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Weak Kripke structures and LTL
- IN: PROCEEDINGS OF THE 22ND INTERNATIONAL CONFERENCE ON CONCURRENCY THEORY
, 2011
"... We revisit the complexity of the model checking problem for formulas of linear-time temporal logic (LTL). We show that the classic PSPACE-hardness result is actually limited to a subclass of the Kripke frames, which is characterized by a simple structural condition: the model checking problem is o ..."
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We revisit the complexity of the model checking problem for formulas of linear-time temporal logic (LTL). We show that the classic PSPACE-hardness result is actually limited to a subclass of the Kripke frames, which is characterized by a simple structural condition: the model checking problem is only PSPACE-hard if there exists a strongly con-nected component with two distinct cycles. If no such component exists, the problem is in coNP. If, additionally, the model checking problem can be decomposed into a polynomial number of finite path checking problems, for example if the frame is a tree or a directed graph with constant depth, or the frame has an SCC graph of constant depth, then the complexity reduces further to NC.
On P/NP Dichotomies for EL Subsumption under Relational Constraints
"... Abstract. We consider the problem of characterising relational constraints under which TBox reasoning in EL is tractable. We obtain P vs. coNP-hardness dichotomies for tabular constraints and constraints imposed on a single reflexive role. 1 ..."
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Abstract. We consider the problem of characterising relational constraints under which TBox reasoning in EL is tractable. We obtain P vs. coNP-hardness dichotomies for tabular constraints and constraints imposed on a single reflexive role. 1
On modal definability of Horn formulas
"... In this short paper we give a criterion of modal definability of a first-order universal Horn sentence with exactly one positive atom in terms of its graph. As a consequence we obtain that every modal logic axiomatized by a single modal Horn formula (i.e. of the form K+ φ where φ is a modal Horn for ..."
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In this short paper we give a criterion of modal definability of a first-order universal Horn sentence with exactly one positive atom in terms of its graph. As a consequence we obtain that every modal logic axiomatized by a single modal Horn formula (i.e. of the form K+ φ where φ is a modal Horn formula) is Kripke complete. Modal definability of first-order formulas has been intensively studied in modal logic, and even applied to automatic reasoning [9]. On the one hand, it has a nice Goldblatt-Thomason characterization [4], on the other hand, the problem “decide whether a first-order formula is modally definable ” is in general undecidable [2]. But the cause of this undecidability is in the undecidability of first-order logic, so when we restrict attention to a fragment with decidable implication, we are likely to obtain an algorithmic criterion for modal definability, as in this paper. Also this research is motivated by scrutinizing Theorem 5.9 of [3] saying that if L1 and L2 are Kripke complete and Horn axiomatizable unimodal logics, then L1 × L2 = [L1, L2] and studying whether Horn axiomatizability implies Kripke completeness. We give the positive answer to the last question for the case of a single universal Horn sentence with exactly one positive atom, but in general this problem seems to be open. Consider the classical first-order language LfΛ in the signature consisting of only binary predicates Rλ indexed by a finite set Λ. An atom is a formula of the form xiRλxj, where xi and xj are object variables and λ ∈ Λ. Universal Horn sentences (in short, Horn formulas) are closed (i.e. without free variables) formulas of the form ∀x1...∀xn(ψ → φ), where ψ is a conjunction of atoms and φ is an atom. Allowing ∨ in ψ as in [3] is equivalent to considering conjunctions of such formulas. Universal Horn sentences can be represented by tuples of the form D = (WD, (RDλ: λ ∈ Λ), α, β, λ0), where W D = {x1,..., xn} is a finite set, R
A Does Treewidth Help in Modal Satisfiability?
"... Many tractable algorithms for solving the Constraint Satisfaction Problem (Csp) have been developed using the notion of the treewidth of some graph derived from the input Csp instance. In particular, the incidence graph of the Csp instance is one such graph. We introduce the notion of an incidence g ..."
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Many tractable algorithms for solving the Constraint Satisfaction Problem (Csp) have been developed using the notion of the treewidth of some graph derived from the input Csp instance. In particular, the incidence graph of the Csp instance is one such graph. We introduce the notion of an incidence graph for modal logic formulas in a certain normal form. We investigate the parameterized complexity of modal satisfiability with the modal depth of the formula and the treewidth of the incidence graph as parameters. For various combinations of Euclidean, reflexive, symmetric and transitive models, we show either that modal satisfiability is Fixed Parameter Tractable (Fpt), or that it is W[1]-hard. In particular, modal satisfiability in general models is Fpt, while it is W[1]-hard in transitive models. As might be expected, modal satisfiability in transitive and Euclidean models is Fpt.
Finite Satisfiability of Modal Logic over Horn Definable Classes of Frames
"... Modal logic plays an important role in various areas of computer science, including verification and knowledge representation. In many practical applications it is natural to consider some restrictions of classes of admissible frames. Traditionally classes of frames are defined by modal axioms. Howe ..."
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Modal logic plays an important role in various areas of computer science, including verification and knowledge representation. In many practical applications it is natural to consider some restrictions of classes of admissible frames. Traditionally classes of frames are defined by modal axioms. However, many important classes of frames, e.g. the class of transitive frames or the class of Euclidean frames, can be defined in a more natural way by first-order formulas. In a recent paper it was proved that the satisfiability problem for modal logic over the class of frames defined by a universally quantified, first-order Horn formula is decidable. In this paper we show that also the finite satisfiability problem for modal logic over such classes is decidable. Keywords: modal logic, decidability, finite satisfiability 1
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"... elementarily generated modal logics Abstract. In this paper we consider the normal modal logics of elementary classes defined by first-order formulas of the form ∀x0∃x1...∃xn∧xiRλxj. We prove that many properties of these logics, such as finite axiomatisability, elementarity, axiomatisability by a s ..."
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elementarily generated modal logics Abstract. In this paper we consider the normal modal logics of elementary classes defined by first-order formulas of the form ∀x0∃x1...∃xn∧xiRλxj. We prove that many properties of these logics, such as finite axiomatisability, elementarity, axiomatisability by a set of canonical formulas or by a single generalised Sahlqvist formula, together with modal definability of the initial formula, either simultaneously hold or simultaneously do not hold.
