Symmetries in Modal Logics

Consider modal propositional formulae, constructed using proposition-letters, connectives and the modal operators □ and ⋄. The semantic structures are frames, i.e., pairs <W, R> with R ⊆ W2. Let F, V be variables ranging respectively over frames and functions from the set of proposition-letters into the powerset of W. Then the relationmay be defined, for arbitrary formulae α, following the Kripke truth-definition. From this relation we may further defineNow, to every modal formula α there corresponds some property Pα of R. A particular example is obtained by considering the well-known translation of modal formulae into formulae of monadic second-order logic with a single binary first-order predicate. For these particular Pα we havefor all F and w ∈ W. These formulae Pα are, however, rather intractable and more convenient ones can often be found. An especially interesting case occurs when Pα may be taken to be some first-order formula. For example, it can be seen thatfor all F and w ∈ W. It is customary to talk about a related correspondence, namely when for all F we haveNote that this correspondence holds whenever the first one above holds.

There are not many global results on modal logics. One of these is the following theorem by Sahlqvist on completeness and correspondence for a wide class of modal formulae (including many well known logics, e.g. D, T, B, S4, K4, S5, …) (see [S]).Sahlqvist's Theorem. Let A be any modal formula which is equivalent to a conjunction of formulae of the form □m(A1 → A2), where m ≥ 0, A2 is positive and A1 is obtained from propositional variables and their negations applying ∧, ∨, ♢, and □ in such a way that no positive occurrence of a variable is in a subformula of the form B1 ∨ B2 or ♢ B1 within the scope of some □. Then A corresponds effectively to a first order formula, and L + A is canonical whenever Lis a canonical logic.A formula A satisfying the above conditions is henceforth called a Sahlqvist formula. Unfortunately, till now, the only complete proof was the original proof of Sahlqvist (a proof of the correspondence half has also been given by van Benthem [vB]). It is so complicated and long that even in an advanced textbook of modal logic [HC] it has not found a place. Here, by considering general frames as topological spaces, an attitude which we developed in [TD], we give a proof of Sahlqvist's theorem simplified to such an extent that one can easily grasp the key idea on which it is based and apply the resulting algorithm to specific modal formulae in a straightforward manner, suitable even for implementation on a personal computer. This key idea also improves on previous preliminary work in the same direction (see [S1], [S2]).

In many practical settings it is useful to find a small unsatisfiable subset of a given unsatisfiable set of constraints. We study this problem from a parameterized complexity perspective, taking the size of the unsatisfiable subset as the natural parameter where the set of constraints is either (i) given a set of clauses, i.e., a formula in conjunctive normal Form (CNF), or (ii) as an instance of the Constraint Satisfaction Problem (CSP). In general, the problem is fixed-parameter in tractable. For an instance of the propositional satisfiability problem (SAT), it was known to be W[1]-complete. We establish A[2]-completeness for CSP instances, where A[2]-hardness prevails already for the Boolean case. With these fixed-parameter intractability results for the general case in mind, we consider various restricted classes of inputs and draw a detailed complexity landscape. It turns out that often Boolean CSP and CNF formulas behave similarly, but we also identify notable exceptions to this rule. The main part of this article is dedicated to classes of inputs that are induced by Boolean constraint languages that Schaefer [1978] identified as the maximal constraint languages with a tractable satisfiability problem. We show that for the CSP setting, the problem of finding small unsatisfiable subsets remains fixed-parameter intractable for all Schaefer languages for which the problem is non-trivial. We show that this is also the case for CNF formulas with the exception of the class of bijunctive (Krom) formulas, which allows for an identification of a small unsatisfiable subset in polynomial time. In addition, we consider various restricted classes of inputs with bounds on the maximum number of times that a variable occurs (the degree), bounds on the arity of constraints, and bounds on the domain size. For the case of CNF formulas, we show that restricting the degree is enough to obtain fixed-parameter tractability, whereas for the case of CSP instances, one needs to restrict the degree, the arity, and the domain size simultaneously to establish fixed-parameter tractability. Finally, we relate the problem of finding small unsatisfiable subsets of a set of constraints to the problem of identifying whether a given variable-value assignment is entailed or forbidden already by a small subset of constraints. Moreover, we use the connection between the two problems to establish similar parameterized complexity results also for the latter problem.

In the maximum satisfiability problem (max-sat) we are given a propositional formula in conjunctive normal form and have to find an assignment that satisfies as many clauses as possible. We study the parallel parameterized complexity of various versions of max-sat and provide the first constant-time algorithms parameterized either by the solution size or by the allowed excess relative to some guarantee. For the dual parameterized version where the parameter is the number of clauses we are allowed to leave unsatisfied, we present the first parallel algorithm for max-2sat (known as almost-2sat). The difficulty in solving almost-2sat in parallel comes from the fact that the iterative compression method, originally developed to prove that the problem is fixed-parameter tractable at all, is inherently sequential. We observe that a graph flow whose value is a parameter can be computed in parallel and develop a parallel algorithm for the vertex cover problem parameterized above the size of a given matching. Finally, we study the parallel complexity of max-sat parameterized by the vertex cover number, the treedepth, the feedback vertex set number, and the treewidth of the input’s incidence graph. While max-sat is fixedparameter tractable for all of these parameters, we show that they allow different degrees of possible parallelization. For all four we develop dedicated parallel algorithms that are constructive, meaning that they output an optimal assignment – in contrast to results that can be obtained by parallel meta-theorems, which often only solve the decision version.

Boolean satisfiability (SAT) and its extensions are becoming a core technology for the analysis of systems. The SAT-based approach divides into three steps: encoding, preprocessing, and search. It is often argued that by encoding arbitrary Boolean formulas in conjunctive normal form (CNF), structural properties of the original problem are not reflected in the CNF. This should result in the fact that CNF-level preprocessing and SAT solver techniques have an inherent disadvantagecompared to related techniques applicable on the level of more structural SAT instance representations such as Boolean circuits. In this work we study the effect of a CNF-level simplification technique called blocked clause elimination (BCE). We show that BCE is surprisingly effective both in theory and in practice on CNFs resulting from a standard CNF encoding for circuits: without explicit knowledge of the underlying circuit structure, it achieves the same level of simplification as a combination of circuit-level simplifications and previously suggested polarity-based CNF encodings. Experimentally, we show that by applying BCE in preprocessing, further formula reduction and faster solving can be achieved, giving promise for applying BCE to speed up solvers.

We presentAbstract CNF2dDNNF, a calculus describing an approach for compiling a formula in conjunctive normal form (CNF) into deterministic negation normal form (d-DNNF). It combines component-based reasoning with a model enumeration approach based on conflict-driven clause learning (CDCL) with chronological backtracking. Its properties, such as soundness and termination, carry over to implementations which can be modeled by it. We provide a correctness proof and a detailed example. The main conceptual differences to currently available tools targeting d-DNNF compilation are discussed and future research directions presented. The aim of this work is to lay the theoretical foundation for a novel method for d-DNNF compilation. To the best of our knowledge, our approach is the first knowledge compilation method using CDCL with chronological backtracking.

In this paper, we discuss the computational complexity of Read-once resolution (ROR) with respect to Horn formulas. Recall that a Horn formula is a boolean formula in conjunctive normal form (CNF), such that each clause has at most one positive literal. Horn formulas find applications in a number of domains such as program verification and logic programming. It is well-known that deduction in ProLog is based on unification, which in turn is based on resolution and instantiation. Resolution is a sound and complete procedure to check whether a boolean formula in CNF is satisfiable. Although inefficient in general, resolution has been used widely in theorem provers, on account of its simplicity and ease of implementation. This paper focuses on two variants of resolution, viz., Read-once resolution and Unit Read-once resolution (UROR). Both these variants are sound, but incomplete. In this paper, the goal is to check for the existence of proofs (refutations) of Horn formulas under these variants. We also discuss the computational complexity of determining optimal length proofs where appropriate.

Minimally Unsatisfiable Subformulas (MUS) find a wide range of practical applications, including product configuration, knowledge-based validation, and hardware and software design and verification. MUSes also find application in recent Maximum Satisfiability algorithms and in CNF formula redundancy removal. Besides direct applications in Propositional Logic, algorithms for MUS extraction have been applied to more expressive logics. This paper proposes two algorithms for computation of MUSes of propositional formulas in Conjunctive Normal Form (CNF). The first algorithm is optimal in its class, meaning that it requires the smallest number of calls to a SAT solver. The second algorithm extends earlier work, but implements a number of new techniques. Among these, this paper analyzes in detail the technique of recursive model rotation, which provides significant performance gains in practice. Experimental results, obtained on representative practical benchmarks, indicate that the new algorithms achieve significant performance gains with respect to state of the art MUS extraction algorithms.

The SAT-Problem fot boolean formulas in conjunctive normal form often arises in the area of artificial intelligence, its solution is known as mechanical theorem proving. Most mechanical theorem provers perform this by using a variant of the resolution principle. The time and space complexity of resolution strongly depends on the class of formulas. Horn formulas, where each clause may contain at most one positive literal, represent the most important class where resolution proofs with polynomial length exist.

There are a number of board games such as Checkers [2], Go [5], and Gobang [8], which are known to be PSPACE-hard. This means that the problem to determine the player having a winning strategy in a given situation on an nA—n board of one of these games is as hard to solve as everything computable in polynomial space. PSPACE-completeness has been previously proven for some combinatorial games played on graphs or by logical formulas [1, 9]. In this paper we will show that the same holds for the game of Hex. The crucial point of the proof is to establish PSPACE-hardness for a generalization of Hex played on planar graphs. This will be done by showing that the problem, whether a given quantified Boolean formula in conjunctive normal form is true, is polynomial time-reducible to the decision problem for generalized Hex. In order to do this we will use methods, which were also used to prove PSPACE-completeness of planar Geography in [5]. Therefore our proof is quite different from the proof provided by Even and Tarjan [1], who showed PSPACE-completeness of generalized Hex played on arbitrary graphs. Since it is easy to see that the decision problem for Hex is in PS PACE, the decision problem for Hex is PSPACE-complete.

SAT-based ATPG turned out to be a robust alternative to classical structural ATPG algorithms such as FAN. The number of unclassified faults can be significantly reduced using a SAT-based ATPG approach. In contrast to structural ATPG, SAT solvers work on a Boolean formula in conjunctive normal form (CNF). This results in some disadvantages for SAT solvers when applied to ATPG, e.g. CNF transformation time and loss of structural knowledge. As a result, SAT-based ATPG algorithms are very robust for hard-to-test faults, but suffer from the overhead for easy-to-test faults. We propose the SAT technique dynamic clause activation (DCA) in order to reduce the run time gap between structural and SAT-based ATPG algorithms and, at the same time, retain the high level of robustness. Using DCA, the SAT solver works on a partial formula of a logic circuit which is dynamically extended during the search process using structural knowledge. Furthermore, efficient dynamic learning techniques can be easily integrated within the proposed technique. The approach is evaluated on large industrial circuits.

We consider boolean formulas in conjunctive normal form (CNF). If all clauses are large, it needs many clauses to obtain an unsatisfiable formula; moreover, these clauses have to interleave. We review quantitative results for the amount of interleaving required, many of which rely on the Lovász Local Lemma, a probabilistic lemma with many applications in combinatorics.In positive terms, we are interested in simple combinatorial conditions which guarantee for a CNF formula to be satisfiable. The criteria obtained are nontrivial in the sense that even though they are easy to check, it is by far not obvious how to compute a satisfying assignment efficiently in case the conditions are fulfilled; until recently, it was not known how to do so. It is also remarkable that while deciding satisfiability is trivial for formulas that satisfy the conditions, a slightest relaxation of the conditions leads us into the territory of NP-completeness.Several open problems remain, some of which we mention in the concluding section.

An analytic criterion for CSAT

Subtractive reductions and complete problems for counting complexity classes

Functional unification (FU) grammar is a general linguistic formalism based on the merging of feature-sets. An informal outline is given of how the definition of derivation within FU grammar can be used to represent the satisfiability of an arbitrary logical formula in conjunctive normal form. This suggests that the generation of a structure from an arbitrary arbitrary FU grammar is NP-hard, which is an undesirably high level of computational complexity.