Finite Model Property Research Articles

Spatial logic of tangled closure operators and modal mu-calculus

There has been renewed interest in recent years in McKinsey and Tarski's interpretation of modal logic in topological spaces and their proof that S4 is the logic of any separable dense-in-itself metric space. Here we extend this work to the modal mu-calculus and to a logic of tangled closure operators that was developed by Fernández-Duque after these two languages had been shown by Dawar and Otto to have the same expressive power over finite transitive Kripke models. We prove that this equivalence remains true over topological spaces.We extend the McKinsey–Tarski topological ‘dissection lemma’. We also take advantage of the fact (proved by us elsewhere) that various tangled closure logics with and without the universal modality ∀ have the finite model property in Kripke semantics. These results are used to construct a representation map (also called a d-p-morphism) from any dense-in-itself metric space X onto any finite connected locally connected serial transitive Kripke frame.This yields completeness theorems over X for a number of languages: (i) the modal mu-calculus with the closure operator ◇; (ii) ◇ and the tangled closure operators 〈t〉 (in fact 〈t〉 can express ◇); (iii) ◇,∀; (iv) ◇,∀,〈t〉; (v) the derivative operator 〈d〉; (vi) 〈d〉 and the associated tangled closure operators 〈dt〉; (vii) 〈d〉,∀; (viii) 〈d〉,∀,〈dt〉. Soundness also holds, if: (a) for languages with ∀, X is connected; (b) for languages with 〈d〉, X validates the well-known axiom G1. For countable languages without ∀, we prove strong completeness. We also show that in the presence of ∀, strong completeness fails if X is compact and locally connected.

Undecidability of the transitive graded modal logic with converse

We extend the language of the modal logic $\mathbf{K4}$ of transitive frames with two sorts of modalities. In addition to the usual possibility modality (which means that a formula holds in some successor of a given point), we consider graded modalities (a formula holds in at least $n$ successors) and converse graded modalities (a formula holds in at least $n$ predecessors). We show that the resulting logic, $\mathbf{GrIK4}$ , is both locally and globally undecidable. The same result is obtained for all logics between $\mathbf{GrIK4}$ and its reflexive companion $\mathbf{GrIS4}$ and for some other modal logics. As a consequence, for the ‘unrestricted version’ of the description logic $\mathcal{SIQ}$ , the problem of concept satisfiability (even with respect to the empty terminology) is undecidable. We also give a survey of results on the local and global decidability, complexity, and the finite model property for fragments of $\mathbf{GrIK4}$ .

Filtrations of generalized Veltman models

The filtration method is often used to prove the finite model property of modal logics. We adapt this technique to the generalized Veltman semantics for interpretability logics. In order to preserve the defining properties of generalized Veltman models, we use bisimulations to define adequate filtrations. We give an alternative proof of the finite model property of interpretability logic with respect to Veltman models, and we prove the finite model property of the systems and with respect to generalized Veltman models.

Expanding $$\hbox {FL}_{ew}$$ FL e w with a Boolean connective

We expand FLew with a unary connective whose algebraic counterpart is the operation that gives the greatest complemented element below a given argument. We prove that the expanded logic is conservative and has the Finite Model Property. We also prove that the corresponding expansion of the class of residuated lattices is an equational class.

Bimodal Logics with a “Weakly Connected” Component without the Finite Model Property

There are two known general results on the finite model property (fmp) of commutators [L0,L1] (bimodal logics with commuting and confluent modalities). If L is finitely axiomatizable by modal formulas having universal Horn first-order correspondents, then both [L,K] and [L,S5] are determined by classes of frames that admit filtration, and so they have the fmp. On the negative side, if both L0 and L1 are determined by transitive frames and have frames of arbitrarily large depth, then [L0,L1] does not have the fmp. In this paper we show that commutators with a “weakly connected” component often lack the fmp. Our results imply that the above positive result does not generalize to universally axiomatizable component logics, and even commutators without “transitive” components such as [K3,K] can lack the fmp. We also generalize the above negative result to cases where one of the component logics has frames of depth one only, such as [S4.3,S5] and the decidable product logic S4.3×S5. We also show cases when already half of commutativity is enough to force infinite frames.

Cofinal Stable Logics

We generalize the \({(\wedge, \vee)}\)-canonical formulas to \({(\wedge, \vee)}\)-canonical rules, and prove that each intuitionistic multi-conclusion consequence relation is axiomatizable by \({(\wedge, \vee)}\)-canonical rules. This yields a convenient characterization of stable superintuitionistic logics. The \({(\wedge, \vee)}\)-canonical formulas are analogues of the \({(\wedge,\to)}\)-canonical formulas, which are the algebraic counterpart of Zakharyaschev’s canonical formulas for superintuitionistic logics (si-logics for short). Consequently, stable si-logics are analogues of subframe si-logics. We introduce cofinal stable intuitionistic multi-conclusion consequence relations and cofinal stable si-logics, thus answering the question of what the analogues of cofinal subframe logics should be. This is done by utilizing the \({(\wedge,\vee,\neg)}\)-reduct of Heyting algebras. We prove that every cofinal stable si-logic has the finite model property, and that there are continuum many cofinal stable si-logics that are not stable. We conclude with several examples showing the similarities and differences between the classes of stable, cofinal stable, subframe, and cofinal subframe si-logics.

Skolemization in intermediate logics with the finite model property

An alternative Skolemization method, which removes strong quantiers from formulas, is presented that is sound and complete with respect to intermediate predicate logics with the nite model property. For logics without constant domains the method makes use of an existence predicate, while for logics with constant domains no additional predicate is necessary. In both cases an analogue of Hebrand’s theorem is obtained and it is proved that the one-variable fragment of a logic with the nite model property is decidable once the propositional fragment of the logic is. It is also shown that universal constant domain logics with the nite model property have interpolation once their propositional fragment has. For logics without constant domains some of these results, but with far more complicated proofs, have been obtained in (Iemho, 2010).

Finite Frames Fail: How Infinity Works Its Way into the Semantics of Admissibility

Many intermediate logics, even extremely well-behaved ones such as IPC, lack the finite model property for admissible rules. We give conditions under which this failure holds. We show that frames which validate all admissible rules necessarily satisfy a certain closure condition, and we prove that this condition, in the finite case, ensures that the frame is of width 2. Finally, we indicate how this result is related to some classical results on finite, free Heyting algebras.

On interval modal logic with “after” relation

This paper is devoted to study of the logic corresponding to intervals of the real line, where the modality is interpreted as "after." Since this logic is finitely axiomatizable, the proof of the finite model property given in the paper implies its decidability. Also, a description of the class of finite rooted Kripke frames corresponding to this logic is provided.

Almost structural completeness; an algebraic approach

A deductive system is structurally complete if all of its admissible inference rules are derivable. For several important systems, like the modal logic S5, failure of structural completeness is caused only by the underivability of a passive rule, i.e., a rule whose premise is not unifiable by any substitution. Neglecting passive rules leads to the notion of almost structural completeness, that means, to the derivability of admissible non-passive rules. We investigate almost structural completeness for quasivarieties and varieties of general algebras. The results apply to all algebraizable deductive systems.Firstly, various characterizations of almost structurally complete quasivarieties are presented. Two of them are general: the one expressed with finitely presented algebras, and the one expressed with subdirectly irreducible algebras. The next one is restricted to quasivarieties with the finite model property and equationally definable principal relative congruences, where the condition is verifiable on finite subdirectly irreducible algebras. Some connections with exact and projective unification are included.Secondly, examples of almost structurally complete varieties are provided. Particular emphasis is put on varieties of closure algebras, that are known to constitute adequate semantics for normal extensions of the modal logic S4. A certain infinite family of such almost structurally complete, but not structurally complete, varieties is constructed. Every variety from this family has a finitely presented unifiable algebra which does not embed into any free algebra for this variety. Hence unification is not unitary there. This shows that almost structural completeness is strictly weaker than projective unification for varieties of closure algebras.

STABLE CANONICAL RULES

AbstractWe introduce stable canonical rules and prove that each normal modal multi-conclusion consequence relation is axiomatizable by stable canonical rules. We apply these results to construct finite refutation patterns for modal formulas, and prove that each normal modal logic is axiomatizable by stable canonical rules. We also define stable multi-conclusion consequence relations and stable logics and prove that these systems have the finite model property. We conclude the paper with a number of examples of stable and nonstable systems, and show how to axiomatize them.

Finite model property of modal logics of finite depth

Finite model property of modal logics of finite depth

Non-Fregean Propositional Logic with Quantifiers

We study the non-Fregean propositional logic with propositional quantifiers, denoted by SCIQ. We prove that SCIQ does not have the finite model property and that it is undecidable. We also present examples of how to interpret in SCIQ various mathematical theories, such as the theory of groups, rings, and fields, and we characterize the spectra of SCIQ-sentences. Finally, we present a translation of SCIQ into a classical two-sorted first-order logic, and we use the translation to prove some model-theoretic properties of SCIQ.

Finite model property of modal logics of finite depth

Finite model property of modal logics of finite depth

Modal Consequence Relations Extending S4.3: An Application of Projective Unification

We characterize all finitary consequence relations over S4.3, both syntactically, by exhibiting so-called (admissible) passive rules that extend the given logic, and semantically, by providing suitable strongly adequate classes of algebras. This is achieved by applying an earlier result stating that a modal logic L extending S4 has projective unification if and only if L contains S4.3. In particular, we show that these consequence relations enjoy the strong finite model property, and are finitely based. In this way, we extend the known results by Bull and Fine, from logics, to consequence relations. We also show that the lattice of consequence relations over S4.3 (the lattice of quasivarieties of S4.3-algebras) is countable and distributive and it forms a Heyting algebra.

A bi-intuitionistic modal logic: Foundations and automation

The paper introduces a bi-intuitionistic modal logic, called BISKT, with two adjoint pairs of tense operators. The semantics of BISKT is defined using Kripke models in which the set of worlds carries a pre-order relation as well as an accessibility relation, and the two relations are linked by a stability condition. A special case of these models arises from graphs in which the worlds are interpreted as nodes and edges of graphs, and formulae represent subgraphs. The pre-order is the incidence structure of the graphs. We present a comprehensive study of the logic, giving decidability, complexity and correspondence results. We also show the logic has the effective finite model property. We present a sound, complete and terminating tableau calculus for the logic and use the MetTeL system to explore implementations of different versions of the calculus. An experimental evaluation gave good results for satisfiable problems using predecessor blocking.

Alternation-Free Weighted Mu-Calculus: Decidability and Completeness

In this paper we introduce WMC, a weighted version of the alternation-free modal mu-calculus for weighted transition systems. WMC subsumes previously studied weighted extensions of CTL and resembles previously proposed time-extended versions of the modal mu-calculus. We develop, in addition, a symbolic semantics for WMC and demonstrate that the notion of satisfiability coincides with that of symbolic satisfiability. This central result allows us to prove two major meta-properties of WMC. The first is decidability of satisfiability for WMC. In contrast to the classical modal mu-calculus, WMC does not possess the finite model-property. Nevertheless, the finite model property holds for the symbolic semantics and decidability readily follows; and this contrasts to resembling logics for timed transitions systems for which satisfiability has been shown undecidable. As a second main contribution, we provide a complete axiomatization, which applies to both semantics. The completeness proof is non-standard, since the logic is non-compact, and it involves the notion of symbolic models.

On the Modal μ-Calculus Over Finite Symmetric Graphs

Abstract In this paper we consider the alternation hierarchy of the modal μ-calculus over finite symmetric graphs and show that in this class the hierarchy is infinite. The μ-calculus over the symmetric class does not enjoy the finite model property, hence this result is not a trivial consequence of the strictness of the hierarchy over symmetric graphs. We also find a lower bound and an upper bound for the satisfiability problem of the μ-calculus over finite symmetric graphs.

A logic of plausible justifications

In this work, we combine features from Justification Logics and Logics of Plausibility-Based Beliefs to build a logic for Multi-Agent Systems where each agent can explicitly state his justification for believing in a given sentence. Our logic is a normal modal logic based on the standard Kripke semantics, where we provide a semantic definition for the evidence terms and define the notion of plausible evidence for an agent, based on plausibility relations in the model. As we deal with beliefs, justifications can be faulty and unreliable. In our logic, agents can disagree not only over whether a sentence is true or false, but also on whether some evidence is a valid justification for a sentence or not. After defining our logic and its semantics, we provide a strongly complete axiomatic system for it, show that it has the finite model property, analyze the complexity of its Model-Checking Problem and show that its Satisfiability Problem has the same complexity as the one from basic modal logics. Thus, this logic seems to be a good first step for the development of a dynamic logic that can model the processes of argumentation and debate in Multi-Agent Systems.

On the Priorean temporal logic with [d] over the real line

We consider the language with temporal operators G,H and coderivative operator [d]. G means ‘always in the future’, H means ‘always in the past’ and [d] means ‘around now but possibly not now’. We finitely axiomatize the logic of this language interpreted over the real line, we prove that this logic is PSPACE-complete, and that it is neither strongly complete nor Kripke-complete (and consequently that it fails the finite model property).