The standard reasoning problem, concept satisfiability, in the basic description logic ALC is PSPACE-complete, and it is EXPTIME-complete in the presence of unrestricted axioms. Several fragments of ALC, notably logics in the FL, EL, and DL-Lite family, have an easier satisfiability problem; sometimes it is even tractable. All these fragments restrict the use of Boolean operators in one way or another. We look at systematic and more general restrictions of the Boolean operators and establish the complexity of the concept satisfiability problem in the presence of axioms. We separate tractable from intractable cases.
The standard reasoning problem, concept satisfiability, in the basic description logic ALC is PSPACE-complete, and it is EXPTIME-complete in the presence of unrestricted axioms. Several fragments of ALC, notably logics in the FL, EL, and DL-Lite families, have an easier satisfiability problem; sometimes it is even tractable. We classify the complexity of the standard satisfiability problems for all possible Boolean and quantifier fragments of ALC in the presence of general axioms.
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, 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 connecti…
We introduce a description logic ALCQIO_{b,Re} which adds reachability assertions to ALCQIO, a sub-logic of the two-variable fragment of first order logic with counting quantifiers. ALCQIO_{b,Re} is well-suited for applications in software verification and shape analysis. Shape analysis requires expressive logics which can express reachability and have good computational properties. We show that ALCQIO_{b,Re} can describe complex data structures with a high degree of sharing and allows compositions such as list of trees. We show that the finite satisfiability and implication problems of ALCQIO…
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, 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 connecti…
In 1978, Schaefer proved his famous dichotomy theorem for generalized satisfiability problems. He defined an infinite number of propositional satisfiability problems (nowadays usually called Boolean constraint satisfaction problems) and showed that all these satisfiability problems are either in P or NP-complete. In recent years, similar results have been obtained for quite a few other problems for Boolean constraints.Almost all of these problems are variations of the satisfiability problem. In this paper, we address a problem that is not a variation of satisfiability, namely, the isomorphism …
Stochastic Boolean Satisfiability (SSAT) is a logical formalism to model decision problems with uncertainty, such as Partially Observable Markov Decision Process (POMDP) for verification of probabilistic systems. SSAT, however, is limited by its descriptive power within the PSPACE complexity class. More complex problems, such as the NEXPTIME-complete Decentralized POMDP (Dec-POMDP), cannot be succinctly encoded with SSAT. To provide a logical formalism of such problems, we extend the Dependency Quantified Boolean Formula (DQBF), a representative problem in the NEXPTIME-complete class, to its s…
This paper depicts an algorithm for solving the Decision Boolean Satisfiability Problem using the binary numerical properties of a Special Decision Satisfiability Problem, parallel execution, object oriented, and short termination. The two operations: expansion and simplification are used to explains why using algebra grows the resolution steps. It is proved that its complexity has an upper bound of $2^{n-1}$ where $n$ is the number of logical variables of the given problem.
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, Gopalan et al. in 2006 studied connectivity properties of the solution graph and related complexity issues for constraint satisfaction problems in Schaefer's framework. They found dichotomies for the diameter of connected components and for the complexity of the st-connectivity question, and conjectured a…
We study the computational complexity of model checking and satisfiability problems of polyadic modal logics extended with permutations and Boolean operators on accessibility relations. First, we show that the combined complexity of the model checking problem for the resulting logic is PTime-complete. Secondly, we show that the satisfiability problem of polyadic modal logic extended with negation on accessibility relations is ExpTime-complete. Finally, we show that the satisfiability problem of polyadic modal logic with permutations and Boolean operators on accessibility relations is ExpTime-c…
In the last two decades, modal and description logics have been applied to numerous areas of computer science, including knowledge representation, formal verification, database theory, distributed computing and, more recently, semantic web and ontologies. For this reason, the problem of automated reasoning in modal and description logics has been thoroughly investigated. In particular, many approaches have been proposed for efficiently handling the satisfiability of the core normal modal logic K(m), and of its notational variant, the description logic ALC. Although simple in structure, K(m)/AL…
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. For this implicitly defined graph, we here study the st-connectivity and connectivity problems. Building on the work of Gopalan et al. ("The Connectivity of Boolean Satisfiability: Computational and Structural Dichotomies", 2006/2009), we first investigate satisfiability problems given by CSPs, more exactly CNF(S)-formulas with constants (as considered in Schaefer's famous …
Boolean satisfiability ({\SAT}) has played a key role in diverse areas spanning testing, formal verification, planning, optimization, inferencing and the like. Apart from the classical problem of checking boolean satisfiability, the problems of generating satisfying uniformly at random, and of counting the total number of satisfying assignments have also attracted significant theoretical and practical interest over the years. Prior work offered heuristic approaches with very weak or no guarantee of performance, and theoretical approaches with proven guarantees, but poor performance in practice…
In this report, we describe three encodings of the multiple constant multiplication (MCM) problem to pseudo-boolean satisfiability (PBS), and introduce an algorithm to solve the MCM problem optimally. To the best of our knowledge, the proposed encodings and the optimization algorithm are the first formalization of the MCM problem in a PBS manner. This report evaluates the complexity of the problem size and the performance of several PBS solvers over three encodings.
The stochastic Boolean satisfiability (SSAT) problem has been introduced by Papadimitriou in 1985 when adding a probabilistic model of uncertainty to propositional satisfiability through randomized quantification. SSAT has many applications, among them probabilistic bounded model checking (PBMC) of symbolically represented Markov decision processes. This article identifies a notion of Craig interpolant for the SSAT framework and develops an algorithm for computing such interpolants based on a resolution calculus for SSAT. As a potential application area of this novel concept of Craig interpola…
The subsumption problem with respect to terminologies in the description logic ALC is EXPTIME-complete. We investigate the computational complexity of fragments of this problem by means of allowed Boolean operators. Hereto we make use of the notion of clones in the context of Post's lattice. Furthermore we consider all four possible quantifier combinations for each fragment parameterized by a clone. We will see that depending on what quantifiers are available the classification will be either tripartite or a quartering.
This paper develops a parallel computational solver for computing all satifying assignments of a Boolean system of equations defined by Boolean functions of several variables. While there are we known solvers for satisfiability of Boolean formulas in CNF form, these are designed primarily for deciding satisfiability of the formula and do not address the problem of finding all satisfying solutions. Moreover development of parallel solvers for satisfiability problems is still an unfinished problem of Computer Science. The solver proposed in this paper is aimed at representing all solutions of Bo…
The paper explores the correspondence between balanced incomplete block designs (BIBD) and certain linear CNF formulas by identifying the points of a block design with the clauses of the Boolean formula and blocks with Boolean variables. Parallel classes in BIBDs correspond to XSAT solutions in the corresponding formula. This correspondence allows for transfers of results from one field to the other. As a new result we deduce from known satisfiability theorems that the problem of finding a parallel class in a partially balanced incomplete block design is NP-complete.
We consider a variant of the Boolean satisfiability problem where a subset E of the propositional variables appearing in formula Fsat encode a symmetric, transitive, binary relation over N elements. Each of these relational variables, e[i,j], for 1 <= i < j <= N, expresses whether or not the relation holds between elements i and j. The task is to either find a satisfying assignment to Fsat that also satisfies all transitivity constraints over the relational variables (e.g., e[1,2] & e[2,3] ==> e[1,3]), or to prove that no such assignment exists. Solving this satisfiability problem is the final…
Satisfiability checking for Linear Temporal Logic (LTL) is a fundamental step in checking for possible errors in LTL assertions. Extant LTL satisfiability checkers use a variety of different search procedures. With the sole exception of LTL satisfiability checking based on bounded model checking, which does not provide a complete decision procedure, LTL satisfiability checkers have not taken advantage of the remarkable progress over the past 20 years in Boolean satisfiability solving. In this paper, we propose a new LTL satisfiability-checking framework that is accelerated using a Boolean SAT …
We give the first cut-free ExpTime (optimal) tableau decision procedure for checking satisfiability of a knowledge base in the description logic SHI, which extends the description logic ALC with transitive roles, inverse roles and role hierarchies.
We propose a version of WalkSAT algorithm, named as BetaWalkSAT. This method uses probabilistic reasoning for biasing the starting state of the local search algorithm. Beta distribution is used to model the belief over boolean values of the literals. Our results suggest that, the proposed BetaWalkSAT algorithm can outperform other uninformed local search approaches for complex boolean satisfiability problems.
Satisfiability solvers are increasingly playing a key role in software verification, with particularly effective use in the analysis of security vulnerabilities. String processing is a key part of many software applications, such as browsers and web servers. These applications are susceptible to attacks through malicious data received over network. Automated tools for analyzing the security of such applications, thus need to reason about strings. For efficiency reasons, it is desirable to have a solver that treats strings as first-class types. In this paper, we present some theories of strings…
In 1978, Schaefer proved his famous dichotomy theorem for generalized satisfiability problems. He defined an infinite number of propositional satisfiability problems, showed that all these problems are either in P or NP-complete, and gave a simple criterion to determine which of the two cases holds. This result is surprising in light of Ladner's theorem, which implies that there are an infinite number of complexity classes between P and NP-complete (under the assumption that P is not equal to NP). Schaefer also stated a dichotomy theorem for quantified generalized Boolean formulas, but this th…
Technische Universität Dresden（德累斯顿工业大学）；University of Wrocław（弗罗茨瓦夫大学）
In logic-based knowledge representation, query answering has essentially replaced mere satisfiability checking as the inferencing problem of primary interest. For knowledge bases in the basic description logic ALC, the computational complexity of conjunctive query (CQ) answering is well known to be ExpTime-complete and hence not harder than satisfiability. This does not change when the logic is extended by certain features (such as counting or role hierarchies), whereas adding others (inverses, nominals or transitivity together with role-hierarchies) turns CQ answering exponentially harder. We…
TU Dresden（德累斯顿工业大学）；University of Wrocław（弗罗茨瓦夫大学）
We introduce and investigate the expressive description logic (DL) ALCSCC++, in which the global and local cardinality constraints introduced in previous papers can be mixed. On the one hand, we prove that this does not increase the complexity of satisfiability checking and other standard inference problems. On the other hand, the satisfiability problem becomes undecidable if inverse roles are added to the languages. In addition, even without inverse roles, conjunctive query entailment in this DL turns out to be undecidable. We prove that decidability of querying can be regained if global and …
Schaefer introduced a framework for generalized satisfiability problems on the Boolean domain and characterized the computational complexity of such problems. We investigate an algebraization of Schaefer's framework in which the Fourier transform is used to represent constraints by multilinear polynomials in a unique way. The polynomial representation of constraints gives rise to a relaxation of the notion of satisfiability in which the values to variables are linear operators on some Hilbert space. For the case of constraints given by a system of linear equations over the two-element field, t…
Given a set U of alternatives, a choice (correspondence) on U is a contractive map c defined on a family Omega of nonempty subsets of U. Semantically, a choice c associates to each menu A in Omega a nonempty subset c(A) of A comprising all elements of A that are deemed selectable by an agent. A choice on U is total if its domain is the powerset of U minus the empty set, and partial otherwise. According to the theory of revealed preferences, a choice is rationalizable if it can be retrieved from a binary relation on U by taking all maximal elements of each menu. It is well-known that rationaliz…
It is well known that modal satisfiability is PSPACE-complete (Ladner 1977). However, the complexity may decrease if we restrict the set of propositional operators used. Note that there exist an infinite number of propositional operators, since a propositional operator is simply a Boolean function. We completely classify the complexity of modal satisfiability for every finite set of propositional operators, i.e., in contrast to previous work, we classify an infinite number of problems. We show that, depending on the set of propositional operators, modal satisfiability is PSPACE-complete, coNP-…
Boolean satisfiability (SAT) has an extensive application domain in computer science, especially in electronic design automation applications. Circuit synthesis, optimization, and verification problems can be solved by transforming original problems to SAT problems. However, the SAT problem is known as NP-complete, which means there is no efficient method to solve it. Therefore, an efficient SAT solver to enhance the performance is always desired. We propose a hardware acceleration method for SAT problems. By surveying the properties of SAT problems and the decoding of low-density parity-check…
United Technologies Research Center（联合技术研究中心）；SRI International（SRI国际）
Given a Boolean formula $ϕ(x)$ in conjunctive normal form (CNF), the density of states counts the number of variable assignments that violate exactly $e$ clauses, for all values of $e$. Thus, the density of states is a histogram of the number of unsatisfied clauses over all possible assignments. This computation generalizes both maximum-satisfiability (MAX-SAT) and model counting problems and not only provides insight into the entire solution space, but also yields a measure for the \emph{hardness} of the problem instance. Consequently, in real-world scenarios, this problem is typically infeas…
The aim of this work is to provide a family of qualitative theories for spatial change in general, and for motion of spatial scenes in particular. To achieve this, we consider a spatio-temporalisation MTALC(D_x), of the well-known ALC(D) family of Description Logics (DLs) with a concrete domainan. In particular, the concrete domain D_x is generated by a qualitative spatial Relation Algebra (RA) x. We show the important result that satisfiability of an MTALC(D_x) concept with respect to a weakly cyclic TBox is decidable in nondeterministic exponential time, by reducing it to the emptiness probl…
Cairo University（开罗大学）；Faculty of Science（理学院）；Department of Mathematics（数学系）
In this paper, we extend the Maximum Satisfiability (MaxSAT) problem to Łukasiewicz logic. The MaxSAT problem for a set of formulae Φ is the problem of finding an assignment to the variables in Φ that satisfies the maximum number of formulae. Three possible solutions (encodings) are proposed to the new problem: (1) Disjunctive Linear Relations (DLRs), (2) Mixed Integer Linear Programming (MILP) and (3) Weighted Constraint Satisfaction Problem (WCSP). Like its Boolean counterpart, the extended fuzzy MaxSAT will have numerous applications in optimization problems that involve vagueness.
Université Sorbonne Paris Nord（索邦巴黎北大学）；INSERM（法国国家健康与医学研究院）
We present in this paper a reformulation of the usual set-theoretical semantics of the description logic $\mathcal{ALC}$ with general TBoxes by using categorical language. In this setting, $\mathcal{ALC}$ concepts are represented as objects, concept subsumptions as arrows, and memberships as logical quantifiers over objects and arrows of categories. Such a category-based semantics provides a more modular representation of the semantics of $\mathcal{ALC}$. This feature allows us to define a sublogic of $\mathcal{ALC}$ by dropping the interaction between existential and universal restrictions, w…
We explore the relationships between Description Logics and Set Theory. The study is carried on using, on the set-theoretic side, a very rudimentary axiomatic set theory Omega, consisting of only four axioms characterizing binary union, set difference, inclusion, and the power-set. An extension of ALC, ALC^Omega, is then defined in which concepts are naturally interpreted as sets living in Omega-models. In ALC^Omega not only membership between concepts is allowed---even admitting circularity---but also the power-set construct is exploited to add metamodeling capabilities. We investigate transl…
We consider here Linear Temporal Logic (LTL) formulas interpreted over \emph{finite} traces. We denote this logic by LTLf. The existing approach for LTLf satisfiability checking is based on a reduction to standard LTL satisfiability checking. We describe here a novel direct approach to LTLf satisfiability checking, where we take advantage of the difference in the semantics between LTL and LTLf. While LTL satisfiability checking requires finding a \emph{fair cycle} in an appropriate transition system, here we need to search only for a finite trace. This enables us to introduce specialized heuri…
University of Southampton（南安普顿大学）；Technical University of Lisbon（里斯本理工大学）；INESC-ID（葡萄牙国家系统与计算机工程研究所）
Symmetries are intrinsic to many combinatorial problems including Boolean Satisfiability (SAT) and Constraint Programming (CP). In SAT, the identification of symmetry breaking predicates (SBPs) is a well-known, often effective, technique for solving hard problems. The identification of SBPs in SAT has been the subject of significant improvements in recent years, resulting in more compact SBPs and more effective algorithms. The identification of SBPs has also been applied to pseudo-Boolean (PB) constraints, showing that symmetry breaking can also be an effective technique for PB constraints. Th…
On the Semantic Web, metadata and ontologies are used to enable computers to read data. The Web Ontology Language (OWL) has been proposed as a standard ontological language, and various inference systems for this language have been studied. Description logics are regarded as the theoretical foundations of OWL; they provide the syntax and semantics of a formal language for describing ontologies and knowledge bases. In addition, tableau algorithms for description logics have been developed as the standard reasoning algorithms for decidable problems. However, tableau algorithms generate inefficie…
Recently there have been some unexpected results concerning Fuzzy Description Logics (FDLs) with General Concept Inclusions (GCIs). They show that, unlike the classical case, the DL ALC with GCIs does not have the finite model property under Lukasiewicz Logic or Product Logic and, specifically, knowledge base satisfiability is an undecidable problem for Product Logic. We complete here the analysis by showing that knowledge base satisfiability is also an undecidable problem for Lukasiewicz Logic.
Given the family $P$ of all nonempty subsets of a set $U$ of alternatives, a choice over $U$ is a function $c \colon Ω\to P$ such that $Ω\subseteq P$ and $c(B) \subseteq B$ for all menus $B \in Ω$. A choice is total if $Ω= P$, and partial otherwise. In economics, an agent is considered rational whenever her choice behavior satisfies suitable axioms of consistency, which are properties quantified over menus. Here we address the following lifting problem: Given a partial choice satisfying one or more axioms of consistency, is it possible to extend it to a total choice satisfying the same axioms?…
In a seminal paper from 1985, Sistla and Clarke showed that satisfiability for Linear Temporal Logic (LTL) is either NP-complete or PSPACE-complete, depending on the set of temporal operators used. If, in contrast, the set of propositional operators is restricted, the complexity may decrease. This paper undertakes a systematic study of satisfiability for LTL formulae over restricted sets of propositional and temporal operators. Since every propositional operator corresponds to a Boolean function, there exist infinitely many propositional operators. In order to systematically cover all possible…
University of Catania（卡塔尼亚大学）；Scuola Superiore di Catania（卡塔尼亚高等研究院）；Vrije Universiteit Amsterdam（阿姆斯特丹自由大学）；University of Trieste（的里雅斯特大学）
As a contribution to quantitative set-theoretic inferencing, a translation is proposed of conjunctions of literals of the forms $x=y\setminus z$, $x \neq y\setminus z$, and $z =\{x\}$, where $x,y,z$ stand for variables ranging over the von Neumann universe of sets, into unquantified Boolean formulae of a rather simple conjunctive normal form. The formulae in the target language involve variables ranging over a Boolean ring of sets, along with a difference operator and relators designating equality, non-disjointness and inclusion. Moreover, the result of each translation is a conjunction of lit…
