Showing papers on "Normal modal logic published in 2012"
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TL;DR: It is shown that modal nested sequents and prexed modal tableau systems are notational variants of each other, roughly in the same way that Gentzen sequent calculi and tableaus are notationally variants.
73 citations
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TL;DR: It is shown here that the apparent problem arises from an objectionable notion of derivability from assumptions in an axiomatic system, and a necessitation rule with this restriction permits a proof of the deduction theorem in its usual formulation.
Abstract: Various sources in the literature claim that the deduction theorem does not hold for normal modal or epistemic logic, whereas others present versions of the deduction theorem for several normal modal systems. It is shown here that the apparent problem arises from an objectionable notion of derivability from assumptions in an axiomatic system. When a traditional Hilbert-type system of axiomatic logic is generalized into a system for derivations from assumptions, the necessitation rule has to be modified in a way that restricts its use to cases in which the premiss does not depend on assumptions. This restriction is entirely analogous to the restriction of the rule of universal generalization of first-order logic. A necessitation rule with this restriction permits a proof of the deduction theorem in its usual formulation. Other suggestions presented in the literature to deal with the problem are reviewed, and the present solution is argued to be preferable to the other alternatives. A contraction- and cut-free sequent calculus equivalent to the Hilbert system for basic modal logic shows the standard failure argument untenable by proving the underivability of \({\square\,A}\) from A.
61 citations
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TL;DR: It is shown that every unifiable formula has a projective unifier in L iff L contains S4.3, and that all normal modal logics L containing S 4.3 are almost structurally complete, i.e. all (structural) admissible rules having unifiable premises are derivable in L.
Abstract: A projective unifier for a modal formula A, over a modal logic L, is a unifier σ for A (i.e. a substitution making A a theorem of L) such that the equivalence of σ with the identity map is the consequence of A. Each projective unifier is a most general unifier for A. Let L be a normal modal logic containing S4. We show that every unifiable formula has a projective unifier in L iff L contains S4.3. The syntactic proof is effective. As a corollary, we conclude that all normal modal logics L containing S4.3 are almost structurally complete, i.e. all (structural) admissible rules having unifiable premises are derivable in L. Moreover, L is (hereditarily) structurally complete iff L contains McKinsey axiom M .
42 citations
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TL;DR: This work defines two notions of bisimulation that allow for truth invariance results and provides game semantics and, for the more interesting and complicated notion, is able to provide characteristic formulae and prove a Hennessy–Milner-type theorem.
Abstract: We examine the notion of bisimulation and its ramifications, in the context of the family of Heyting-valued modal languages introduced by M. Fitting. Each modal language in this family is built on an underlying space of truth values, a Heyting algebra H. All the truth values are directly represented in the language, which is interpreted on relational frames with an H-valued accessibility relation. We define two notions of bisimulation that allow us to obtain truth invariance results. We provide game semantics and, for the more interesting and complicated notion, we are able to provide characteristic formulae and prove a Hennessy–Milner-type theorem. If the underlying algebra H is finite, Heyting-valued modal models can be equivalently reformulated to a form relevant to epistemic situations with many interrelated experts. Our definitions and results draw inspiration from this formulation, which is of independent interest to Knowledge Representation applications.
39 citations
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TL;DR: It is shown how a cut-free hypersequent calculus for 2D modal logic not only captures the logic precisely, but may be used to address issues in the epistemology and metaphysics of the authors' modal concepts.
39 citations
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27 Aug 2012TL;DR: This paper presents several implementations of fully automated theorem provers for first-order modal logics based on different proof calculi, among them the standard sequent calculus, a prefixed tableau calculus, an embedding into simple type theory, an instance-based method, and a Prefixed connection calculus.
Abstract: While there is a broad literature on the theory of first-order modal logics, little is known about practical reasoning systems for them. This paper presents several implementations of fully automated theorem provers for first-order modal logics based on different proof calculi. Among these calculi are the standard sequent calculus, a prefixed tableau calculus, an embedding into simple type theory, an instance-based method, and a prefixed connection calculus. All implementations are tested and evaluated on the new QMLTP problem library for first-order modal logic.
36 citations
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26 Jun 2012TL;DR: The Quantified Modal Logic Theorem Proving library provides a platform for testing and evaluating automated theorem proving systems for first-order modal logics and a small number of problems for multi-modal logic are included as well.
Abstract: The Quantified Modal Logic Theorem Proving (QMLTP) library provides a platform for testing and evaluating automated theorem proving (ATP) systems for first-order modal logics. The main purpose of the library is to stimulate the development of new modal ATP systems and to put their comparison onto a firm basis. Version 1.1 of the QMLTP library includes 600 problems represented in a standardized extended TPTP syntax. Status and difficulty rating for all problems were determined by running comprehensive tests with existing modal ATP systems. In the presented version 1.1 of the library the modal logics K, D, T, S4 and S5 with constant, cumulative and varying domains are considered. Furthermore, a small number of problems for multi-modal logic are included as well.
36 citations
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21 Jan 2012TL;DR: In this paper, the authors show that model checking for MDL formulae over Kripke structures is NP-complete and further consider fragments of MDL obtained by restricting the set of allowed propositional and modal connectives.
Abstract: Modal dependence logic (MDL) was introduced recently by Vaananen. It enhances the basic modal language by an operator = (·). For propositional variables p 1 ,…,p n the atomic formula = (p 1 ,…,p n −1 ,p n ) intuitively states that the value of p n is determined solely by those of p 1 ,…,p n −1 .
We show that model checking for MDL formulae over Kripke structures is NP -complete and further consider fragments of MDL obtained by restricting the set of allowed propositional and modal connectives. It turns out that several fragments, e.g., the one without modalities or the one without propositional connectives, remain NP -complete.
We also consider the restriction of MDL where the length of each single dependence atom is bounded by a number that is fixed for the whole logic. We show that the model checking problem for this bounded MDL is still NP -complete. Furthermore we almost completely classifiy the computational complexity of the model checking problem for all restrictions of propositional and modal operators for both unbounded as well as bounded MDL.
An extended version of this article can be found on arXiv.org [3].
ACM Subject Classifiers: F.2.2 Complexity of proof procedures; F.4.1 Modal logic; D.2.4 Model checking.
35 citations
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TL;DR: In this article, the modal Sahlqvist formulas for modal mu-calculus have been defined, and a correspondence theorem for them has been proved for the case of modal sahlqvists.
Abstract: We define analogues of modal Sahlqvist formulas for the modal mu-calculus, and prove a correspondence theorem for them.
34 citations
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TL;DR: A general constructive method of proving the realization of a modal logic in an appropriate justification logic by means of cut-free modal nested sequent systems and obtains a modular realization theorem that provides several justification counterparts based on various axiomatizations of amodal logic.
31 citations
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TL;DR: The decomposition uses the structural operational semantics that underlies the process algebra to derive congruence formats for two weak and rooted weak semantics: branching and @h-bisimilarity.
Abstract: We present a method for decomposing modal formulas for processes with the internal action @t. To decide whether a process algebra term satisfies a modal formula, one can check whether its subterms satisfy formulas that are obtained by decomposing the original formula. The decomposition uses the structural operational semantics that underlies the process algebra. We use this decomposition method to derive congruence formats for two weak and rooted weak semantics: branching and @h-bisimilarity.
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TL;DR: The probabilistic modal µ-calculus with independent product and coproduct (PCTL-with-independent product) as mentioned in this paper is an extension of PCTL that allows a play to be split into concurrent subplays.
Abstract: The probabilistic modal {\mu}-calculus is a fixed-point logic designed for expressing properties of probabilistic labeled transition systems (PLTS's). Two equivalent semantics have been studied for this logic, both assigning to each state a value in the interval [0,1] representing the probability that the property expressed by the formula holds at the state. One semantics is denotational and the other is a game semantics, specified in terms of two-player stochastic parity games. A shortcoming of the probabilistic modal {\mu}-calculus is the lack of expressiveness required to encode other important temporal logics for PLTS's such as Probabilistic Computation Tree Logic (PCTL). To address this limitation we extend the logic with a new pair of operators: independent product and coproduct. The resulting logic, called probabilistic modal {\mu}-calculus with independent product, can encode many properties of interest and subsumes the qualitative fragment of PCTL. The main contribution of this paper is the definition of an appropriate game semantics for this extended probabilistic {\mu}-calculus. This relies on the definition of a new class of games which generalize standard two-player stochastic (parity) games by allowing a play to be split into concurrent subplays, each continuing their evolution independently. Our main technical result is the equivalence of the two semantics. The proof is carried out in ZFC set theory extended with Martin's Axiom at an uncountable cardinal.
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TL;DR: This work gives a model-theoretic definition of the operator K and shows that this definition excapes the paradox, though it is validated only under restrictive conditions on the models.
Abstract: The present work is motivated by two questions. (1) What should an intuitionistic epistemic logic look like? (2) How should one interpret the knowledge operator in a Kripke-model for it? In what follows we outline an answer to (2) and give a model-theoretic definition of the operator K. This will shed some light also on (1), since it turns out that K, defined as we do, fulfills the properties of a necessity operator for a normal modal logic. The interest of our construction also lies in a better insight into the intuitionistic solution to Fitch’s paradox, which is discussed in the third section. In particular we examine, in the light of our definition, DeVidi and Solomon’s proposal of formulating the verification thesis as \(\phi \rightarrow
eg
eg K\phi\). We show, as our main result, that this definition excapes the paradox, though it is validated only under restrictive conditions on the models.
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TL;DR: This paper introduces a substitution-rewrite approach based on Ackermann?s Lemma to second-order quantifier elimination in modal logic, and considers two applications: computing first-order frame correspondence properties for modal axioms and rules, and rewriting second- order modal problems to equivalent simpler forms.
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04 Jun 2012TL;DR: A new logic for reasoning about the interaction between knowledge and action is introduced, in which each agent in a system is assumed to perceive some subset of the overall set of Boolean variables in the system; these variables give rise to epistemic indistinguishability relations.
Abstract: The last decade has been witness to a rapid growth of interest in logics intended to support reasoning about the interactions between knowledge and action Typically, logics combining dynamic and epistemic components contain ontic actions (which change the state of the world, eg, switching a light on) or epistemic actions (which affect the information possessed by agents, eg, making an announcement) We introduce a new logic for reasoning about the interaction between knowledge and action, in which each agent in a system is assumed to perceive some subset of the overall set of Boolean variables in the system; these variables give rise to epistemic indistinguishability relations, in that two states are considered indistinguishable to an agent if all the variables visible to that agent have the same value in both states In the dynamic component of the logic, we introduce actions r(p, i) and c(p, i): the effect of r(p, i) is to reveal variable p to agent i; the effect of c(p, i) is to conceal p from i By using these dynamic operators, we can represent and reason about how the knowledge of agents changes when parts of their environment are concealed from them, or by revealing parts of their environment to them Our main technical result is a sound and complete axiomatisation for our logic
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TL;DR: The class of BK-lattices is introduced, it is shown that this class coincides with the abstract closure of the class of twist-structures, and it forms a variety that is dually isomorphic to the lattice of extensions of Belnapian modal logic BK.
Abstract: Earlier algebraic semantics for Belnapian modal logics were defined in terms of twist-structures over modal algebras. In this paper we introduce the class of BK-lattices, show that this class coincides with the abstract closure of the class of twist-structures, and it forms a variety. We prove that the lattice of subvarieties of the variety of BK-lattices is dually isomorphic to the lattice of extensions of Belnapian modal logic BK. Finally, we describe invariants determining a twist-structure over a modal algebra.
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25 Jun 2012TL;DR: It is shown that the satisfiability problem for the two-dimensional extension KxK of unimodal K is nonelementary, hereby confirming a conjecture of Marx and Mikulas from 2001.
Abstract: We show that the satisfiability problem for the two-dimensional extension KxK of unimodal K is nonelementary, hereby confirming a conjecture of Marx and Mikulas from 2001. Our lower bound technique allows us to derive further lower bounds for many-dimensional modal logics for which only elementary lower bounds were previously known. We also derive nonelementary lower bounds on the sizes of Feferman-Vaught decompositions w.r.t. product for any decomposable logic that is at least as expressive as unimodal K. Finally, we study the sizes of Feferman-Vaught decompositions and formulas in Gaifman normal form for fixed-variable fragments of first-order logic.
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TL;DR: This paper shows that, under reasonable additional restrictions, these reflected fragments are NP-complete, thereby proving a matching lower bound, and provides a uniform proof that the corresponding full pure justification logics are Π 2 p -hard, reproving and generalizing an earlier result by Milnikel.
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16 Jul 2012TL;DR: This paper addresses the issue of possible world semantics for (modal) defeasible logics and defines proof theoretically based on the proof conditions for the logic.
Abstract: Defeasible Deontic Logic is a simple and computationally efficient approach for the representation of normative reasoning. Traditionally defeasible logics are defined proof theoretically based on the proof conditions for the logic. While several logic programming, operational and argumentation semantics have been provided for defeasible logics, possible world semantics for (modal) defeasible logics remained elusive. In this paper we address this issue.
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01 Jan 2012TL;DR: This talk outlines a consistency proof for Peano arithmetic based on RC and state a simple combinatorial statement, the so-called Worm principle, that was suggested by the use of GLP but is even more directly related to the Reflection Calculus.
Abstract: Several interesting applications of provability logic in proof theory made use of a polymodal logic GLP due to Giorgi Japaridze. This system, although decidable, is not very easy to handle. In particular, it is not Kripke complete. It is complete w.r.t. neighborhood semantics, however this could only be established recently by rather complicated techniques [1]. In this talk we will advocate the use of a weaker system, called Reflection Calculus, which is much simpler than GLP, yet expressive enough to regain its main proof-theoretic applications, and more. From the point of view of modal logic, RC can be seen as a fragment of polymodal logic consisting of implications of the form A→ B, where A and B are formulas built-up from > and the variables using just ∧ and the diamond modalities. In this paper we formulate it in a somewhat more succinct self-contained format. Further, we state its arithmetical interpretation, and provide some evidence that RC is much simpler than GLP. We then outline a consistency proof for Peano arithmetic based on RC and state a simple combinatorial statement, the so-called Worm principle, that was suggested by the use of GLP but is even more directly related to the Reflection Calculus.
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TL;DR: In this article, the authors investigate semantics for an intuitionistic modal logic in which the possibility modality does not distribute over disjunction, and show that a relational model can be represented as a neighborhood model, and the converse direction holds under a slight restriction.
Abstract: We investigate semantics for an intuitionistic modal logic in which the “possibility” modality does not distribute over disjunction. In particular, the main aim of this paper is to study such intuitionistic modal logic as a variant of classical non-normal modal logic. We first give a neighborhood semantics together with a sound and complete axiomatization. Next, we study relationships between our approach and the relational (Kripke-style) semantics considered in the literature. It is shown that a relational model can be represented as a neighborhood model, and the converse direction holds under a slight restriction. Also, by considering degenerate cases of neighborhood and relational semantics, we demonstrate that a certain classical monotone modal logic has relational semantics, and can be embedded into a classical normal bimodal logic.
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01 Jan 2012TL;DR: In this article, an automated theorem prover for first-order modal logic is presented for the constant, cumulative, and varying domain of the modal logics D, T, S4, and S5.
Abstract: This paper presents an implementation of an automated theorem prover for first-order modal logic that works for the constant, cumulative, and varying domain of the modal logics D, T, S4, and S5. It is based on the connection calculus for classical logic and uses prefixes representing world paths and a prefix unification algorithm to capture the restrictions given by the Kripke semantics of the standard modal logics. This permits a modular and elegant treatment of the considered modal logics and yields an efficient implementation. Details of the calculus, the implementation and performance results on the QMLTP problem library are presented.
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04 Jul 2012
TL;DR: A simple fragment of CTL is proposed that is able to express properties about accumulated weights along maximal runs of multiweighted modal automata by equipped with a game-based semantics and guaranteeing both soundness and completeness.
Abstract: Multiweighted modal automata provide a specification theory for multiweighted transition systems that have recently attracted interest in the context of energy games. We propose a simple fragment of CTL that is able to express properties about accumulated weights along maximal runs of multiweighted modal automata. Our logic is equipped with a game-based semantics and guarantees both soundness (formula satisfaction is propagated to the modal refinements) as well as completeness (formula non-satisfaction is propagated to at least one of its implementations). We augment our theory with a summary of decidability and complexity results of the generalized model checking problem, asking whether a specification -- abstracting the whole set of its implementations -- satisfies a given formula.
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18 Jun 2012TL;DR: It is seen how Turing progressions are closely related to the closed fragment of GLP, polymodal provability logic, and natural well-orders in GLP that characterize certain aspects of these Turing progressIONS are studied.
Abstract: We see how Turing progressions are closely related to the closed fragment of GLP, polymodal provability logic. Next we study natural well-orders in GLP that characterize certain aspects of these Turing progressions.
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TL;DR: The purpose of this paper is to describe a set of temporal alethic–deontic systems, i.e. systems that include temporal, Alethic and deontic operators.
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TL;DR: A prefixed tableau system for one of the major logics of this kind S4LPN, which combines the Logic of Proofs and epistemic logic S4 with an explicit negative introspection principle is introduced, and is shown to be sound and complete with respect to Fitting-style semantics.
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TL;DR: A hybrid modal logic is introduced for this purpose and it is proved that it can be represented using points and arrows by introducing higher order arrows: the switches.
Abstract: The notion of reactive graph generalises the one of graph by allowing the base accessibility relation to change when its edges are traversed. Can we represent these more general structures using points and arrows? We prove this can be done by introducing higher order arrows: the switches. The possibility of expressing the dependency of the future states of the accessibility relation on individual transitions by the use of higher-order relations, that is, coding meta-relational concepts by means of relations, strongly suggests the use of modal languages to reason directly about these structures. We introduce a hybrid modal logic for this purpose and prove its completeness.
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TL;DR: A nested sequent system with syntactic cut-elimination which is incomplete for the modal mu-calculus, but complete with respect to a restricted language that allows only fixed points of a certain form.
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01 Jan 2012
TL;DR: In particular, there is little consensus on the way to understand what it is to prove a statement like □A as discussed by the authors, and no such consensus on what the basic items of deduction in modal vocabulary are.
Abstract: Modal logic is traditionally the logic obtained by adding to basic propositional logic, like classical logic, the concepts of necessity (□) and possibility (◊). There is a wide consensus on which are the main systems of modal logic—systems such as K, KT, KB, S4, S5 and the provability logic GL—and their canonical interpretation, Kripke models. Beyond that, there is little consensus. In particular, there is little consensus on the way to understand what it is to prove a statement like □A. While we have a systematic and rigorous formal account of truth conditions of modal statements (in Kripke models with points and accessibility relations with different properties, underwriting different principles governing □ and ◊ and their interaction), we have no such consensus on what the basic items of deduction in modal vocabulary are.
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TL;DR: This paper offers to use temporal and modal language formulas to represent arguments in the nodes of a network using Kripke semantics, and introduces a new key concept of usability of an argument.
Abstract: The traditional Dung networks depict arguments as atomic and study the relationships of attack between them. This can be generalised in two ways. One is to consider various forms of attack, support, feedback, etc. Another is to add content to nodes and put there not just atomic arguments but more structure, e.g. proofs in some logic or simply just formulas from a richer language. This paper offers to use temporal and modal language formulas to represent arguments in the nodes of a network. The suitable semantics for such networks is Kripke semantics. We also introduce a new key concept of usability of an argument. This is the beginning of a continuing research for adding contents to the nodes of an argumentation network. This research will allow us to address notions like ‘what does it exactly mean for a node to attack another’ or ‘what does it mean for a network to be consistent’ or ‘can we give proper proof rules to manipulate networks’, and more.