scholarly journals Recursive enumerability and elementary frame definability in predicate modal logic

2019 ◽  
Vol 30 (2) ◽  
pp. 549-560 ◽  
Author(s):  
Mikhail Rybakov ◽  
Dmitry Shkatov
Keyword(s):  
Modal Logic ◽  
Modal Logics ◽  
The Other ◽  
Kripke Semantics ◽  
First Order ◽  
Elementary Class ◽  
Recursive Enumerability ◽  
Kripke Frames ◽  
Recursively Enumerable ◽  
The Relationship

Abstract We investigate the relationship between recursive enumerability and elementary frame definability in first-order predicate modal logic. On one hand, it is well known that every first-order predicate modal logic complete with respect to an elementary class of Kripke frames, i.e. a class of frames definable by a classical first-order formula, is recursively enumerable. On the other, numerous examples are known of predicate modal logics, based on ‘natural’ propositional modal logics with essentially second-order Kripke semantics, that are either not recursively enumerable or Kripke incomplete. This raises the question of whether every Kripke complete, recursively enumerable predicate modal logic can be characterized by an elementary class of Kripke frames. We answer this question in the negative, by constructing a normal predicate modal logic which is Kripke complete, recursively enumerable, but not complete with respect to an elementary class of frames. We also present an example of a normal predicate modal logic that is recursively enumerable, Kripke complete, and not complete with respect to an elementary class of rooted frames, but is complete with respect to an elementary class of frames that are not rooted.

Algorithmic properties of first-order modal logics of finite Kripke frames in restricted languages

2020 ◽  
Vol 30 (7) ◽  
pp. 1305-1329 ◽  
Author(s):  
Mikhail Rybakov ◽  
Dmitry Shkatov
Keyword(s):  
Possible Worlds ◽  
Modal Logics ◽  
Finite Frames ◽  
First Order ◽  
Kripke Frames ◽  
Finite Frame ◽  
Recursively Enumerable ◽  
Finite Set ◽  
Individual Variables ◽  
Natural Classes

Abstract We study the effect of restricting the number of individual variables, as well as the number and arity of predicate letters, in languages of first-order predicate modal logics of finite Kripke frames on the logics’ algorithmic properties. A finite frame is a frame with a finite set of possible worlds. The languages we consider have no constants, function symbols or the equality symbol. We show that most predicate modal logics of natural classes of finite Kripke frames are not recursively enumerable—more precisely, $\varPi ^0_1$-hard—in languages with three individual variables and a single monadic predicate letter. This applies to the logics of finite frames of the predicate extensions of the sublogics of propositional modal logics $\textbf{GL}$, $\textbf{Grz}$ and $\textbf{KTB}$—among them, $\textbf{K}$, $\textbf{T}$, $\textbf{D}$, $\textbf{KB}$, $\textbf{K4}$ and $\textbf{S4}$.

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.

On axiomatising products of Kripke frames

2000 ◽  
Vol 65 (2) ◽  
pp. 923-945 ◽  
Author(s):  
Ágnes Kurucz
Keyword(s):  
Modal Logic ◽  
The Other ◽  
Modal Language ◽  
First Order ◽  
Kripke Frames ◽  
Other Hand ◽  
The Many ◽  
By Products

AbstractIt is shown that the many-dimensional modal logic Kn, 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, Kn is determined by a class of frames satisfying a single first-order sentence.

scholarly journals Fine’s Theorem on First-Order Complete Modal Logics

2020 ◽  
pp. 316-334
Author(s):  
Robert Goldblatt
Keyword(s):  
Modal Logic ◽  
Modal Logics ◽  
Modal Algebra ◽  
First Order ◽  
Kripke Frames ◽  
Canonical Frame ◽  
History Of ◽  
Order Completeness

Fine’s influential Canonicity Theorem states that if a modal logic is determined by a first-order definable class of Kripke frames, then it is valid in its canonical frames. This article reviews the background and context of this result, and the history of its impact on further research. It then develops a new characterization of when a logic is canonically valid, providing a precise point of distinction with the property of first-order completeness. The ultimate point is that the construction of the canonical frame of a modal algebra does not commute with the ultrapower construction.

scholarly journals KRIPKE COMPLETENESS OF STRICTLY POSITIVE MODAL LOGICS OVER MEET-SEMILATTICES WITH OPERATORS

2019 ◽  
Vol 84 (02) ◽  
pp. 533-588 ◽  
Author(s):  
STANISLAV KIKOT ◽  
AGI KURUCZ ◽  
YOSHIHITO TANAKA ◽  
FRANK WOLTER ◽  
MICHAEL ZAKHARYASCHEV
Keyword(s):  
Computer Science ◽  
Monotone Operators ◽  
Modal Logics ◽  
Consequence Relations ◽  
First Order ◽  
Completeness Problem ◽  
Relational Structures ◽  
Kripke Frames

AbstractOur concern is the completeness problem for spi-logics, that is, sets of implications between strictly positive formulas built from propositional variables, conjunction and modal diamond operators. Originated in logic, algebra and computer science, spi-logics have two natural semantics: meet-semilattices with monotone operators providing Birkhoff-style calculi and first-order relational structures (aka Kripke frames) often used as the intended structures in applications. Here we lay foundations for a completeness theory that aims to answer the question whether the two semantics define the same consequence relations for a given spi-logic.

scholarly journals EVERYONE KNOWS THAT SOMEONE KNOWS: QUANTIFIERS OVER EPISTEMIC AGENTS

2019 ◽  
Vol 12 (2) ◽  
pp. 255-270 ◽  
Author(s):  
PAVEL NAUMOV ◽  
JIA TAO
Keyword(s):  
Modal Logic ◽  
Epistemic Logic ◽  
Possible Worlds ◽  
Completeness Theorem ◽  
Kripke Semantics ◽  
Possible Worlds Semantics ◽  
Distributed Knowledge ◽  
First Order ◽  
Soundness And Completeness ◽  
First Order Modal Logic

AbstractModal logic S5 is commonly viewed as an epistemic logic that captures the most basic properties of knowledge. Kripke proved a completeness theorem for the first-order modal logic S5 with respect to a possible worlds semantics. A multiagent version of the propositional S5 as well as a version of the propositional S5 that describes properties of distributed knowledge in multiagent systems has also been previously studied. This article proposes a version of S5-like epistemic logic of distributed knowledge with quantifiers ranging over the set of agents, and proves its soundness and completeness with respect to a Kripke semantics.

Thermodynamics of Aluminum Deoxidization Equilibria in GCr18Mo Bearing-Steel

2018 ◽  
Vol 382 ◽  
pp. 80-85 ◽  
Author(s):  
Xin Su ◽  
Shu Qiang Guo ◽  
Meng Ran Qiao ◽  
Hong Yan Zheng ◽  
Li Bin Qin
Keyword(s):  
Bearing Steel ◽  
Second Order ◽  
The Other ◽  
Interaction Parameters ◽  
Interaction Coefficients ◽  
Concentration Region ◽  
Order Interaction ◽  
First Order ◽  
The Relationship ◽  
Reaction Equilibrium

Based on the predecessors of thermodynamic data, the relationship between aluminum contents and oxygen contents of the aluminum deoxidization reaction was calculated. And the influence of activity coefficient to the reaction equilibrium in bearing-steel is analyzed. First-order and second-order interaction coefficients were used to calculate and draw the equilibrium curves, respectively. The effects of different temperature and different interaction parameters on the deoxidization equilibrium curves were studied. And through the curve the influence of the change of aluminum contents to the activity can be known. The trend of the curve with first-order interaction parameters is consistent with the curve with first-order and second-order interaction parameters at the low Al concentration region. And the oxygen contents of curve with first-order interaction parameters are higher than the other curve at the high Al concentration region

Liberated Brouwerian Modal Logic

Dialogue ◽  
1974 ◽  
Vol 13 (3) ◽  
pp. 505-514 ◽  
Author(s):  
Charles G. Morgan
Keyword(s):  
Modal Logic ◽  
Modal Logics ◽  
Possible World ◽  
Modal Systems ◽  
Relationship Of ◽  
The Relationship

In an attempt to “purify” logic of existential presuppositions, attention has recently focused on modal logics, where one usually assumes that at least one possible world exists. Systems very analogous to some of the standard modal systems have been developed which drop this presupposition. We will here treat the removal of the existential assumption from Brouwerian modal logic and discuss the relationship of the system so derived to other modal systems.

scholarly journals Here and There with Arithmetic

2021 ◽  
pp. 1-15
Author(s):  
VLADIMIR LIFSCHITZ
Keyword(s):  
Programming Language ◽  
Propositional Logic ◽  
Answer Set Programming ◽  
The Other ◽  
Arithmetic Operations ◽  
First Order ◽  
The Relationship ◽  
Answer Set

Abstarct In the theory of answer set programming, two groups of rules are called strongly equivalent if, informally speaking, they have the same meaning in any context. The relationship between strong equivalence and the propositional logic of here-and-there allows us to establish strong equivalence by deriving rules of each group from rules of the other. In the process, rules are rewritten as propositional formulas. We extend this method of proving strong equivalence to an answer set programming language that includes operations on integers. The formula representing a rule in this language is a first-order formula that may contain comparison symbols among its predicate constants, and symbols for arithmetic operations among its function constants. The paper is under consideration for acceptance in TPLP.

scholarly journals Neighbourhood semantics for graded modal logic

2021 ◽  
Author(s):  
Jinsheng Chen ◽  
Hans Van Ditmarsch ◽  
Giuseppe Greco ◽  
Apostolos Tzimoulis
Keyword(s):  
Modal Logic ◽  
First Order ◽  
Kripke Frames ◽  
Definition Of

We introduce a class of neighbourhood frames for graded modal logic using an operation from Kripke frames to neighbourhood frames. This class of neighbourhood frames is shown to be first-order definable but not modally definable. We also obtain a new definition of graded bisimulation by modifying the definition of monotonic bisimulation.  

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