Showing papers on "Normal modal logic published in 2000"

On an intuitionistic modal logic

Abstract: In this paper we consider an intuitionistic variant of the modal logic S4 (which we call IS4). The novelty of this paper is that we place particular importance on the natural deduction formulation of IS4— our formulation has several important metatheoretic properties. In addition, we study models of IS4— not in the framework of Kirpke semantics, but in the more general framework of category theory. This allows not only a more abstract definition of a whole class of models but also a means of modelling proofs as well as provability.

Knowledge-Driven versus Data-Driven Logics

Abstract: The starting point of this work is the gap between two distinct traditions in information engineering: knowledge representation and data-driven modelling. The first tradition emphasizes logic as a tool for representing beliefs held by an agent. The second tradition claims that the main source of knowledge is made of observed data, and generally does not use logic as a modelling tool. However, the emergence of fuzzy logic has blurred the boundaries between these two traditions by putting forward fuzzy rules as a Janus-faced tool that may represent knowledge, as well as approximate non-linear functions representing data. This paper lays bare logical foundations of data-driven reasoning whereby a set of formulas is understood as a set of observed facts rather than a set of beliefs. Several representation frameworks are considered from this point of view: classical logic, possibility theory, belief functions, epistemic logic, fuzzy rule-based systems. Mamdani‘s fuzzy rules are recovered as belonging to the data-driven view. In possibility theory a third set-function, different from possibility and necessity plays a key role in the data-driven view, and corresponds to a particular modality in epistemic logic. A bi-modal logic system is presented which handles both beliefs and observations, and for which a completeness theorem is given. Lastly, our results may shed new light in deontic logic and allow for a distinction between explicit and implicit permission that standard deontic modal logics do not often emphasize.

A Benchmark Method for the Propositional Modal Logics K, KT, S4

Abstract: A lot of methods have been proposed – and sometimes implemented – for proof search in the propositional modal logics K, KT, and S4 It is difficult to compare the usefulness of these methods in practice, since in most cases no or only a few execution times have been published We try to improve this unsatisfactory situation by presenting a set of benchmark formulas Note that we do not just list formulas, but give a method that allows us to compare different provers today and in the future As a starting point we give the results we obtained when we applied this benchmark method to the Logics Workbench (LWB) We hope that the discussion of postulates concerning ATP benchmark helps to obtain improved benchmark methods for other logics, too

Resolution-based methods for modal logics

Abstract: In this paper we give an overview of resolution methods for extended propositional modal logics. We adopt the standard translation approach and consider different resolution refinements which provide decision procedures for the resulting clause sets. Our procedures are based on ordered resolution and selection-based resolution. The logics that we cover are multi-modal logics defined over relations closed under intersection, union, converse and possibly complementation.

On the Complexity of Explicit Modal Logics

Abstract: Explicit modal logic was introduced by S. Artemov. Whereas the traditional modal logic uses atoms □F with a possible semantics "F is provable", the explicit modal logic deals with atoms of form t:F, where t is a proof polynomial denoting a specific proof of a formula F. Artemov found the explicit modal logic LP in this new format and built an algorithm that recovers explicit proof polynomials corresponding to modalities in every derivation in K. Godel's modal provability calculus S4. In this paper we study the complexity of LP as well as the complexity of explicit counterparts of the modal logics K, D, T, K4, D4 found by V. Brezhnev. The main result: the satisfiability problem for each of these explicit modal logics belongs to the class Σ2p of the polynomial hierarchy. Similar problem for the original modal logics is known to be PSPACE-complete. Therefore, explicit modal logics have much better upper complexity bounds than the original modal logics.

Modal Logics and Philosophy

Abstract: Unlike most modal logic textbooks, which are both forbidding mathematically and short on philosophical discussion, Modal Logics and Philosophy places its emphasis firmly on showing how useful modal logic can be as a tool for formal philosophical analysis. In Part 1 of the book, the reader is introduced to some standard systems of modal logic and encouraged through a series of exercises to become proficient in manipulating these logics. The emphasis is on possible world semantics for modal logics and the semantic emphasis is carried into the formal method, Jeffrey-style truth-trees. Standard truth-trees are extended in a simple and transparent way to take possible worlds into account. Part 2 systematically explores the applications of modal logic to philosophical issues such as truth, time, processes, knowledge and belief, obligation and permission. The second edition sees the addition of two new chapters on conditionals. The first, in Part 1, presents the formalities of a range of conditional logics, and the second, in Part 2, discusses some of the philosophical issues raised by them. Other chapters have been revised and updated, including some reordering of content in Part 1, to strengthen the book as a fully comprehensive introduction to modal logics and their application suitable for course use.

An abstract algebraic logic approach to tetravalent modal logics

Abstract: This paper contains a joint study of two sentential logics that combine a many-valued character, namely tetravalence, with a modal character; one of them is normal and the other one quasinormal. The method is to study their algebraic counterparts and their abstract models with the tools of Abstract Algebraic Logic, and particularly with those of Brown and Suszko's theory of abstract logics as recently developed by Font and Jansana in their “A General Algebraic Semantics for Sentential Logics”. The logics studied here arise from the algebraic and lattice-theoretical properties we review of Tetravalent Modal Algebras, a class of algebras studied mainly by Loureiro, and also by Figallo. Landini and Ziliani, at the suggestion of the late Antonio Monteiro.

Constructing the Least Models for Positive Modal Logic Programs

Abstract: We give algorithms to construct the least L-model for a given positive modal logic program P, where L can be one of the modal logics KD, T, KDB, B, KD4, S4, KD5, KD45, and S5. If L ∈ {KD5,KD45,S5}, or L ∈ {KD,T,KDB,B} and the modal depth of P is finitely bounded, then the least L-model of P can be constructed in PTIME and coded in polynomial space. We also show that if P has no flat models then it has the least models in KB, K5, K45, and KB5. As a consequence, the problem of checking the satisfiability of a set of modal Horn formulae with finitely bounded modal depth in KD, T, KB, KDB, or B is decidable in PTIME. The known result that the problem of checking the satisfiability of a set of Horn formulae in K5, KD5, K45, KD45, KB5, or S5 is decidable in PTIME is also studied in this work via a different method.

An overview of rough set semantics for modal and quantifier logics

Abstract: In this paper, we would like to present some logics with semantics based on rough set theory and related notions. These logics are mainly divided into two classes. One is the class of modal logics and the other is that of quantifier logics. For the former, the approximation space is based on a set of possible worlds, whereas in the latter, we consider the set of variable assignments as the universe of approximation. In addition to surveying some well-known results about the links between logics and rough set notions, we also develop some new applied logics inspired by rough set theory.

Independence: Logics and Concurrency

Abstract: We consider Hintikka et al's 'independence-friendly first-order logic' We apply it to a modal logic setting, defining a notion of 'independent' modal logic, and we examine the associated fixpoint logics

Products of modal logics. Part 2: relativised quantifiers in classical logic

Abstract: In the rst part of this paper we introduced products of modal logics and proved basic results on their axiomatisability and the f.m.p. In this continuation paper we prove a stronger result - the product f.m.p. holds for products of modal logics in which some of the modalities are reflexive or serial. This theorem is applied in classical rst-order logic; we identify a new Square Fragment (SF) of the classical logic, where the basic predicates are binary and all quantiers are relativised, and for which we show the f.m.p. in the classical sense. Also we prove that SF not included in Guarded Fragment (and in Packed Fragment) and that it can be embedded into the equational theory of relation algebras. 1

Two-Phase Deontic Logic

Abstract: We show that for the adequate representation of some examples of normative reasoning a combination of different operators is needed, where each operator validates different inference rules. The combination of different modal operators imposes the restriction on the proof theory of the logic that a proof rule can be blocked in a derivation due to the fact that another proof rule has been used earlier in the derivation. In this paper we only use two operators and therefore we call the restriction the two-phase approach in the proof theory, which we formalize in two-phase labeled deontic logic (2ldl) and in two-phase dyadic deontic logic (2dl). The preference-based semantics of 2dl is based on an explicit deontic preference ordering between worlds, representing different degrees of ideality. The two different modal operators represent two different usages of the preference ordering, called minimizing and ordering. 1. Why deontic logic derivations must consist of two phases 1.1. Van Fraassen’s paradox Van Fraassen (1973) presents a logical analysis of dilemmas. In a logic that formalizes reasoning about dilemmas we cannot accept the conjunction rule, because it derives©(p∧¬p) from the dilemma©p∧©¬p, whereas ‘ought implies can’ ¬ © (p ∧ ¬p). On the other hand we do not want to reject the conjunction rule in all cases. For example, we want to derive ©(p ∧ q) from ©p ∧ ©q when p and q are distinct propositional atoms. That is, we have to add a restriction on the conjunction rule such that we only derive ©(α1 ∧α2) from ©α1 and ©α2 if α1 ∧ α2 is consistent. Van Fraassen calls the latter inference pattern Consistent Aggregation, which we write as the restricted conjunction rule (rand). He encounters a problem in the formalization of obliga2 L. VAN DER TORRE AND Y. TAN tions, and wonders if he needs a language in which he can talk directly about the imperatives as well. A variant of this problem is illustrated in the following example. Example 1 (Van Fraassen’s paradox). Assume a monadic deontic logic without nested modal operators1 in which dilemmas like ©p ∧ ©¬p are consistent, but which validates ¬©⊥, where ⊥ stands for any contradiction like p∧¬p. Moreover, assume that it satisfies replacement of logical equivalents and at least the following two inference patterns Restricted Conjunction rule (rand), also called consistent aggregation, and Weakening (w), where ↔ 3φ can loosely be read as φ is possible (or propositionally consistent). rand: ©α1,©α2, ↔ 3(α1 ∧ α2) ©(α1 ∧ α2) w: ©α1 ©(α1 ∨ α2) Moreover, assume the two premises ‘Honor thy father or thy mother!’ ©(f ∨ m) and ‘Honor not thy mother!’ ©¬m. The derivation of Figure 1 illustrates how the desired conclusion ‘thou shalt honor thy father’ ©(f ∨m) ©¬m ©(f ∧ ¬m) rand ©f w Figure 1. Van Fraassen’s paradox (1) ©f can be derived from the premises. Unfortunately, the derivation of Figure 2 illustrates that we cannot accept restricted conjunction and ©p ©(f ∨ p) w ©¬p ©(f ∧ ¬p) rand ©f w Figure 2. Van Fraassen’s paradox (2) weakening in a monadic deontic logic, because we can derive the counterintuitive obligation ©f from a deontic dilemma ©p ∧ ©¬p. The point of this paradox is that every ©(β), of which ©(f) is a special case, would be derivable. Van Fraassen asks himself whether restricted conjunction can be formalized, and he observes interesting technical questions. In this paper we pursue some of these technical questions. 2dl.tex; 15/06/2001; 17:08; no v.; p.2 TWO-PHASE DEONTIC LOGIC 3 ‘But can this happy circumstance be reflected in the logic of the ought-statements alone? Or can it be expressed only in a language in which we can talk directly about the imperatives as well? This is an important question, because it is the question whether the inferential structure of the ‘ought’ language game can be stated in so simple a manner that it can be grasped in and by itself. Intuitively, we want to say: there are simple cases, and in the simple cases the axiologist’s logic is substantially correct even if it is not in general – but can we state precisely when we find ourselves in such a simple case? These are essentially technical questions for deontic logic, and I shall not pursue them here.’ (van Fraassen, 1973) As far as we know, there is no discussion on Van Fraassen’s paradox in the deontic logic literature.2 We analyze Van Fraassen’s paradox in Example 1 by forbidding application of rand after w has been applied. This blocks the counterintuitive derivation in Figure 2 and it does not block the intuitive derivation in Figure 1, as we show below. Our formalization of two-phase reasoning works as follows. In the logic, the two phases are represented by two different types of obligations, written as phase-1 obligations ©1 and phase-2 obligations ©2 . The premises are phase-1 obligations, the conclusions are phase-2 obligations and the two phases are linked to each other with the following inference pattern rel.

On Axiomatising Products of Kripke Frames

Abstract: It is shown that the many-dimensional modal logic K n , determined by products of n -many Kripke frames, is not finitely axiomatisable in the n -modal language, for any n > 2. On the other hand, K n is determined by a class of frames satisfying a single first-order sentence.

Modal Logics for Topological Spaces

Abstract: In this thesis we shall present two logical systems, MP and MP, for the purpose of reasoning about knowledge and effort. These logical systems will be interpreted in a spatial context and therefore, the abstract concepts of knowledge and effort will be defined by concrete mathematical concepts.

A modal logic for epistemic tests

Abstract: We study a modal logic of knowledge and action, focussing on test actions. Such knowledge-gathering actions increase the agents' knowledge. We propose a semantics, and associatean axiomatics and a rewriting-based proof procedure.

Back and forth between guarded and modal logics

Abstract: Guarded fixed point logic /spl mu/GF extends the guarded fragment by means of least and greatest fixed points, and thus plays the same role within the domain of guarded logics as the modal /spl mu/-calculus plays within the modal domain. We provide a semantic characterisation of /spl mu/GF within an appropriate fragment of second-order logic, in terms of invariance under guarded bisimulation. The corresponding characterisation of the modal /spl mu/-calculus, due to D. Janin and I. Walukiewicz (1999), is lifted from the modal to the guarded domain by means of model theoretic translations. At the methodological level, these translations make the intuitive analogy between modal and guarded logics available as a tool in the analysis of the guarded domain.

Normal Forms and Proofs in Combined Modal and Temporal Logics

Abstract: In this paper we present a framework for the combination of modal and temporal logic. This framework allows us to combine different normal forms, in particular, a separated normal form for temporal logic and a first-order clausal form for modal logics. The calculus of the framework consists of temporal resolution rules and standard first-order resolution rules.

Arrow Logic and Infinite Counting

Abstract: We consider arrow logics (ie, propositional multi-modal logics having three -- a dyadic, a monadic, and a constant -- modal operators) augmented with various kinds of infinite counting modalities, such as 'much more', 'of good quantity', 'many times' It is shown that the addition of these modal operators to weakly associative arrow logic results in finitely axiomatizable and decidable logics, which fail to have the finite base property

The nondeterministic information logic NIL is PSPACE-complete

Abstract: The nondeterministic information logic NIL has been introduced by Orlowska and Pawlak in 1984 as a logic for reasoning about total information systems with the similarity, the forward inclusion and the backward inclusion relations. In 1987, Vakarelov provides the first first-order characterization of structures derived from information systems and this has been done with the semantical structures of NIL. Since then, various extensions of NIL have been introduced and many issues for information logics about decidability and Hilbert-style proof systems have been solved. However, computational complexity issues have been seldom attacked in the literature mainly because the information logics are propositional polymodal logics with interdependent modal connectives. We show that NIL satisfiability is a PSPACE-complete problem. PSPACE-hardness is shown to be an easy consequence of PSPACE-hardness of the well-known modal logic S4. The main difficulty is to show that NIL satisfiability is in PSPACE. To do so we present an original construction that extends various previous works by Ladner (1977), Halpern and Moses (1992) and Spaan (1993).

The Semantics of Modal Predicate Logic I. Counterpart-Frames

Abstract: We introduce a new semantics for modal predicate logic, with respect to which a rich class of first-order modal logics is complete, namely all normal first-order modal logics that are extensions of free quantified K. This logic is defined by combining positive free logic with equality PFL . = and the propositional modal logic K. We then uniformly construct—for each modal predicate logic L—a canonical model whose theory is exactly L. This proves completeness with respect to so-called modalstructures. We add some remarks on canonicity and frame-completeness and finally show that if suitable modal algebras of ‘admissible interpretations’ are added to modal predicate frames, general frame-completeness is gained.

Bimodal Logics for Reasoning About Continuous Dynamics.

Abstract: We study a propositional bimodal logic consisting of two S4 modalities and [a], together with the interaction axiom scheme h ai ' → h ai '. In the intended semantics, the plain is given the McKinsey-Tarski interpretation as the interior operator of a topology, while the labelled [a] is given the standard Kripke semantics using a reflexive and transitive binary relation Ra. The interaction axiom expresses the property that the Ra relation is lower semi-continuous with respect to the topology. The class of topological Kripke frames axiomatised by the logic includes all frames over Euclidean space where Ra is the positive flow relation of a differential equation. We establish the completeness of the axiomatisation with respect to the intended class of topological Kripke frames, and investigate tableau calculi for the logic, although decidability is still an open question.

Fibred modal tableaux

Abstract: We describe a general and uniform tableau methodology for multi-modal logics arising from Gabbay’s methodology of fibring and Governatori’s labelled tableau system KEM.

Embeddings of propositional monomodal logics

Abstract: The aim of this paper is to investigate the expressibility of classical propositional monomodal logics. To this end, a notion of embedding of one logic into another is introduced, which is, roughly, a translation preserving theoremhood. This enables to measure the expressibility of a logic by a (finite or infinite) number of logics embeddable into it. This measure is calculated here for a large family of modal logics including K, K4, KB, K5, GL, T, S4, B, S5, Grz, and provability logics. It is also shown that some of these logics (e.g., all normal logics containing the symmetry axiom except for the logics Triv, Ver, and the intersection of these two) are not embeddable into some others (e.g., K, K4, K5, GL, T, S4, Grz).

Agent-BDI Logic

Abstract: In this paper, it is demonstrated that the logic tool used in agent formalized d epiction should be the mixed modal logic which has both normal and non-normal m odal operators. Then a logic system A-BI is built for Agent-BDI logic and its semantics and axiom system are discussed. Especially for non-normal modal opera tors a new semantic interpretation based on Kripke's normal possible worlds is p resented. It is proved that A-BI logic system is sound and complete. A-BI logi c appropriately depicts the essence and relation of belief and intension, and ca n be used as logic tool in formalized research on agent.