scholarly journals THE EXPRESSIVE POWER OF MEMORY LOGICS

2011 ◽  
Vol 4 (2) ◽  
pp. 290-318 ◽  
Author(s):  
CARLOS ARECES ◽  
DIEGO FIGUEIRA ◽  
SANTIAGO FIGUEIRA ◽  
SERGIO MERA
Keyword(s):  
Modal Logic ◽  
Expressive Power ◽  
Modal Logics ◽  
Hybrid Logic ◽  
Satisfiability Problem ◽  
Current Node

We investigate the expressive power of memory logics. These are modal logics extended with the possibility to store (or remove) the current node of evaluation in (or from) a memory, and to perform membership tests on the current memory. From this perspective, the hybrid logic ℋℒ (↓), for example, can be thought of as a particular case of a memory logic where the memory is an indexed list of elements of the domain.This work focuses in the case where the memory is a set, and we can test whether the current node belongs to the set or not. We prove that, in terms of expressive power, the memory logics we discuss here lie between the basic modal logic ${\cal K}$ and ℋℒ (↓). We show that the satisfiability problem of most of the logics we cover is undecidable. The only logic with a decidable satisfiability problem is obtained by imposing strong constraints on which elements can be memorized.

scholarly journals Coalgebraic Reasoning with Global Assumptions in Arithmetic Modal Logics

2022 ◽  
Vol 23 (2) ◽  
pp. 1-34
Author(s):  
Clemens Kupke ◽  
Dirk Pattinson ◽  
Lutz Schröder
Keyword(s):  
Modal Logic ◽  
Upper Bound ◽  
Practical Reasoning ◽  
Linear Time ◽  
Modal Logics ◽  
Hybrid Logic ◽  
Complexity Bound ◽  
Elimination Algorithm ◽  
Polynomial Inequalities ◽  
Global Caching

We establish a generic upper bound ExpTime for reasoning with global assumptions (also known as TBoxes) in coalgebraic modal logics. Unlike earlier results of this kind, our bound does not require a tractable set of tableau rules for the instance logics, so that the result applies to wider classes of logics. Examples are Presburger modal logic, which extends graded modal logic with linear inequalities over numbers of successors, and probabilistic modal logic with polynomial inequalities over probabilities. We establish the theoretical upper bound using a type elimination algorithm. We also provide a global caching algorithm that potentially avoids building the entire exponential-sized space of candidate states, and thus offers a basis for practical reasoning. This algorithm still involves frequent fixpoint computations; we show how these can be handled efficiently in a concrete algorithm modelled on Liu and Smolka’s linear-time fixpoint algorithm. Finally, we show that the upper complexity bound is preserved under adding nominals to the logic, i.e., in coalgebraic hybrid logic.

A Modal Defeasible Reasoner of Deontic Logic for the Semantic Web

2011 ◽  
Vol 7 (1) ◽  
pp. 18-43 ◽  
Author(s):  
Efstratios Kontopoulos ◽  
Nick Bassiliades ◽  
Guido Governatori ◽  
Grigoris Antoniou
Keyword(s):  
Modal Logic ◽  
Semantic Web ◽  
Deontic Logic ◽  
Expressive Power ◽  
Modal Logics ◽  
Cognitive State ◽  
Privacy Requirements ◽  
Defeasible Logic ◽  
Conflicting Information ◽  
Web Information

Defeasible logic is a non-monotonic formalism that deals with incomplete and conflicting information, whereas modal logic deals with the concepts of necessity and possibility. These types of logics play a significant role in the emerging Semantic Web, which enriches the available Web information with meaning, leading to better cooperation between end-users and applications. Defeasible and modal logics, in general, and, particularly, deontic logic provide means for modeling agent communities, where each agent is characterized by its cognitive profile and normative system, as well as policies, which define privacy requirements, access permissions, and individual rights. Toward this direction, this article discusses the extension of DR-DEVICE, a Semantic Web-aware defeasible reasoner, with a mechanism for expressing modal logic operators, while testing the implementation via deontic logic operators, concerned with obligations, permissions, and related concepts. The motivation behind this work is to develop a practical defeasible reasoner for the Semantic Web that takes advantage of the expressive power offered by modal logics, accompanied by the flexibility to define diverse agent behaviours. A further incentive is to study the various motivational notions of deontic logic and discuss the cognitive state of agents, as well as the interactions among them.

scholarly journals Constructive interpolation in hybrid logic

2003 ◽  
Vol 68 (2) ◽  
pp. 463-480 ◽  
Author(s):  
Patrick Blackburn ◽  
Maarten Marx
Keyword(s):  
Modal Logic ◽  
Strong Evidence ◽  
Interpolation Property ◽  
Modal Logics ◽  
Hybrid Logic ◽  
Basic Algorithm ◽  
Technical Problems ◽  
First Order ◽  
Elementary Class ◽  
Interpolation Algorithms

AbstractCraig's interpolation lemma (if φ → ψ is valid, then φ → θ and θ → ψ 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.

A Modal Defeasible Reasoner of Deontic Logic for the Semantic Web

Semantic Web ◽  
2013 ◽  
pp. 140-167
Author(s):  
Efstratios Kontopoulos ◽  
Nick Bassiliades ◽  
Guido Governatori ◽  
Grigoris Antoniou
Keyword(s):  
Modal Logic ◽  
Semantic Web ◽  
Deontic Logic ◽  
Expressive Power ◽  
Modal Logics ◽  
Cognitive State ◽  
Privacy Requirements ◽  
Defeasible Logic ◽  
Conflicting Information ◽  
Web Information

Defeasible logic is a non-monotonic formalism that deals with incomplete and conflicting information, whereas modal logic deals with the concepts of necessity and possibility. These types of logics play a significant role in the emerging Semantic Web, which enriches the available Web information with meaning, leading to better cooperation between end-users and applications. Defeasible and modal logics, in general, and, particularly, deontic logic provide means for modeling agent communities, where each agent is characterized by its cognitive profile and normative system, as well as policies, which define privacy requirements, access permissions, and individual rights. Toward this direction, this article discusses the extension of DR-DEVICE, a Semantic Web-aware defeasible reasoner, with a mechanism for expressing modal logic operators, while testing the implementation via deontic logic operators, concerned with obligations, permissions, and related concepts. The motivation behind this work is to develop a practical defeasible reasoner for the Semantic Web that takes advantage of the expressive power offered by modal logics, accompanied by the flexibility to define diverse agent behaviours. A further incentive is to study the various motivational notions of deontic logic and discuss the cognitive state of agents, as well as the interactions among them.

scholarly journals An analogue of Bull's theorem for Hybrid Logic

10.29007/bhm3 ◽  
2018 ◽  
Author(s):  
Claudette Robinson ◽  
Willem Conradie
Keyword(s):  
Modal Logic ◽  
Expressive Power ◽  
Hybrid Logic ◽  
Current Paper ◽  
Algebraic Methods ◽  
Normal Extension ◽  
Finite Model ◽  
Kripke Models ◽  
Original Result ◽  
The World

Hybrid logic extends modal logic with a special sort of variables, called nominals, which are evaluated to singletons in Kripke models by valuations, thus acting as names for states in models. Various syntactic mechanisms for exploiting and enhancing the expressive power gained through the addition of nominals can be included, most characteristically the satisfaction operator, @_ip, allowing one to express that p holds at the world named by a nominal i.R.A. Bull famously proved that each normal extension of S4.3 has the finite modelproperty. In the current paper, we prove a hybrid analogue of Bull's result. Like the proof of Bull's original result, ours is algebraic, and thus our secondary aim with this work is to illustrate the usefulness of algebraic methods within hybrid logic research, a field where such methods have been largely ignored.

scholarly journals Admissible Inference Rules and Semantic Property of Modal Logics

2021 ◽  
Vol 37 ◽  
pp. 104-117
Author(s):  
V.V. Rimatskiy ◽  
Keyword(s):  
Modal Logic ◽  
Inference Rule ◽  
Expressive Power ◽  
Modal Logics ◽  
Human Reasoning ◽  
Inference Rules ◽  
Semantic Property ◽  
Admissible Rule ◽  
Admissible Inference Rules ◽  
Deductive Power

Firstly semantic property of nonstandart logics were described by formulas which are peculiar to studied a models in general, and do not take to consideration a variable conditions and a changing assumptions. Evidently the notion of inference rule generalizes the notion of formulas and brings us more flexibility and more expressive power to model human reasoning and computing. In 2000-2010 a few results on describing of explicit bases for admissible inference rules for nonstandard logics (S4, K4, H etc.) appeared. The key property of these logics was weak co-cover property. Beside the improvement of deductive power in logic, an admissible rule are able to describe some semantic property of given logic. We describe a semantic property of modal logics in term of admissibility of given set of inference rules. We prove that modal logic over logic 𝐺𝐿 enjoys weak co-cover property iff all given rules are admissible for logic.

A Modal Semantics of Negation in Logic Programming

1992 ◽  
Vol 16 (3-4) ◽  
pp. 231-262
Author(s):  
Philippe Balbiani
Keyword(s):  
Modal Logic ◽  
Logic Programming ◽  
Modal Logics ◽  
Kripke Models ◽  
Logic Programs ◽  
Modal Semantics ◽  
Soundness And Completeness ◽  
Modal Completion ◽  
Well Foundedness

The beauty of modal logics and their interest lie in their ability to represent such different intensional concepts as knowledge, time, obligation, provability in arithmetic, … according to the properties satisfied by the accessibility relations of their Kripke models (transitivity, reflexivity, symmetry, well-foundedness, …). The purpose of this paper is to study the ability of modal logics to represent the concepts of provability and unprovability in logic programming. The use of modal logic to study the semantics of logic programming with negation is defended with the help of a modal completion formula. This formula is a modal translation of Clack’s formula. It gives soundness and completeness proofs for the negation as failure rule. It offers a formal characterization of unprovability in logic programs. It characterizes as well its stratified semantics.

scholarly journals Modal context restriction for multiagent BDI logics

2021 ◽  
Author(s):  
Marcin Dziubiński
Keyword(s):  
Multiagent Systems ◽  
Modal Logics ◽  
Satisfiability Problem ◽  
Language Restriction ◽  
Multimodal Logics ◽  
Fix Point

AbstractWe present and discuss a novel language restriction for modal logics for multiagent systems, called modal context restriction, that reduces the complexity of the satisfiability problem from EXPTIME complete to NPTIME complete. We focus on BDI multimodal logics that contain fix-point modalities like common beliefs and mutual intentions together with realism and introspection axioms. We show how this combination of modalities and axioms affects complexity of the satisfiability problem and how it can be reduced by restricting the modal context of formulas.

scholarly journals Canonical formulas via locally finite reducts and generalized dualities

10.29007/hgbj ◽  
2018 ◽  
Author(s):  
Nick Bezhanishvili
Keyword(s):  
Modal Logic ◽  
Intuitionistic Logic ◽  
Algebraic Approach ◽  
Modal Logics ◽  
Substructural Logic ◽  
Locally Finite ◽  
Canonical Formulas

The method of canonical formulas is a powerful tool for investigating intuitionistic and modal logics. In this talk I will discuss an algebraic approach to this method. I will mostly concentrate on the case of intuitionistic logic. But I will also review the case of modal logic and possible generalizations to substructural logic.

PROOF SYSTEMS FOR VARIOUS FDE-BASED MODAL LOGICS

2019 ◽  
Vol 13 (4) ◽  
pp. 720-747
Author(s):  
SERGEY DROBYSHEVICH ◽  
HEINRICH WANSING
Keyword(s):  
Modal Logic ◽  
Sequent Calculus ◽  
Axiom System ◽  
Modal Logics ◽  
Cut Elimination ◽  
Proof Systems ◽  
Image Position ◽  
Axiomatic Extension

AbstractWe present novel proof systems for various FDE-based modal logics. Among the systems considered are a number of Belnapian modal logics introduced in Odintsov & Wansing (2010) and Odintsov & Wansing (2017), as well as the modal logic KN4 with strong implication introduced in Goble (2006). In particular, we provide a Hilbert-style axiom system for the logic $BK^{\square - } $ and characterize the logic BK as an axiomatic extension of the system $BK^{FS} $. For KN4 we provide both an FDE-style axiom system and a decidable sequent calculus for which a contraction elimination and a cut elimination result are shown.

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