Algorithmic problems concerning first-order definability of modal formulas on the class of all finite frames

Studia Logica ◽  
1995 ◽  
Vol 55 (3) ◽  
pp. 421-448 ◽  
Author(s):  
A. V. Chagrov ◽  
L. A. Chagrova
Keyword(s):  
Finite Frames ◽  
First Order ◽  
Algorithmic Problems ◽  
First Order Definability

Simulating polyadic modal logics by monadic ones

2003 ◽  
Vol 68 (2) ◽  
pp. 419-462 ◽  
Author(s):  
George Goguadze ◽  
Carla Piazza ◽  
Yde Venema
Keyword(s):  
Modal Logics ◽  
Finite Model ◽  
Model Property ◽  
Finite Model Property ◽  
First Order ◽  
First Order Definability

AbstractWe 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 Λsim in the diamond language with each logic Λ 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.

An undecidable problem in correspondence theory

1991 ◽  
Vol 56 (4) ◽  
pp. 1261-1272 ◽  
Author(s):  
L. A. Chagrova
Keyword(s):  
Correspondence Theory ◽  
First Order ◽  
Undecidable Problem ◽  
First Order Definability ◽  
Effective Procedures

In this paper we prove undecidability of first-order definability of propositional formulas. The main result is proved for intuitionistic formulas, but it remains valid for other kinds of propositional formulas by analogous arguments or with the help of various translations.For general background on correspondence theory the reader is referred to van Benthem [1], [2] (see [3] for a survey of recent results).The method for the proofs of undecidability in this paper will be to simulate calculations of a Minsky machine by intuitionistic formulas. §3 concerns this simulation. Effective procedures for the construction of simulating modal formulas can be found in the literature (cf. [4]).The principal results of the paper are in §4. §5 gives some further undecidability results, that will be proved in another paper by modification of the method of this paper.I am indebted to the referee for drawing my attention to some errors in an earlier version of this paper.

scholarly journals Functional first order definability of LRTP

2010 ◽  
Vol 14 (48) ◽  
Author(s):  
José Raymundo Marcial Romero ◽  
José Antonio Hernández
Keyword(s):  
First Order ◽  
First Order Definability

First-order definability

2004 ◽  
pp. 151-174
Author(s):  
Richard Lassaigne ◽  
Michel de Rougemont
Keyword(s):  
First Order ◽  
First Order Definability

Algorithmic problems and hierarchies of first-order languages

1987 ◽  
Vol 26 (4) ◽  
pp. 241-252
Author(s):  
Yu. M. Vazhenin
Keyword(s):  
First Order ◽  
Algorithmic Problems

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 $\textbf{S4}$.

First-order definability in modal logic

1975 ◽  
Vol 40 (1) ◽  
pp. 35-40 ◽  
Author(s):  
R. I. Goldblatt
Keyword(s):  
Modal Logic ◽  
Binary Relation ◽  
Second Order ◽  
Order Conditions ◽  
Kripke Models ◽  
Order Condition ◽  
First Order ◽  
Axiom Systems ◽  
Set Of Points ◽  
First Order Definability

In the early days of the development of Kripke-style semantics for modal logic a great deal of effort was devoted to showing that particular axiom systems were characterised by a class of models describable by a first-order condition on a binary relation. For a time the approach seemed all encompassing, but recent work by Thomason [6] and Fine [2] has shown it to be somewhat limited—there are logics not determined by any class of Kripke models at all. In fact it now seems that modal logic is basically second-order in nature, in that any system may be analysed in terms of structures having a nominated class of second-order individuals (subsets) that serve as interpretations of propositional variables (cf. [7]). The question has thus arisen as to how much of modal logic can be handled in a first-order way, and precisely which modal sentences are determined by first-order conditions on their models. In this paper we present a model-theoretic characterisation of this class of sentences, and show that it does not include the much discussed LMp → MLp.Definition 1. A modal frame ℱ = 〈W, R〉 consists of a set W on which a binary relation R is defined. A valuation V on ℱ is a function that associates with each propositional variable p a subset V(p) of W (the set of points at which p is “true”).

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