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.

Finite Kripke models and predicate logics of provability

1990 ◽  
Vol 55 (3) ◽  
pp. 1090-1098 ◽  
Author(s):  
Sergei Artemov ◽  
Giorgie Dzhaparidze
Keyword(s):  
Modal Logic ◽  
Kripke Model ◽  
Finite Model ◽  
Kripke Models ◽  
Model Property ◽  
Logical Formula ◽  
Finite Model Property ◽  
Modal Predicate Logics ◽  
The One ◽  
Arithmetical Interpretation

AbstractThe paper proves a predicate version of Solovay's well-known theorem on provability interpretations of modal logic:If a closed modal predicate-logical formula R is not valid in some finite Kripke model, then there exists an arithmetical interpretation f such that PA ⊬ fR.This result implies the arithmetical completeness of arithmetically correct modal predicate logics with the finite model property (including the one-variable fragments of QGL and QS). The proof was obtained by adding “the predicate part” as a specific addition to the standard Solovay construction.

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 An Arithmetically Complete Predicate Modal Logic

2021 ◽  
Author(s):  
George Tourlakis ◽  
Yunge Hao
Keyword(s):  
Modal Logic ◽  
Current Paper ◽  
Kripke Models ◽  
First Order ◽  
Semantic Completeness ◽  
First Order Modal Logic ◽  
Conservation Theorem ◽  
Order Extension

This paper investigates a first-order extension of GL called \(\textup{ML}^3\). We outline briefly the history that led to \(\textup{ML}^3\), its key properties and some of its toolbox: the \emph{conservation theorem}, its cut-free Gentzenisation, the ``formulators'' tool. Its semantic completeness (with respect to finite reverse well-founded Kripke models) is fully stated in the current paper and the proof is retold here. Applying the Solovay technique to those models the present paper establishes its main result, namely, that \(\textup{ML}^3\) is arithmetically complete. As expanded below, \(\textup{ML}^3\) is a first-order modal logic that along with its built-in ability to simulate general classical first-order provability―"\(\Box\)" simulating the the informal classical "\(\vdash\)"―is also arithmetically complete in the Solovay sense.

scholarly journals Losing connection: the modal logic of definable link deletion

2020 ◽  
Vol 30 (3) ◽  
pp. 715-743
Author(s):  
Dazhu Li
Keyword(s):  
Modal Logic ◽  
Characterization Theorem ◽  
Expressive Power ◽  
Finite Model ◽  
Model Property ◽  
Tree Model ◽  
First Order ◽  
Language Semantics ◽  
New Type ◽  
Player Game

Abstract In this article, we start with a two-player game that models communication under adverse circumstances in everyday life and study it from the perspective of a modal logic of graphs, where links can be deleted locally according to definitions available to the adversarial player. We first introduce a new language, semantics and some typical validities. We then formulate a new type of first-order translation for this modal logic and prove its correctness. Then, a novel notion of bisimulation is proposed that leads to a characterization theorem for the logic as a fragment of first-order logic, and a further investigation is made of its expressive power against hybrid modal languages. Next, we discuss how to axiomatize this logic of link deletion, using dynamic-epistemic logics as a contrast. Finally, we show that our new modal logic lacks both the tree model property and the finite model property and that its satisfiability problem is undecidable.

scholarly journals Formula size games for modal logic and μ-calculus

2019 ◽  
Vol 29 (8) ◽  
pp. 1311-1344 ◽  
Author(s):  
Lauri T Hella ◽  
Miikka S Vilander
Keyword(s):  
Modal Logic ◽  
Optimal Strategy ◽  
Order Logic ◽  
First Order Logic ◽  
Kripke Models ◽  
First Order ◽  
Modal Operators ◽  
Crucial Difference ◽  
Definition Of ◽  
Person Game

Abstract We propose a new version of formula size game for modal logic. The game characterizes the equivalence of pointed Kripke models up to formulas of given numbers of modal operators and binary connectives. Our game is similar to the well-known Adler–Immerman game. However, due to a crucial difference in the definition of positions of the game, its winning condition is simpler, and the second player does not have a trivial optimal strategy. Thus, unlike the Adler–Immerman game, our game is a genuine two-person game. We illustrate the use of the game by proving a non-elementary succinctness gap between bisimulation invariant first-order logic $\textrm{FO}$ and (basic) modal logic $\textrm{ML}$. We also present a version of the game for the modal $\mu $-calculus $\textrm{L}_\mu $ and show that $\textrm{FO}$ is also non-elementarily more succinct than $\textrm{L}_\mu $.

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 The Study of The Validity of The Surface Structure of Holy Quran

2019 ◽  
Vol 19 (1) ◽  
Author(s):  
Esmaeil Rezaei
Keyword(s):  
Data Collection ◽  
Surface Structure ◽  
Islamic Law ◽  
Current Paper ◽  
Data Collection Tool ◽  
Human Reason ◽  
Human Happiness ◽  
The World ◽  
Holy Quran ◽  
The Holy Quran

The validity of the surface structure of the Holy Qur'an is one of the most important Qur'anic debates, which also affects the issues of Islamic law. The meaning of surface structure of the Qur'an is that the surface structure of the Qur'an is understandable to every reader and listener, as well as any human text, and that which man understands the surface structure of the Qur'an by means of verbal and illogical indications. The current paper examines the validity of the surface structure of the Holy Quran, following expanding the conceptual subject of the research.  Given the thematic documents presented, this is a descriptive-analytical paper with library sources data collection tool. The findings indicate that: The most important reason for believing in the validity of the Holy Qur'an surface structure is that this holy scripture includes the plan of human happiness in the world and the Hereafter. Therefore, if the surface structure of the Holy Quran is not justified, how one can use its surface structure to answer difficult questions that limited human reason is incapable of answering. In addition, the Holy Qur'an, although revelation and has been revealed by God, but because it is revealed in the form of words, it is governed by the principles and rules governing all languages, among which are the validity of their surface structure for those familiar with the language. Keabsahan struktur permukaan Al-Qur'an adalah salah satu perdebatan Al-Qur'an yang paling penting karena  mempengaruhi masalah hukum Islam. Makna struktur permukaan Al-Qur'an adalah bahwa struktur permukaan Al-Qur'an dapat dipahami oleh setiap pembaca dan pendengar, serta teks manusia mana pun, dan apa yang dipahami manusia oleh struktur permukaan Al-Qur'an dengan cara indikasi verbal dan tidak logis. Penelitian ini  meneliti validitas struktur permukaan Al-Qur'an, dengan memperluas subjek konseptual penelitian. Mengingat dokumen tematik yang disajikan, pendekatan deskriptif-analitis dengan alat pengumpulan data sumber perpustakaan. Temuan menunjukkan bahwa: alasan paling penting untuk percaya pada validitas struktur permukaan Al-Qur'an adalah bahwa kitab suci ini mencakup rencana kebahagiaan manusia di dunia dan akhirat. Karena itu, jika struktur permukaan Al-Qur'an tidak benar, bagaimana seseorang dapat menggunakan struktur permukaannya untuk menjawab pertanyaan-pertanyaan sulit yang membatasi akal manusia. Selain itu, Alquran, meskipun wahyu dari Allah,  tetapi karena diturunkan dalam bentuk kata-kata, diatur oleh prinsip dan aturan yang mengatur semua bahasa, di antaranya adalah validitas struktur permukaan untuk mereka yang terbiasa dengan bahasa.

scholarly journals Unifying Class-Based Representation Formalisms

1999 ◽  
Vol 11 ◽  
pp. 199-240 ◽  
Author(s):  
D. Calvanese ◽  
M. Lenzerini ◽  
D. Nardi
Keyword(s):  
Knowledge Representation ◽  
Description Logic ◽  
Description Logics ◽  
Expressive Power ◽  
Finite Model ◽  
Functional Dependencies ◽  
Semantic Data ◽  
Representation Systems ◽  
Frame Systems ◽  
Essential Set

The notion of class is ubiquitous in computer science and is central in many formalisms for the representation of structured knowledge used both in knowledge representation and in databases. In this paper we study the basic issues underlying such representation formalisms and single out both their common characteristics and their distinguishing features. Such investigation leads us to propose a unifying framework in which we are able to capture the fundamental aspects of several representation languages used in different contexts. The proposed formalism is expressed in the style of description logics, which have been introduced in knowledge representation as a means to provide a semantically well-founded basis for the structural aspects of knowledge representation systems. The description logic considered in this paper is a subset of first order logic with nice computational characteristics. It is quite expressive and features a novel combination of constructs that has not been studied before. The distinguishing constructs are number restrictions, which generalize existence and functional dependencies, inverse roles, which allow one to refer to the inverse of a relationship, and possibly cyclic assertions, which are necessary for capturing real world domains. We are able to show that it is precisely such combination of constructs that makes our logic powerful enough to model the essential set of features for defining class structures that are common to frame systems, object-oriented database languages, and semantic data models. As a consequence of the established correspondences, several significant extensions of each of the above formalisms become available. The high expressiveness of the logic we propose and the need for capturing the reasoning in different contexts forces us to distinguish between unrestricted and finite model reasoning. A notable feature of our proposal is that reasoning in both cases is decidable. We argue that, by virtue of the high expressive power and of the associated reasoning capabilities on both unrestricted and finite models, our logic provides a common core for class-based representation formalisms.

Canonical formulas for K4. Part I: Basic results

1992 ◽  
Vol 57 (4) ◽  
pp. 1377-1402 ◽  
Author(s):  
Michael Zakharyaschev
Keyword(s):  
Fundamental Result ◽  
Normal Extension ◽  
Finite Model ◽  
Model Property ◽  
Finite Model Property ◽  
Canonical Formulas ◽  
Morphic Image ◽  
Normal Logic ◽  
A New Technique ◽  
Almost All

This paper presents a new technique for handling modal logics with transitive frames, i.e. extensions of the modal system K4. In effect, the technique is based on the following fundamental result, to be obtained below in §3.Given a formula φ, we can effectively construct finite frames 1, …, n which completely characterize the set of all transitive general frames refuting φ. More exactly, an arbitrary general frame refutes φ iff contains a (not necessarily generated) subframe such that (1) i, for some i ϵ {1, …, n}, is a p-morphic image of (after Fine [1985] we say is subreducible to i), (2) is cofinal in , and (3) every point in that is not in does not get into “closed domains” which are uniquely determined in i, by φ.This purely technical result has, as it turns out, rather unexpected and profound consequences. For instance, it follows at once that if φ determines no closed domains in the frames 1, …, n associated with it, then the normal extension of K4 generated by φ has the finite model property and so is decidable. Moreover, every normal logic axiomatizable by any (even infinite) set of such formulas φ also has the finite model property. This observation would not possibly merit any special attention, were it not for the fact that the class of such logics contains almost all the standard systems within the field of K4 (at least all those mentioned by Segerberg [1971] or Bull and Segerberg [1984]), all logics containing S4.3, all subframe logics of Fine [1985], and a continuum of other logics as well.

MODAL LOGICS OF METRIC SPACES

2014 ◽  
Vol 8 (1) ◽  
pp. 178-191 ◽  
Author(s):  
GURAM BEZHANISHVILI ◽  
DAVID GABELAIA ◽  
JOEL LUCERO-BRYAN
Keyword(s):  
Modal Logic ◽  
Metric Space ◽  
Metric Spaces ◽  
Modal Logics ◽  
Finite Model ◽  
Finite Model Property ◽  
Classic Result ◽  
Topological Closure ◽  
Image Position

AbstractIt is a classic result (McKinsey & Tarski, 1944; Rasiowa & Sikorski, 1963) that if we interpret modal diamond as topological closure, then the modal logic of any dense-in-itself metric space is the well-known modal system S4. In this paper, as a natural follow-up, we study the modal logic of an arbitrary metric space. Our main result establishes that modal logics arising from metric spaces form the following chain which is order-isomorphic (with respect to the ⊃ relation) to the ordinal ω + 3:$S4.Gr{z_1} \supset S4.Gr{z_2} \supset S4.Gr{z_3} \supset \cdots \,S4.Grz \supset S4.1 \supset S4.$It follows that the modal logic of an arbitrary metric space is finitely axiomatizable, has the finite model property, and hence is decidable.

Sign in / Sign up

Export Citation Format

Share Document