Showing papers on "Normal modal logic published in 2003"

Deontic interpreted systems

Abstract: We investigate an extension of the formalism of interpreted systems by Halpern and colleagues to model the correct behaviour of agents. The semantical model allows for the representation and reasoning about states of correct and incorrect functioning behaviour of the agents, and of the system as a whole. We axiomatise this semantic class by mapping it into a suitable class of Kripke models. The resulting logic, KD45ni-j, is a stronger version of KD, the system often referred to as Standard Deontic Logic. We extend this formal framework to include the standard epistemic notions defined on interpreted systems, and introduce a new doubly-indexed operator representing the knowledge that an agent would have if it operates under the assumption that a group of agents is functioning correctly. We discuss these issues both theoretically and in terms of applications, and present further directions of work.

Reasoning About Space: The Modal Way

Abstract: We investigate the topological interpretation of modal logic in modern terms, using a new notion of bisimulation. We look at modal logics with interesting topological content, presenting, among others, a new proof of McKinsey and Tarski’s theorem on completeness of S4 with respect to the real line, and a completeness proof for the logic of finite unions of convex sets of reals. We conclude with a broader picture of extended modal languages of space, for which the main logical questions are still wide open.

Monotonic modal logics

Abstract: Monotonic modal logics form a generalisation of normal modal logics in which the additivity of the diamond modality has been weakened to monotonicity: 3p∨3q → 3(p∨q). This generalisation means that Kripke structures no longer form an adequate semantics. Instead monotonic modal logics are interpreted over monotonic neighbourhood structures, that is, neighbourhood structures where the neighbourhood function is closed under supersets. As specific examples of monotonic modal logics we mention Game Logic, Coalition Logic and the Alternating-Time Temporal Logic. This thesis presents results on monotonic modal logics in a general framework. The topics covered include model constructions and truth invariance, definability and correspondence theory, the canonical model construction, algebraic duality (for monotonic neighbourhood frames), coalgebraic semantics, Craig interpolation via superamalgamation, and simulations of monotonic modal logics by bimodal normal ones. The main contributions are: generalisations of the Sahlqvist correspondence and canonicity theorems, a detailed account of algebraic duality via canonical extensions, an analogue of the Goldblatt-Thomason theorem on definable frame classes, results on the relationship between bisimulation and coalgebraic notions of structural equivalence, Craig interpolation results, and a simulation construction which preserves descriptiveness of general frames.

Model Checking and Satisfiability for Sabotage Modal Logic

Abstract: We consider the sabotage modal logic SML which was suggested by van Benthem. SML is the modal logic equipped with a ‘transition-deleting’ modality and hence a modal logic over changing models. It was shown that the problem of uniform model checking for this logic is PSPACE-complete. In this paper we show that, on the other hand, the formula complexity and the program complexity are linear, resp., polynomial time. Further we show that SML lacks nice model-theoretic properties such as bisimulation invariance, the tree model property, and the finite model property. Finally we show that the satisfiability problem for SML is undecidable. Therefore SML seems to be more related to FO than to usual modal logic.

Regression in Modal Logic

Abstract: In this work we propose an encoding of Reiter's Situation Calculus solution to the frame problem into the framework of a simple multimodal logic of actions. In particular we present the modal counterpart of the regression technique. This gives us a theorem proving method for a relevant fragment of our modal logic.

A Fuzzy Modal Logic for Belief Functions

Abstract: In this paper we introduce a new logical approach to reason explicitly about Dempster-Shafer belief functions. We adopt the following view: one just starts with Boolean formulas ϕ and a belief function on them; the belief of ϕ is taken to be the truth degree of the (fuzzy) proposition Bϕ standing for "ϕ is believed". For our complete axiomatization (Hylbert-style) we use one of the possible definitions of belief, namely as probability of (modal) necessity. This enables us to define a logical system combining the modal logic S5 with an already proposed fuzzy logic approach to reason about probabilities. In particular, our fuzzy logic is the logic LII½ which puts Lukasiewicz and Product fuzzy logics together.

Preference semantics for deontic logic part I — Simple models

Abstract: This paper presents determination results for some deontic logics with respect to a simple preference-based semantics, in which possible worlds are ranked by comparative value but none need be sup- posed to be best or top-ranked among alternatives. This kind of semantics is useful for defining deontic logics that allow for conflicts of obligation. Monadic standard deontic logic (SDL) is determined by the class of frames in which the preference ranking is reflexive, transitive and connected. The weak deontic logic P, which allows for normative conflicts, is determined by the class of all preference frames. These results are extended to corresponding dyadic deontic logics that formalize the logic of conditional obligation and the logic of preference itself.

Modal probability, belief, and actions

Abstract: We investigate a modal logic of probability with a unary modal operator expressing that a proposition is more probable than its negation Such an operator is not closed under conjunction, and its modal logic is therefore non-normal Within this framework we study the relation of probability with other modal concepts: belief and action We focus on the evolution of belief, and propose an integration of revision For that framework we give a regression algorithm

Game logic is strong enough for parity games

Abstract: We investigate the expressive power of Parikh's Game Logic interpreted in Kripke structures, and show that the syntactical alternation hierarchy of this logic is strict This is done by encoding the winning condition for parity games of rank n It follows that Game Logic is not captured by any finite level of the modal μ-calculus alternation hierarchy Moreover, we can conclude that model checking for the μ-calculus is efficiently solvable iff this is possible for Game Logic

Modal Epistemic and Doxastic Logic

Abstract: Knowledge has always been a topic central to philosophy (cf. e.g. [Glymour, 1992]). Since ancient times philosophers have been interested in the way knowledge comes to us and in what way it relates to reality, the world in which we live. As is the case with so many things, during this century also the topic of knowledge has become the subject of formal investigations. Questions arose such as what the logical properties of knowledge are, and in order to come up with answers to these, logics have been devised to study these questions in a formal setting. These logics are now generally called ‘epistemic logics’, i.e., logics pertaining to ‘knowledge’. Mostly, also the notion of ‘belief, which is sometimes thought of as a weaker form of knowledge (but this is debated among philosophers (cf. [Gettier, 1963; Pollock, 1986; Voorbraak, 1993]) is considered, and we will also incorporate this notion in our treatment. Sometimes logics of belief are referred to with the special term ‘doxastic logics’, but we will just use the term ‘epistemic logic(s)’ for logics of knowledge and belief. Jaakko Hintikka [1962] was the first who proposed a modal logic approach to knowledge and belief. We follow the tradition of most recent treatments of modal logics, including those of knowledge and belief, by adopting possible world semantics in the style of Kripke [1963].

A Sahlqvist Theorem for Relevant Modal Logics

Abstract: Kripke-completeness of every classical modal logic with Sahlqvist formulas is one of the basic general results on completeness of classical modal logics. This paper shows a Sahlqvist theorem for modal logic over the relevant logic Bin terms of Routley-Meyer semantics. It is shown that usual Sahlqvist theorem for classical modal logics can be obtained as a special case of our theorem.

Euclidean Hierarchy in Modal Logic

Abstract: For a Euclidean space $$\mathbb{R}^n $$ , let L n denote the modal logic of chequered subsets of $$\mathbb{R}^n $$ For every n ≥ 1, we characterize L n using the more familiar Kripke semantics, thus implying that each L n is a tabular logic over the well-known modal system Grz of Grzegorczyk We show that the logics L n form a decreasing chain converging to the logic L ∞ of chequered subsets of $$\mathbb{R}^\infty $$ As a result, we obtain that L ∞ is also a logic over Grz, and that L ∞ has the finite model property We conclude the paper by extending our results to the modal language enriched with the universal modality

A Fixpoint Semantics and an SLD-Resolution Calculus for Modal Logic Programs

Abstract: We propose a modal logic programming language called MProlog, which is as expressive as the general modal Horn fragment. We give a fixpoint semantics and an SLD-resolution calculus for MProlog in all of the basic serial modal logics KD, T, KDB, B, KD4, S4, KD5, KD45, and S5. For an MProlog program P and for L being one of the mentioned logics, we define an operator T_{L,P}, which has the least fixpoint I_{L,P}. This fixpoint is a set of formulae, which may contain labeled forms of the modal operator d, and is called the least L-model generator of P. The standard model of I_{L,P} is shown to be a least L-model of P. The SLD-resolution calculus for MProlog is designed with a similar style as for classical logic programming. It is sound and complete. We also extend the calculus for MProlog in the almost serial modal logics KB, K5, K45, and KB5.

Belief Change: from Situation Calculus to Modal Logic

Abstract: We propose a translation into Modal Logic of the ideas that formalise belief change in the Situation Calculus. This translation is extended to the case of revision. In the conclusion is presented a set of open issues.

Mechanised Reasoning and Model Generation for Extended Modal Logics

Abstract: The approach presented in this overview paper exploits that modal logics can be seen to be fragments of first-order logic and deductive methods can be developed and studied within the framework of first-order resolution. We focus on a class of extended modal logics very similar in spirit to propositional dynamic logic and closely related to description logics. We review and discuss the development of decision procedures for decidable extended modal logics and look at methods for automatically generating models.

Constructive interpolation in hybrid logic

Abstract: Craig’s interpolation lemma (if $φ\rightarrowψ$ is valid, then $φ\rightarrowθ$ and $θ\rightarrowψ$ are valid, for θ a formula constructed using only primitive symbols which occur both in φ and ψ) fails for many propositional and first order modal logics. The interpolation property is often regarded as a sign of well-matched syntax and semantics. Hybrid logicians claim that modal logic is missing important syntactic machinery, namely tools for referring to worlds, and that adding such machinery solves many technical problems. The paper presents strong evidence for this claim by defining interpolation algorithms for both propositional and first order hybrid logic. These algorithms produce interpolants for the hybrid logic of every elementary class of frames satisfying the property that a frame is in the class if and only if all its point-generated subframes are in the class. In addition, on the class of all frames, the basic algorithm is conservative: on purely modal input it computes interpolants in which the hybrid syntactic machinery does not occur.

XPath and modal logics of finite DAG's

Abstract: XPath, CTL and the modal logics proposed by Blackburn et al, Palm and Kracht are variable free formalisms to describe and reason about (finite) trees. XPath expressions evaluated at the root of a tree correspond to existential positive modal formulas. The models of XPath expressions are finite ordered trees, or in the presence of XML’s ID/IDREF mechanism graphs. The ID/IDREF mechanism can be seen as a device for naming nodes. Naming devices have been studied in hybrid logic by nominals. We add nominals to the modal logic of Palm and interpret the language on directed acyclic graphs. We give an algorithm which decides the consequence problem of this logic in exponential time. This yields a complexity result for query containment of the corresponding extension of XPath.

On canonical modal logics that are not elementarily determined

Abstract: There exist modal logics that are validated by their canonical frames but are not sound and complete for any elementary class of frames. Continuum many such bimodal logics are exhibited, including one of each degree of unsolvability, and all with the finite model property. Monomodal examples are also constructed that extend K4 and are related to the proof of non-canonicity of the McKinsey axiom.

Simulating polyadic modal logics by monadic ones

Abstract: We define an interpretation of modal languages with polyadic operators in modal languages that use monadic operators (diamonds) only. We also define a simulation operator which associates a logic $\simL$ in the diamond language with each logic $\La$ in the language with polyadic modal connectives. We prove that this simulation operator transfers several useful properties of modal logics, such as finite/recursive axiomatizability, frame completeness and the finite model property, canonicity and first-order definability.

The Worlds of Possibility: Modal Realism and the Semantics of Modal Logic

Abstract: Perhaps no one has done more in the last 30 years to advance thinking in the metaphysics of modality than has Alvin Plantinga. Collected here are some of his most important essays on this influential subject. Dating back from the late 1960's to the present, they chronicle the development of Plantinga's thoughts about some of the most fundamental issues in metaphysics: what is the nature of abstract objects like possible worlds, properties, propositions, and such phenomena? Are there possible but non-actual objects? Can objects that do not exist exemplify properties? Plantinga gives thorough and penetrating answers to all of these questions and many others. This volume contains some of the best work in metaphysics from the past 30 years, and will remain a source of critical contention and keen interest among philosophers of metaphysics and philosophical logic for years to come.... Download ebook, read file pdf Essays in the Metaphysics of Modality

Preservativity logic: An analogue of interpretability logic for constructive theories

Abstract: In this paper we study the modal behavior of Σ-preservativity, an extension of provability which is equivalent to interpretability for classical superarithmetical theories. We explain the connection between the principles of this logic and some well-known properties of HA, like the disjunction property and its admissible rules. We show that the intuitionistic modal logic given by the preservativity principles of HA known so far, is complete with respect to a certain class of frames.

General Frames for Relevant Modal Logics

Abstract: General frames are often used in classical modal logic. Since they are duals of modal algebras, completeness follows automatically as with algebras but the intuitiveness of Kripke frames is also retained. This paper develops basics of general frames for relevant modal logics by showing that they share many important properties with general frames for classical modal logic.

Combining Dynamic Logic with Doxastic Modal Logics

Abstract: We prove completeness and decidability results for a family of combinations ofpropositional dynamic logic and unimodal doxastic logics in which the modalitiesmay interact. The kind of interactions we consider include three forms of commut-ing axioms, namely, axioms similar to the axiom of perfect recall and the axiom ofno learning from temporal logic and a Church-Rosser axiom. We investigate theinfluence of the substitution rule on the properties of these logics and propose anew semantics for the test operator to avoid unwanted side effects caused by theinteraction of the classic test operator with the extra axioms. Copyright c 2003, University of Manchester. All rights reserved. Reproduction (electronically or byother means) of all or part of this work is permitted for educational or research purposes only, oncondition that no commercial gain is involved.Recent preprints issued by the Department ofComputer Science, ManchesterUniversity, areavailableonWWW viaURL http://www.cs.man.ac.uk/preprints/index.htmlorbyftp fromftp.cs.man.ac.ukin the directory pub/preprints.

Unitary unification of s5 modal logic and its extensions

Abstract: It is shown that all extensions of S5 modal logic, both in the standard formalization and in the formalization with strict implication, as well as all varieties of monadic algebras have unitary unification.

A proof-theoretic study of the correspondence of classical logic and modal logic

Abstract: It is well known that the modal logic S5 can be embedded in the classical predicate logic by interpreting the modal operator in terms of a quantifier. Wajsberg [10] proved this fact in a syntactic way. Mints [7] extended this result to the quantified version of S5; using a purely proof-theoretic method he showed that the quantified S5 corresponds to the classical predicate logic with one-sorted variable. In this paper we extend Mints' result to the basic modal logic S4; we investigate the correspondence between the quantified versions of S4 (with and without the Barcan formula) and the classical predicate logic (with one-sorted variable). We present a purely proof-theoretic proof-transformation method, reducing an LK-proof of an interpreted formula to a modal proof.