Satisfiability Of Formulas Research Articles

Relaxation Search: A Simple Way of Managing Optional Clauses

A number of problems involve managing a set of optional clauses. For example, the soft clauses in a MAXSAT formula are optional—they can be falsified for a cost. Similarly, when computing a Minimum Correction Set for an unsatisfiable formula, all clauses are optional—some can be falsified in order to satisfy the remaining. In both of these cases the task is to find a subset of the optional clauses that achieves some criteria, and whose removal leaves a satisfiable formula. Relaxation search is a simple method of using a standard SAT solver to solve this task. Relaxation search is easy to implement, sometimes requiring only a simple modification of the variable selection heuristic in the SAT solver; it offers considerable flexibility and control over the order in which subsets of optional clauses are examined; and it automatically exploits clause learning to exchange information between the two phases of finding a suitable subset of optional clauses and checking if their removal yields satisfiability. We demonstrate how relaxation search can be used to solve MAXSAT and to compute Minimum Correction Sets. In both cases relaxation search is able to achieve state-of-the-art performance and solve some instances other solvers are not able to solve.

Parity Games and Propositional Proofs

A propositional proof system is weakly automatizable if there is a polynomial time algorithm that separates satisfiable formulas from formulas that have a short refutation in the system, with respect to a given length bound. We show that if the resolution proof system is weakly automatizable, then parity games can be decided in polynomial time. We give simple proofs that the same holds for depth-1 propositional calculus (where resolution has depth 0) with respect to mean payoff and simple stochastic games. We define a new type of combinatorial game and prove that resolution is weakly automatizable if and only if one can separate, by a set decidable in polynomial time, the games in which the first player has a positional winning strategy from the games in which the second player has a positional winning strategy. Our main technique is to show that a suitable weak bounded arithmetic theory proves that both players in a game cannot simultaneously have a winning strategy, and then to translate this proof into propositional form.

STL⁎: Extending signal temporal logic with signal-value freezing operator

To express temporal properties of dense-time real-valued signals, the Signal Temporal Logic (STL) has been defined by Maler et al. The work presented a monitoring algorithm deciding the satisfiability of STL formulae on finite discrete samples of continuous signals. The logic is not expressive enough to sufficiently distinguish oscillatory properties important in biology. In this paper we introduce the extended logic STL⁎ in which STL is augmented with a signal-value freezing operator allowing to express (and distinguish) various dynamic aspects of oscillations. This operator may be nested for further increase of expressiveness. The logic is supported by a monitoring algorithm prototyped in Matlab for the fragment that avoids nesting of the freezing operator. The monitoring procedure for STL⁎ is evaluated on a sample oscillatory signal with varied parameters. Application of the extended logic is demonstrated on a case study of a biological oscillator. We also discuss expressive power of STL with respect to STL⁎.

Random Model Sampling: Making Craig Interpolation Work When It Should Not

One of the most serious problems when doing program analyses is dealing with function calls. While function inlining is the traditional approach to this problem, it nonetheless suffers from the increase in analysis complexity due to the state space explosion. Craig interpolation has been successfully used in recent years in the context of bounded model checking to do function summarization which allows one to replace the complete function body with its succinct summary and, therefore, reduce the complexity. Unfortunately this technique can be applied only to a pair of unsatisfiable formulae. In this work-in-progress paper we present an approach to function summarization based on Craig interpolation that overcomes its limitation by using random model sampling. It captures interesting input/output relations, strengthening satisfiable formulae into unsatisfiable ones and thus allowing the use of Craig interpolation. Preliminary experiments show the applicability of this approach; in our future work we plan to do a full evaluation on real-world examples.

A decomposition based algorithm for maximal contractions

This paper proposes a decomposition based algorithm for revision problems in classical propositional logic. A set of decomposing rules are presented to analyze the satisfiability of formulas. The satisfiability of a formula is equivalent to the satisfiability of a set of literal sets. A decomposing function is constructed to calculate all satisfiable literal sets of a given formula. When expressing the satisfiability of a formula, these literal sets are equivalent to all satisfied models of such formula. A revision algorithm based on this decomposing function is proposed, which can calculate maximal contractions of a given problem. In order to reduce the memory requirement, a filter function is introduced. The improved algorithm has a good performance in both time and space perspectives.

Right-convergence of sparse random graphs

The paper is devoted to the problem of establishing right-convergence of sparse random graphs. This concerns the convergence of the logarithm of number of homomorphisms from graphs or hyper-graphs \(\mathbb{G }_N, N\ge 1\) to some target graph \(W\). The theory of dense graph convergence, including random dense graphs, is now well understood (Borgs et al. in Ann Math 176:151–219, 2012; Borgs et al. in Adv Math 219:1801–1851, 2008; Chatterjee and Varadhan in Eur J Comb 32:1000–1017, 2011; Lovász and Szegedy in J Comb Theory Ser B 96:933–957, 2006), but its counterpart for sparse random graphs presents some fundamental difficulties. Phrased in the statistical physics terminology, the issue is the existence of the limits of appropriately normalized log-partition functions, also known as free energy limits, for the Gibbs distribution associated with \(W\). In this paper we prove that the sequence of sparse Erdös-Rényi graphs is right-converging when the tensor product associated with the target graph \(W\) satisfies a certain convexity property. We treat the case of discrete and continuous target graphs \(W\). The latter case allows us to prove a special case of Talagrand’s recent conjecture [more accurately stated as level III Research Problem 6.7.2 in his recent book (Talagrand in Mean Field Models for Spin Glasses: Volume I: Basic examples. Springer, Berlin, 2010)], concerning the existence of the limit of the measure of a set obtained from \(\mathbb{R }^N\) by intersecting it with linearly in \(N\) many subsets, generated according to some common probability law. Our proof is based on the interpolation technique, introduced first by Guerra and Toninelli (Commun Math Phys 230:71–79, 2002) and developed further in (Abbe and Montanari in On the concentration of the number of solutions of random satisfiability formulas, 2013; Bayati et al. in Ann Probab Conference version in Proceedings of 42nd Ann. Symposium on the Theory of Computing (STOC), 2010; Contucci et al. in Antiferromagnetic Potts model on the Erdös-Rényi random graph, 2011; Franz and Leone in J Stat Phys 111(3/4):535–564, 2003; Franz et al. in J Phys A Math Gen 36:10967–10985, 2003; Montanari in IEEE Trans Inf Theory 51(9):3221–3246, 2005; Panchenko and Talagrand in Probab Theory Relat Fields 130:312–336, 2004). Specifically, Bayati et al. (Ann Probab Conference version in Proceedings of 42nd Ann. Symposium on the Theory of Computing (STOC), 2010) establishes the right-convergence property for Erdös-Rényi graphs for some special cases of \(W\). In this paper most of the results in Bayati et al. (Ann Probab Conference version in Proceedings of 42nd Ann. Symposium on the Theory of Computing (STOC), 2010) follow as a special case of our main theorem.

Decision procedures for some strong hybrid logics

Hybrid logics are extensions of standard modal logics, which significantly increase the expressive power of the latter. Since most of hybrid logics are known to be decidable, decision procedures for them is a widely investigated field of research. So far, several tableau calculi for hybrid logics have been presented in the literature. In this paper we introduce a sound, complete and terminating tableau calculus T H(@,E,D, ♦ −) for hybrid logics with the satisfaction operators, the universal modality, the difference modality and the inverse modality as well as the corresponding sequent calculus S H(@,E,D, ♦ −) . They not only uniformly cover relatively wide range of various hybrid logics but they are also conceptually simple and enable effective search for a minimal model for a satisfiable formula. The main novelty is the exploitation of the unrestricted blocking mechanism introduced as an explicit, sound tableau rule.

Semidefinite resolution and exactness of semidefinite relaxations for satisfiability

This paper explores new connections between the satisfiability problem and semidefinite programming. We show how the process of resolution in satisfiability is equivalent to a linear transformation between the feasible sets of the relevant semidefinite programming problems. We call this transformation semidefinite programming resolution, and we demonstrate the potential of this novel concept by using it to obtain a direct proof of the exactness of the semidefinite formulation of satisfiability without applying Lasserre’s general theory for semidefinite relaxations of 0/1 problems. In particular, our proof explicitly shows how the exactness of the semidefinite formulation for any satisfiability formula can be interpreted as the implicit application of a finite sequence of resolution steps to verify whether the empty clause can be derived from the given formula.

On the concentration of the number of solutions of random satisfiability formulas

ABSTRACTLet Z(F) be the number of solutions of a random k‐satisfiability formula F with n variables and clause density α. Assume that the probability that F is unsatisfiable is for some . We show that (possibly excluding a countable set of “exceptional” α's) the number of solutions concentrates, i.e., there exists a non‐random function such that, for any , we have with high probability. In particular, the assumption holds for all , which proves the above concentration claim in the whole satisfiability regime of random 2‐SAT. We also extend these results to a broad class of constraint satisfaction problems. © 2013 Wiley Periodicals, Inc. Random Struct. Alg., 45, 362–382, 2014

TRANSITIVE PRIMAL INFON LOGIC

AbstractPrimal infon logic was introduced in 2009 in connection with access control. In addition to traditional logic constructs, it contains unary connectives p said indispensable in the intended access control applications. Propositional primal infon logic is decidable in linear time, yet suffices for many common access control scenarios. The most obvious limitation on its expressivity is the failure of the transitivity law for implication: $x \to y$ and $y \to z$ do not necessarily yield $x \to z$. Here we introduce and investigate equiexpressive “transitive” extensions TPIL and TPIL* of propositional primal infon logic as well as their quote-free fragments TPIL0 and TPIL0* respectively. We prove the subformula property for TPIL0* and a similar property for TPIL*; we define Kripke models for the four logics and prove the corresponding soundness-and-completeness theorems; we show that, in all these logics, satisfiable formulas have small models; but our main result is a quadratic-time derivation algorithm for TPIL*.

A Tableau Based Decision Procedure for an Expressive Fragment of Hybrid Logic with Binders, Converse and Global Modalities

In this paper we provide the first (as far as we know) direct calculus deciding satisfiability of formulae in negation normal form in the fragment of FHL (full hybrid logic with the binder, including the global and converse modalities), where no occurrence of a universal operator is in the scope of a binder. By means of a satisfiability preserving translation of formulae, the calculus can be turned into a satisfiability decision procedure for the fragment $\textsf{FHL}\setminus\Box \mathord\downarrow\Box$ , i.e. formulae in negation normal form where no occurrence of the binder is both in the scope of and contains in its scope a universal operator. The calculus is based on tableaux and termination is achieved by means of a form of anywhere blocking with indirect blocking. Direct blocking is a relation between nodes in a tableau branch, holding whenever the respective labels (formulae) are equal up to (a proper form of) nominal renaming. Indirect blocking is based on a partial order on the nodes of a tableau branch, which arranges them into a tree-like structure.

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.

On Expressing and Monitoring Oscillatory Dynamics

To express temporal properties of dense-time real-valued signals, the Signal Temporal Logic (STL) has been defined by Maler et al. The work presented a monitoring algorithm deciding the satisfiability of STL formulae on finite discrete samples of continuous signals. The logic has been used to express and analyse biological systems, but it is not expressive enough to sufficiently distinguish oscillatory properties important in biology. In this paper we define the extended logic STL* in which STL is augmented with a signal-value freezing operator allowing us to express (and distinguish) detailed properties of biological oscillations. The logic is supported by a monitoring algorithm prototyped in Matlab. The monitoring procedure of STL* is evaluated on a biologically-relevant case study.

Deductive temporal reasoning with constraints

When modelling realistic systems, physical constraints on the resources available are often required. For example, we might say that at most N processes can access a particular resource at any moment, exactly M participants are needed for an agreement, or an agent can be in exactly one mode at any moment. Such situations are concisely modelled where literals are constrained such that at most N, or exactly M, can hold at any moment in time. In this paper we consider a logic which is a combination of standard propositional linear time temporal logic with cardinality constraints restricting the numbers of literals that can be satisfied at any moment in time. We present the logic and show how to represent a number of case studies using this logic. We propose a tableau-like algorithm for checking the satisfiability of formulae in this logic, provide details of a prototype implementation and present experimental results using the prover.

Dynamic topological logic of metric spaces

AbstractDynamic Topological Logic is a modal framework for reasoning about dynamical systems, that is, pairs 〈X, f〉 where X is a topological space and f: X → X a continuous function.In this paper we consider the case where X is a metric space. We first show that any formula which can be satisfied on an arbitrary dynamic topological system can be satisfied on one based on a metric space; in fact, this space can be taken to be countable and have no isolated points. Since any metric space with these properties is homeomorphic to the set of rational numbers, it follows that any satisfiable formula can be satisfied on a system based on ℚ.We then show that the situation changes when considering complete metric spaces, by exhibiting a formula which is not valid in general but is valid on the class of systems based on a complete metric space. While we do not attempt to give a full characterization of the set of valid formulas on this class we do give a relative completeness result; any formula which is satisfiable on a dynamical system based on a complete metric space is also satisfied on one based on the Cantor space.

Applying SMT Solvers to the Test Template Framework

The Test Template Framework (TTF) is a model-based testing method for the Z notation. In the TTF, test cases are generated from test specifications, which are predicates written in Z. In turn, the Z notation is based on first-order logic with equality and Zermelo-Fraenkel set theory. In this way, a test case is a witness satisfying a formula in that theory. Satisfiability Modulo Theory (SMT) solvers are software tools that decide the satisfiability of arbitrary formulas in a large number of built-in logical theories and their combination. In this paper, we present the first results of applying two SMT solvers, Yices and CVC3, as the engines to find test cases from TTF's test specifications. In doing so, shallow embeddings of a significant portion of the Z notation into the input languages of Yices and CVC3 are provided, given that they do not directly support Zermelo-Fraenkel set theory as defined in Z. Finally, the results of applying these embeddings to a number of test specifications of eight cases studies are analysed.

A Combinatorial Analysis for the Critical Clause Tree

In Paturi, Pudlak, Saks, and Zane (Proceedings of the 39th Annual IEEE Symposium on Foundations of Computer Science (FOCS1998), pp. 628---637, 1998) proposed a simple randomized algorithm for finding a satisfying assignment of a k-CNF formula. The main lemma of the paper is as follows: Given a satisfiable k-CNF formula that has a d-isolated satisfying assignment z, the randomized algorithm finds z with probability at least $2^{-(1-\mu_{k}/(k-1)+\epsilon_{k}(d))n}$ , where $\mu_{k}/(k-1)=\sum_{i=1}^{\infty}1/(i((k-1)i+1))$ , and ? k (d)=o d (1). They estimated the lower bound of the probability in an analytical way, and used some asymptotics. In this paper, we analyze the same randomized algorithm, and estimate the probability in a combinatorial way. The lower bound we obtain is a little simpler: $2^{-(1-\mu_{k}(d)/(k-1))n}$ , where $\mu_{k}(d)/(k-1)=\sum_{i=1}^{d}1/(i((k-1)i+1))$ . This value is a little bit larger (i.e., better) than that of Paturi et al. (Proceedings of the 39th Annual IEEE Symposium on Foundations of Computer Science (FOCS1998), pp. 628---637, 1998) although the two values are asymptotically equal when d=?(1).

Regular sets over extended tree structures

We investigate notions of decidability and definability for the Monadic Second-Order Logic over labeled tree structures, and its relations with finite automata using oracles to test input prefixes. A general framework is defined allowing to transfer some MSO-properties from a graph-structure to a labeled tree structure. Transferred properties are the decidability of sentences and the existence of a definable model for every satisfiable formula. A class of finite automata with prefix-oracles is also defined, recognizing exactly languages defined by MSO-formulas in any labeled tree-structure. Applying these results, the well-known equivalence between languages recognized by finite automata, sets of vertices MSO definable in a tree-structure and sets of pushdown contexts generated by pushdown-automata is extended to k -iterated pushdown automata.

Optimization hardness as transient chaos in an analog approach to constraint satisfaction

Constraint-satisfaction problems are among the computationally hardest tasks: solutions are efficiently checkable, but no efficient algorithms are known to compute those solutions. Fresh insight might come from physics. A study mapping optimization hardness onto the phenomena of turbulence and chaos suggests that constraint-satisfaction problems can be tackled using analog devices. Boolean satisfiability1 (k-SAT) is one of the most studied optimization problems, as an efficient (that is, polynomial-time) solution to k-SAT (for k≥3) implies efficient solutions to a large number of hard optimization problems2,3. Here we propose a mapping of k-SAT into a deterministic continuous-time dynamical system with a unique correspondence between its attractors and the k-SAT solution clusters. We show that beyond a constraint density threshold, the analog trajectories become transiently chaotic4,5,6,7, and the boundaries between the basins of attraction8 of the solution clusters become fractal7,8,9, signalling the appearance of optimization hardness10. Analytical arguments and simulations indicate that the system always finds solutions for satisfiable formulae even in the frozen regimes of random 3-SAT (ref. 11) and of locked occupation problems12 (considered among the hardest algorithmic benchmarks), a property partly due to the system’s hyperbolic4,13 character. The system finds solutions in polynomial continuous time, however, at the expense of exponential fluctuations in its energy function.

Satisfiability modulo theories

Checking the satisfiability of logical formulas, SMT solvers scale orders of magnitude beyond custom ad hoc solvers.