L. L. Maksimova

Craig's theorem in superintuitionistic logics and amalgamable varieties of pseudo-boolean algebras

In recent years there have appeared many studies of an interesting and important property of logical theories, the so-called Craig interpolation theorem. Craig proved the interpolation theorem for classical predicate logic in 1957 [9]. SchHtte [23] proved the interpolation theorem for intuitionistic predicate logic, and Gabbay [12] for certain extensions of this logic. In [I0, 13] the interpolation theorem was proved for a number of modal logics, and in [21] for many-valued predicate calculi.

Amalgamation and interpolation in normal modal logics

This is a survey of results on interpolation in propositional normal modal logics. Interpolation properties of these logics are closely connected with amalgamation properties of varieties of modal algebras. Therefore, the results on interpolation are also reformulated in terms of amalgamation.

On maximal intermediate logics with the disjunction property

For intermediate logics, there is obtained in the paper an algebraic equivalent of the disjunction propertyDP. It is proved that the logic of finite binary trees is not maximal among intermediate logics withDP. Introduced is a logicND, which has the only maximal extension withDP, namely, the logicML of finite problems.

Interpolation properties of superintuitionistic logics

AbstractA family of prepositional logics is considered to be intermediate between the intuitionistic and classical ones. The generalized interpolation property is defined and proved is the following.Theorem on interpolation. For every intermediate logic L the following statements are equivalent:(i)Craigs interpolation theorem holds in L,(ii)L possesses the generalized interpolation property,(iii)Robinsons consistency statement is true in L. There are just 7 intermediate logics in which Craigs theorem holds.Besides, Craigs interpolation theorem holds in L iff all the modal companions of L possess Craigs interpolation property restricted to those formulas in which every variable is proceeded by necessity symbol.

On variable separation in modal and superintuitionistic logics

In this paper we find an algebraic equivalent of the Hallden property in modal logics, namely, we prove that the Hallden-completeness in any normal modal logic is equivalent to the so-called super-embedding property of a suitable class of modal algebras. The joint embedding property of a class of algebras is equivalent to the Pseudo-Relevance Property. We consider connections of the above-mentioned properties with interpolation and amalgamation. Also an algebraic equivalent of of the principle of variable separation in superintuitionistic logics will be found.

Problem of restricted interpolation in superintuitionistic and some modal logics

A restricted interpolation property IPR is investigated in modal and superintuitionistic logics. The problem of description of logics with IPR over the intuitionistic logic Int and the modal Grzegorczyk logic Grz is solved. It is proved that in extensions of Int or Grz IPR is equivalent to the projective Beth property PB2. It follows that IPR is decidable over Int and strongly decidable over Grz.

Projective Beth Property in Extensions of Grzegorczyk Logic

All extensions of the modal Grzegorczyk logic Grz possessing projective Beths property PB2 are described. It is proved that there are exactly 13 logics over Grz with PB2. All of them are finitely axiomatizable and have the finite model property. It is shown that PB2 is strongly decidable over Grz, i.e. there is an algorithm which, for any finite system Rul of additional axiom schemes and rules of inference, decides if the calculus Grz+Rul has the projective Beth property.

The decidability of craig’s interpolation property in well-composed J-logics

Under study are the extensions of Johansson’s minimal logic J. We find sufficient conditions for the finite approximability of J-logics in dependence on the form of their axioms. Using these conditions, we prove the decidability of Craig’s interpolation property (CIP) in well-composed J-logics. Previously all J-logics with weak interpolation property (WIP) were described and the decidability of WIP over J was proved. Also we establish the decidability of the problem of amalgamability of well-composed varieties of J-algebras.

Interpolation and Definability over the Logic Gl

In a previous paper [21] all extensions of Johansson’s minimal logic J with the weak interpolation property WIP were described. It was proved that WIP is decidable over J. It turned out that the weak interpolation problem in extensions of J is reducible to the same problem over a logic Gl, which arises from J by adding tertium non datur.In this paper we consider extensions of the logic Gl. We prove that only finitely many logics over Gl have the Craig interpolation property CIP, the restricted interpolation property IPR or the projective Beth property PBP. The full list of Gl-logics with the mentioned properties is found, and their description is given. We note that IPR and PBP are equivalent over Gl. It is proved that CIP, IPR and PBP are decidable over the logic Gl.

Complexity of some problems in positive and related calculi

A problem of recognizing important properties of propositional calculi is considered, and complexity bounds for some decidable properties are found. For a given logical system L, a property P of logical calculi is called decidable over L if there is an algorithm which for any finite set Ax of new axiom schemes decides whether the calculus L + Ax has the property P or not. In Maksimova and Voronkov (Bull Symbol. Logic 6 (2000) 118) the complexity of tabularity, pretabularity, and interpolation problems over the intuitionistic logic (Int) and over modal logic S4 was studied.In the present paper, positive and positively axiomatizable calculi are investigated. We prove NP-completeness of tabularity, DP-hardness of pretabularity and PSPACE-completeness of interpolation and projective Beths property over the positive fragment Int+ of the intuitionistic logic. Some complexity bounds for properties of propositional calculi over the intuitionistic or the minimal logic are found.