Projective Beth Property Research Articles

Unary negation

We study fragments of first-order logic and of least fixed point logic that allow only unary negation: negation of formulas with at most one free variable. These logics generalize many interesting known formalisms, including modal logic and the $\mu$-calculus, as well as conjunctive queries and monadic Datalog. We show that satisfiability and finite satisfiability are decidable for both fragments, and we pinpoint the complexity of satisfiability, finite satisfiability, and model checking. We also show that the unary negation fragment of first-order logic is model-theoretically very well behaved. In particular, it enjoys Craig Interpolation and the Projective Beth Property.

Restricted interpolation over modal logic S4

The problem of restricted interpolation and definability in normal extensions of modal logic S4 is investigated. We specify necessary conditions for the restricted interpolation property IPR in the systems under consideration, and prove that there exist only finitely many logics possessing IPR or the projective Beth property PB2. These logics are all residually finite and recognizable over S4. As a consequence, the restricted interpolation problem and the projective Beth property are decidable over S4.

The projective Beth property in well-composed logics

The interpolation and Beth definability problems are proved decidable in well-composed logics, i.e., in extensions of Johansson’s minimal logic J satisfying an axiom (⊥ → A) ∨ (A → ⊥). In previous studies, all J-logics with the weak interpolation property (WIP) were described and WIP was proved decidable over J. Also it was shown that only finitely many well-composed logics possess Craig’s interpolation property (CIP) and the restricted interpolation property (IPR), and moreover, IPR is equivalent to the projective Beth property (PBP) on the class of logics in question. These results are applied to prove decidability of IPR and PBP in well-composed logics. The decidability of CIP in such logics was stated earlier. Thus all basic versions of the interpolation and Beth properties are decidable on the class of well-composed logics.

Interpolation and the projective Beth property in well-composed logics

We study the interpolation and Beth definability problems in propositional extensions of minimal logic J. Previously, all J-logics with the weak interpolation property (WIP) were described, and it was proved that WIP is decidable over J. In this paper, we deal with so-called well-composed J-logics, i.e., J-logics satisfying an axiom (⊥ → A) ∨ (A → ⊥). Representation theorems are proved for well-composed logics possessing Craig’s interpolation property (CIP) and the restricted interpolation property (IPR). As a consequence, we show that only finitely many well-composed logics share these properties and that IPR is equivalent to the projective Beth property (PBP) on the class of well-composed J-logics.

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.

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.

Restricted interpolation property in superintuitionistic logics

The restricted interpolation property IPR in modal and superintuitionistic logics is investigated. It is proved that in superintuitionistic logics of finite slices and in finite-slice extensions of the Grzegorczyk logic, the property IPR is equivalent to the projective Beth property PB2.

Projective Beth Property in Extensions of Grzegorczyk Logic

All extensions of the modal Grzegorczyk logic Grz possessing projective Beth's 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 Projective Beth Property and Interpolation in Positive and Related Logics

We look at the interplay between the projective Beth property in non-classical logics and interpolation. Previously, we proved that in positive logics as well as in superintuitionistic and modal ones, the projective Beth property PB2 follows from Craig's interpolation property and implies the restricted interpolation property IPR. Here, we show that IPR and PB2 are equivalent in positive logics, and also in extensions of the superintuitionistic logic KC and of the modal logic Grz.2.

Interpolation and Definability in Extensions of the Minimal Logic

We study into the interpolation property and the projective Beth property in extensions of Johansson's minimal logic. A family of logics of some special form is considered. Effective criteria are specified which allow us to verify whether an arbitrary logic in this family has a given property.

Definability in Normal Extensions of S4

A projective Beth property, PB2, in normal modal logics extending S4 is studied. A convenient criterion is furnished for PB2 to be valid in a larger family of extensions of K4. All locally tabular extensions of the Grzegorczyk logic with PB2 are described. Superintuitionistic logics with the projective Beth property that have no modal companions with this property are found.

Restricted Interpolation and the Projective Beth Property in Equational Logic

Interconnections between syntactic and categorical properties of equational theories are established. The notions of restricted interpolation and of restricted amalgamation are introduced and their equivalence proved; interrelations of the above-mentioned properties and the projective Beth property, interpolation, and amalgamation are studied.

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 Beth's 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.

Implicit Definability and Positive Logics

We furnish a description of all positive logics possessing the projective Beth property PBP. Decidability of PBP is proved for the positive calculi extending a positive fragment $$Int^ +$$ of the intuitionistic propositional calculus. The results mentioned are applied to explore extensions of Johansson's minimal logic.

Decidability of the Projective Beth Property in Varieties of Heyting Algebras

Previously, we proved that there are only finitely many varieties of Heyting algebras possessing the projective Beth property and gave an exhaustive list of these. The projective Beth property is equivalent to strong epimorphisms surjectivity (SES). Here, we prove that the projective Beth property and SES are base-decidable on a class of varieties of Heyting algebras.

Intuitionistic logic and implicit definability

It is proved that there are exactly 16 superintuitionistic propositional logics with the projective Beth property. These logics are finitely axiomatizable and have the finite model property. Simultaneously, all varieties of Heyting algebras with strong epimorphisms surjectivity are found.

Algebraic Characterizations of Various Beth Definability Properties

In this paper it will be shown that the Beth definability property corresponds to surjectiveness of epimorphisms in abstract algebraic logic. This generalizes a result by I. Nemeti (cf. [11, Theorem 5.6.10]). Moreover, an equally general characterization of the weak Beth property will be given. This gives a solution to Problem 14 in [20]. Finally, the characterization of the projective Beth property for varieties of modal algebras by L. Maksimova (see [15]) will be shown to hold for the larger class of semantically algebraizable logics.

Superintuitionistic logics and the projective beth property

It is proved that in superintuitionistic logics, the projective Beth property follows from the Craig interpolation property, but the converse does not hold. A criterion is found which allows us to reduce the problem asking whether the projective Beth property is valid in superintuitionistic logics to suitable properties of varieties of pseudoboolean algebras. It is shown that the principle of variable separation follows from the projective Beth property. On the other hand, the interpolation property in a logic L implies the projective Beth property in Δ(L).