times, and that Monotone Sat is [Formula: see text]-complete even if each clause consists of three distinct literals and each variable appears exactly four times in the formula. The latter confirms a conjecture stated in scribe notes [7] of an MIT lecture by Eric Demaine. In addition, we provide hardness results with respect to bounded variable appearances for two variants of Planar Monotone Sat.

Local Restrictions from the Furst-Saxe-Sipser Paper

In their celebrated paper (Furst et al., Math. Syst. Theory 17(1), 13---27 (12)), Furst, Saxe, and Sipser used random restrictions to reveal the weakness of Boolean circuits of bounded depth, establishing that constant-depth and polynomial-size circuits cannot compute the parity function. Such local restrictions have played important roles and have found many applications in complexity analysis and algorithm design over the past three decades. In this article, we give a brief overview of two intriguing applications of local restrictions: the first one is for the Isomorphism Conjecture and the second one is for moderately exponential time algorithms for the Boolean formula satisfiability problem.

Model Checking Software with First Order Logic Specifications Using AIG Solvers

Static verification techniques leverage Boolean formula satisfiability solvers such as SAT and SMT solvers that operate on conjunctive normal form and first order logic formulae, respectively, to validate programs. They force bounds on variable ranges and execution time and translate the program and its specifications into a Boolean formula. They are limited to programs of relatively low complexity for the following reasons. (1) A small increase in the bounds can cause a large increase in the size of the translated formula. (2) Boolean satisfiability solvers are restricted to using optimizations that apply at the level of the formula. Finally, (3) the Boolean formulae often need to be regenerated with higher bounds to ensure the correctness of the translation. We present a method that uses And-Inverter-Graph (AIG) sequential circuits, and AIG synthesis and verification frameworks to validate programs. An AIG is a Boolean formula with memory elements, logically complete negated conjunction gates, and a hierarchical structure. Encoding the validation problem of a program as an AIG (1) typically provides a more succinct representation than a Boolean formulae encoding with no memory elements, (2) preserves the high-level structure of the program, and (3) enables the use of a number of powerful automated analysis techniques that have no counterparts for other Boolean formulae such as CNF. Our method takes an imperative program with a first order logic specification consisting of a precondition and a postcondition pair, and a bound on the program variable ranges, and produces an AIG with a designated output that is ${true}$ when the program violates the specification. Our method uses AIG synthesis reduction techniques to reduce the AIG, and then uses AIG verification techniques to check the satisfiability of the designated output. The results show that our method can validate designs that are not possible with other state of the art techniques, and with bounds that are an order of magnitude larger.

A Model for Phase Transition of Random Answer-Set Programs

The critical behaviors of NP-complete problems have been studied extensively, and numerous results have been obtained for Boolean formula satisfiability (SAT) and constraint satisfaction (CSP), among others. However, few results are known for the critical behaviors of NP-hard nonmonotonic reasoning problems so far; in particular, a mathematical model for phase transition in nonmonotonic reasoning is still missing. In this article, we investigate the phase transition of negative two-literal logic programs under the answer-set semantics. We choose this class of logic programs since it is the simplest class for which the consistency problem of deciding if a program has an answer set is still NP-complete. We first introduce a new model, called quadratic model for generating random logic programs in this class. We then mathematically prove that the consistency problem for this class of logic programs exhibits a phase transition. Furthermore, the phase-transition follows an easy-hard-easy pattern. Given the correspondence between answer sets for negative two-literal programs and kernels for graphs, as a corollary, our result significantly generalizes de la Vega's well-known theorem for phase transition on the existence of kernels in random graphs. We also report some experimental results. Given our mathematical results, these experimental results are not really necessary. We include them here as they suggest that our phase-transition result is more general and likely holds for more general classes of logic programs.

Проектирование вещественных переменных в булевы в методе простой итерации, примененном к задаче выполнимости булевых формул

Проектирование вещественных переменных в булевы в методе простой итерации, примененном к задаче выполнимости булевых формул

A satisfiability algorithm and average-case hardness for formulas over the full binary basis

We present a moderately exponential time algorithm for the satisfiability of Boolean formulas over the full binary basis. For formulas of size at most cn, our algorithm runs in time $${2^{(1-\mu_{c})n}}$$ for some constant μ c > 0. As a byproduct of the running time analysis of our algorithm, we obtain strong average-case hardness of affine extractors for linear-sized formulas over the full binary basis.

Techniques for SAT-based constrained test pattern generation

Testing of digital circuits seems to be a completely mastered part of the design flow, but Constrained Test Patterns Generation (CTPG) is still a highly evolving branch of digital circuits testing. Our previous research on CTPG proved that we can benefit from an implicit representation of test patterns set. The set of test patterns is implicitly represented as a Boolean formula satisfiability problem in CNF, like in common SAT-based ATPGs. However, the CTPG process can be much more memory or time consuming than common TPG, thus some techniques of speeding up the constrained SAT-based test patterns generation are described and analyzed into detail in this paper. These techniques are experimentally evaluated on a real SAT-based algorithm performing a test compression based on overlapping of test patterns. Experiments are performed on ISCAS’85, ’89 and ITC’99 benchmark circuits. Results of the experiments are discussed and recommendations for further development of similar SAT-based tools for CTPG are given.

Robust QBF Encodings for Sequential Circuits with Applications to Verification, Debug, and Test

Formal CAD tools operate on mathematical models describing the sequential behavior of a VLSI design. With the growing size and state-space of modern digital hardware designs, the conciseness of this mathematical model is of paramount importance in extending the scalability of those tools, provided that the compression does not come at the cost of reduced performance. Quantified Boolean Formula satisfiability (QBF) is a powerful generalization of Boolean satisfiability (SAT). It also belongs to the same complexity class as many CAD problems dealing with sequential circuits, which makes it a natural candidate for encoding such problems. This work proposes a succinct QBF encoding for modeling sequential circuit behavior. The encoding is parametrized and further compression is achieved using time-frame windowing. Comprehensive hardware constructions are used to illustrate the proposed encodings. Three notable CAD problems, namely bounded model checking, design debugging and sequential test pattern generation, are encoded as QBF instances to demonstrate the robustness and practicality of the proposed approach. Extensive experiments on OpenCore circuits show memory reductions in the order of 90 percent and demonstrate competitive runtimes compared to state-of-the-art SAT techniques. Furthermore, the number of solved instances is increased by 16 percent. Admittedly, this work encourages further research in the use of QBF in CAD for VLSI.

The Helly property and satisfiability of Boolean formulas defined on set families

In this paper, we study the problem of satisfiability of Boolean formulas φ in conjunctive normal form (CNF) whose literals have the form v ∈ S and express the membership of values to sets S of a given set family S defined on a finite domain D . We establish the following dichotomy result. We show that checking the satisfiability of such formulas (called S - formulas) with three or more literals per clause is NP -complete except the trivial case when the intersection of all sets in S is nonempty. On the other hand, the satisfiability of S -formulas φ containing at most two literals per clause is decidable in polynomial time if S satisfies the Helly property, and is NP -complete otherwise (in the first case, we present an O ( | φ | ⋅ | S | ⋅ | D | ) -time algorithm for deciding if φ is satisfiable). Deciding whether a given set family S satisfies the Helly property can be done in polynomial time. We also overview several well-known examples of Helly families and discuss the consequences of our result to such set families and its relationship with the previous work on the satisfiability of signed formulas in multiple-valued logic.

Constraint satisfaction problems with isolated solutions are hard

We study the phase diagram and the algorithmic hardness of the random ‘locked’constraint satisfaction problems, and compare them to the commonly studied ‘non-locked’problems like satisfiability of Boolean formulae or graph coloring. The special property ofthe locked problems is that clusters of solutions are isolated points. This simplifiessignificantly the determination of the phase diagram, which makes the locked problemsparticularly appealing from the mathematical point of view. On the other hand, weshow empirically that the clustered phase of these problems is extremely hardfrom the algorithmic point of view: the best known algorithms all fail to findsolutions. Our results suggest that the easy/hard transition (for currently knownalgorithms) in the locked problems coincides with the clustering transition. Theseshould thus be regarded as new benchmarks of really hard constraint satisfactionproblems.

On the Freezing of Variables in Random Constraint Satisfaction Problems

The set of solutions of random constraint satisfaction problems (zero energy groundstates of mean-field diluted spin glasses) undergoes several structural phase transitions as the amount of constraints is increased. This set first breaks down into a large number of well separated clusters. At the freezing transition, which is in general distinct from the clustering one, some variables (spins) take the same value in all solutions of a given cluster. In this paper we introduce and study a message passing procedure that allows to compute, for generic constraint satisfaction problems, the sizes of the rearrangements induced in response to the modification of a variable. These sizes diverge at the freezing transition, with a critical behavior which is also investigated in details. We apply the generic formalism in particular to the random satisfiability of boolean formulas and to the coloring of random graphs. The computation is first performed in random tree ensembles, for which we underline a connection with percolation models and with the reconstruction problem of information theory. The validity of these results for the original random ensembles is then discussed in the framework of the cavity method.

A Faster Clause-Shortening Algorithm for SAT with No Restriction on Clause Length

We give a randomized algorithm for testing satisfiability of Boolean formulas in conjunctive normal form with no restriction on clause length. This algorithm uses the clause-shortening approach proposed by Schuler. The running time of the algorithm is O(2^(n(1−1/α))) where α = ln(m/n) + O(ln ln m) and n, m are respectively the number of variables and the number of clauses in the input formula. This bound is asymptotically better than the previously best known 2^(n(1−1/ log(2m))) bound for SAT.

The complexity of maximum matroid–greedoid intersection and weighted greedoid maximization

The maximum intersection problem for a matroid and a greedoid, given by polynomial-time oracles, is shown NP-hard by expressing the satisfiability of boolean formulas in 3-conjunctive normal form as such an intersection. The corresponding approximation problems are shown NP-hard for certain approximation performance bounds. Moreover, some natural parameterized variants of the problem are shown W[P]-hard. The results are in contrast with the maximum matroid–matroid intersection which is solvable in polynomial time by an old result of Edmonds. We also prove that it is NP-hard to approximate the weighted greedoid maximization within 2 n O ( 1 ) where n is the size of the domain of the greedoid.

Simplification Rules for the Coherent Probability Assessment Problem

In this paper we develop a procedure for checking the consistency (coherence) of a partial probability assessment. The general problem (called CPA) is NP-complete, hence, to have a reasonable application some heuristic is needed. Our proposal differs from others because it is based on a skilful use of the logical relations present among the events. In other approaches the consistency problem is reduced directly to the satisfiability of a system of linear constraints. Here, thanks to the characterization of particular configurations and to the elimination of variables, an instance of the problem is reduced to smaller instances. To obtain such results, we introduce a procedure based on rules resembling those given by Davis–Putnam for the satisfiability of Boolean formulas. At the end a particularized description of an actual implementation is given.