Parameterized Modal Satisfiability

We investigate the parameterized computational complexity of the satisfiability problem for modal logic and attempt to pinpoint relevant structural parameters which cause the problem??s combinatorial explosion, beyond the number of propositional variables v. To this end we study the modality depth, a natural measure which has appeared in the literature, and show that, even though modal satisfiability parameterized by v and the modality depth is FPT, the running time??s dependence on the parameters is a tower of exponentials (unless P = NP). To overcome this limitation we propose possible alternative parameters, namely diamond dimension and modal width. We show fixed-parameter tractability results using these measures where the exponential dependence on the parameters is much milder (doubly and singly exponential respectively) than in the case of modality depth thus leading to FPT algorithms for modal satisfiability with much more reasonable running times. We also give lower bound arguments which prove that our algorithms cannot be improved significantly unless the Exponential Time Hypothesis fails.     

A polynomial space construction of tree-like models for logics with local chains of modal connectives

The LA-logics (“logics with Local Agreement”) are polymodal logics defined semantically such that at any world of a model, the sets of successors for the different accessibility relations can be linearly ordered and the accessibility relations are equivalence relations. In a previous work, we have shown that every LA-logic defined with a finite set of modal indices has an NP-complete satisfiability problem. In this paper, we introduce a class of LA-logics with a countably infinite set of modal indices and we show that the satisfiability problem is PSPACE-complete for every logic of such a class. The upper bound is shown by exhibiting a tree structure of the models. This allows us to establish a surprising correspondence between the modal depth of formulae and the number of occurrences of distinct modal connectives. More importantly, as a consequence, we can show the PSPACE-completeness of Gargov's logic DALLA and Nakamura's logic LGM restricted to modal indices that are rational numbers, for which the computational complexity characterization has been open until now. These logics are known to belong to the class of information logics and fuzzy modal logics, respectively.     

Decidability by Resolution for Propositional Modal Logics

The paper shows that satisfiability in a range of popular propositional modal systems can be decided by ordinary resolution procedures. This follows from a general result that resolution combined with condensing, and possibly some additional form of normalization, is a decision procedure for the satisfiability problem in certain so-called path logics. Path logics arise from normal propositional modal logics by the optimized functional translation method. The decision result provides an alternative method of proving decidability for modal logics, as well as closely related systems of artificial intelligence. This alone is not interesting. A more far-reaching consequence of the result has practical value, namely, many standard first-order theorem provers that are based on resolution are suitable for facilitating modal reasoning.     

Evaluating Optimized Decision Procedures for Propositional Modal K (m) Satisfiability

Heavily optimized decision procedures for propositional modal satisfiability are now becoming available. Two systems incorporating such procedures for modal K, DLP and KSATC, are tested on randomly generated CNF formulae with several sets of parameters, varying the maximum modal depth and ratio of propositional variable to modal subformulae. The results show some easy-hard-easy behavior, but there is as yet no sharp peak as in propositional satisfiability.     

Planning as satisfiability: parallel plans and algorithms for plan search

We address two aspects of constructing plans efficiently by means of satisfiability testing: efficient encoding of the problem of existence of plans of a given number t of time points in the propositional logic and strategies for finding plans, given these formulae for different values of t.For the first problem we consider three semantics for plans with parallel operator application in order to make the search for plans more efficient. The standard semantics requires that parallel operators are independent and can therefore be executed in any order. We consider a more relaxed definition of parallel plans which was first proposed by Dimopoulos et al., as well as a normal form for parallel plans that requires every operator to be executed as early as possible. We formalize the semantics of parallel plans emerging in this setting and present translations of these semantics into the propositional logic. The sizes of the translations are asymptotically optimal. Each of the semantics is constructed in such a way that there is a plan following the semantics exactly when there is a sequential plan, and moreover, the existence of a parallel plan implies the existence of a sequential plan with as many operators as in the parallel one.For the second problem we consider strategies based on testing the satisfiability of several formulae representing plans of n time steps for several values of n concurrently by several processes. We show that big efficiency gains can be obtained in comparison to the standard strategy of sequentially testing the satisfiability of formulae for an increasing number of time steps.     

Set Constraints with Intersection

Set constraints are inclusions between expressions denoting sets of trees. The efficiency of their satisfiability test is a central issue in set-based program analysis, their main application domain. We introduce the class of set constraints with intersection (the only operators forming the expressions are constructors and intersection) and show that its satisfiability problem is DEXPTIME-complete. The complexity characterization continues to hold for negative set constraints with intersection (which have positive and negated inclusions). We reduce the satisfiability problem for these constraints to one over the interpretation domain of nonempty sets of trees. Set constraints with intersection over the domain of nonempty sets of trees enjoy the fundamental property of independence of negated conjuncts. This allows us to handle each negated inclusion separately by the entailment algorithm that we devise. We furthermore prove that set constraints with intersection are equivalent to the class of definite set constraints and thereby settle the complexity question of the historically first class for which the decidability question was solved.     

Complexity issues in basic logic

We survey complexity results concerning a family of propositional many-valued logics. In particular, we shall address satisfiability and tautologousness problems for Hájek's Basic Logic BL and for several of its schematic extensions. We shall review complexity bounds obtained from functional representation results, as well as techniques for dealing with non-trivial ordinal sums of continuous t-norms.     

Complexity of Regular Language Matching and Other Decidable Cases of the Satisfiability Problem for Constraints Between Regular Open Terms

In this paper we study the satisfiability problem for language equations and constraints between regular open terms. We prove that: the constraint problem is PSPACE-complete, the regular language matching problem is EXPSPACE-complete even if we ask about satisfiability by regular languages, satisfiability of word-like language equations is in PSPACE, the finite solution problem is EXPSPACE-complete.     

The Unique Horn-Satisfiability problem and quadratic Boolean equations

The unique satisfiability problem for general Boolean expressions has attracted interest in recent years in connection with basic complexity issues [12,13]. We investigate here Unique Horn-Satisfiability, i.e. the subclass of Unique-Sat restricted to Horn expressions. We introduce two operators,reduction andshrinking, each transforming a given Horn expression into another Horn expression involving strictly fewer variables and preserving the unique satisfiability property, if present.Uniquely satisfiable Horn expressions are then characterized as those Horn expressions which can be converted into a formula composed of an empty set of clauses on an empty set of free variables through finitely many applications of the shrink-and-reduce operator.Finally, an algorithm for testing whether a given irreducible Horn formula is uniquely satisfiable is described. Data structures for its implementation are discussed, leading toO(mn) complexity for the general case (m=number of clauses,n=number of variables), hence to linear complexity for dense formulae.     

Building Decision Procedures for Modal Logics from Propositional Decision Procedures: The Case Study of Modal K(m)

The goal of this paper is to propose a new technique for developing decision procedures for propositional modal logics. The basic idea is that propositional modal decision procedures should be developed on top of propositional decision procedures. As a case study, we consider satisfiability in modal K(m), that is modal K with m modalities, and develop an algorithm, called K , on top of an implementation of the Davis–Putnam–Longemann–Loveland procedure. K is thoroughly tested and compared with various procedures and in particular with the state-of-the-art tableau-based system K . The experimental results show that K outperforms K and the other systems of orders of magnitude, highlight an intrinsic weakness of tableau-based decision procedures, and provide partial evidence of a phase transition phenomenon for K(m).     

基于模糊命题模态逻辑的形式推理系统

探讨基于可信度的模糊命题模态逻辑的形式推理,给出相关的模糊Kripke语义描述.其研究目的旨在解决基于模态命题逻辑的模糊推理的能行问题.在研究过程与方法上,以完全形式化的方法将模糊模态逻辑语法和语义统一在一个形式系统中,以模糊约束作为基本表达式,给出推理规则,建立了相应的模糊推理形式系统,并以形式系统中模糊约束集的可满足性来表示模糊推理的有效性,使模糊推理过程变得容易,为最终在计算机上实现基于模态逻辑的模糊推理打下了一定的基础.主要结论是证明了基于可满足性的模糊推理形式系统的可靠性与完备性.     

Quantified epistemic logics for reasoning about knowledge in multi-agent systems

We introduce quantified interpreted systems, a semantics to reason about knowledge in multi-agent systems in a first-order setting. Quantified interpreted systems may be used to interpret a variety of first-order modal epistemic languages with global and local terms, quantifiers, and individual and distributed knowledge operators for the agents in the system. We define first-order modal axiomatisations for different settings, and show that they are sound and complete with respect to the corresponding semantical classes.The expressibility potential of the formalism is explored by analysing two MAS scenarios: an infinite version of the muddy children problem, a typical epistemic puzzle, and a version of the battlefield game. Furthermore, we apply the theoretical results here presented to the analysis of message passing systems [R. Fagin, J. Halpern, Y. Moses, M. Vardi, Reasoning about Knowledge, MIT Press, 1995; L. Lamport, Time, clocks, and the ordering of events in a distributed system, Communication of the ACM 21 (7) (1978) 558–565], and compare the results obtained to their propositional counterparts. By doing so we find that key known meta-theorems of the propositional case can be expressed as validities on the corresponding class of quantified interpreted systems.     

Is computational complexity a barrier to manipulation?

When agents are acting together, they may need a simple mechanism to decide on joint actions. One possibility is to have the agents express their preferences in the form of a ballot and use a voting rule to decide the winning action(s). Unfortunately, agents may try to manipulate such an election by mis-reporting their preferences. Fortunately, it has been shown that it is NP-hard to compute how to manipulate a number of different voting rules. However, NP-hardness only bounds the worst-case complexity. In this survey article, we summarize the evidence for and against computational complexity being a barrier to manipulation. We look both at techniques identified to increase complexity (for example, hybridizing together two or more voting rules), as well as other features that may change the computational complexity of computing a manipulation (for example, if votes are restricted to be single peaked then some of the complexity barriers fall away). We discuss recent theoretical results that consider the average case, as well as simple greedy and approximate methods. We also describe how computational ??phase transitions??, which have been fruitful in identifying hard instances of propositional satisfiability and other NP-hard problems, have provided insight into the hardness of manipulating voting rules in practice. Finally, we consider manipulation of other related problems like stable marriage and tournament problems.     

Propositional semantics for disjunctive logic programs

In this paper we study the properties of the class of head-cycle-free extended disjunctive logic programs (HEDLPs), which includes, as a special case, all nondisjunctive extended logic programs. We show that any propositional HEDLP can be mapped in polynomial time into a propositional theory such that each model of the latter corresponds to an answer set, as defined by stable model semantics, of the former. Using this mapping, we show that many queries over HEDLPs can be determined by solving propositional satisfiability problems. Our mapping has several important implications: It establishes the NP-completeness of this class of disjunctive logic programs; it allows existing algorithms and tractable subsets for the satisfiability problem to be used in logic programming; it facilitates evaluation of the expressive power of disjunctive logic programs; and it leads to the discovery of useful similarities between stable model semantics and Clark's predicate completion.     

命题线性时序逻辑的对偶模型问题的复杂性

定义了一个命题线性时序逻辑的对偶模型的概念.一个公式f的对偶模型是指f的满足以下条件的两个模型(即状态的w序列):在每个位置上这两个模型对原子命题的赋值都是对偶的.然后,对于确定一个公式f是否有对偶模型的判定问题(记为DM)和在一个Kripke-结构中确定是否存在从两个给定状态出发的对偶模型满足给定公式f的判定问题(记为KDM)的复杂性进行了研究.证明了以下结果:对于只含有F("Future")算子的命题线性时序逻辑,DM和KDM都是NP完全的;而对于以下命题线性时序逻辑,DM和KDM都是PSPACE完全的:含有F,X ("Next")算子的逻辑、含有U("Until")算子的逻辑、含有U,S,X算子的逻辑以及由Wolper给出的含有正规语言算子的逻辑(一般称为扩展时序逻辑,简称ETL).     

Implementing the Davis–Putnam Method

The method proposed by Davis, Putnam, Logemann, and Loveland for propositional reasoning, often referred to as the Davis–Putnam method, is one of the major practical methods for the satisfiability (SAT) problem of propositional logic. We show how to implement the Davis–Putnam method efficiently using the trie data structure for propositional clauses. A new technique of indexing only the first and last literals of clauses yields a unit propagation procedure whose complexity is sublinear to the number of occurrences of the variable in the input. We also show that the Davis–Putnam method can work better when unit subsumption is not used. We illustrate the performance of our programs on some quasigroup problems. The efficiency of our programs has enabled us to solve some open quasigroup problems.     

On the computational complexity of satisfiability in propositional logics of programs

The satisfiability problems of propositional algorithmic logic and propositional dynamic logic are shown to be complete in the classes of languages accepted in polynomial space by the deterministic and alternating Turing machines respectively. Explicit upper and lower bounds on the space complexity are calculated. Exponential lower bounds on the space complexity of the satisfiability problems of these logics extended by adding a certain program connective are proved.     

What makes propositional abduction tractable

Abduction is a fundamental form of nonmonotonic reasoning that aims at finding explanations for observed manifestations. This process underlies many applications, from car configuration to medical diagnosis. We study here the computational complexity of deciding whether an explanation exists in the case when the application domain is described by a propositional knowledge base. Building on previous results, we classify the complexity for local restrictions on the knowledge base and under various restrictions on hypotheses and manifestations. In comparison to the many previous studies on the complexity of abduction we are able to give a much more detailed picture for the complexity of the basic problem of deciding the existence of an explanation. It turns out that depending on the restrictions, the problem in this framework is always polynomial-time solvable, NP-complete, coNP-complete, or -complete.

Based on these results, we give an a posteriori justification of what makes propositional abduction hard even for some classes of knowledge bases which allow for efficient satisfiability testing and deduction. This justification is very simple and intuitive, but it reveals that no nontrivial class of abduction problems is tractable. Indeed, tractability essentially requires that the language for knowledge bases is unable to express both causal links and conflicts between hypotheses. This generalizes a similar observation by Bylander et al. for set-covering abduction.