A paraconsistent extension of Sylvan’s logic

We deal with Sylvan’s logic CC_ω. It is proved that this logic is a conservative extension of positive intuitionistic logic. Moreover, a paraconsistent extension of Sylvan’s logic is constructed, which is also a conservative extension of positive intuitionistic logic and has the property of being decidable. The constructed logic, in which negation is defined via a total accessibility relation, is a natural intuitionistic analog of the modal system S5. For this logic, an axiomatization is given and the completeness theorem is proved. Supported by RFBR grant No. 06-01-00358 and by the Council for Grants (under RF President) and State Aid of Fundamental Science Schools, project NSh-4787.2006.1. __________ Translated from Algebra i Logika, Vol. 46, No. 5, pp. 533–547, September–October, 2007.     

A sequent calculus for logic of knowledge and past time: Completeness and decidability

We consider logic of knowledge and past time. This logic involves the discrete-time linear temporal operators next, until, weak yesterday, and since. In addition, it contains an indexed set of unary modal operators agent i knows.We consider the semantic constraint of the unique initial states for this logic. For the logic, we present a sequent calculus with a restricted cut rule. We prove the soundness and completeness of the sequent calculus presented. We prove the decidability of provability in the considered calculus as well. So, this calculus can be used as a basis for automated theorem proving. The proof method for the completeness can be used to construct complete sequent calculi with a restricted cut rule for this logic with other semantical constraints as well. Published in Lietuvos Matematikos Rinkinys, Vol. 46, No. 3, pp. 427–437, July–September, 2006.     

A coding method for a sequent calculus of propositional logic

The paper deals with a coding method for a sequent calculus of the propositional logic. The method is based on the sequent calculus. It allows us to determine if a formula is derivable in the calculus without constructing a derivation tree. The main advantage of the coding method is its compactness in comparison with derivation trees of the sequent calculus. The coding method can be used as a decision procedure for the propositional logic.     

A method of proving interpolation in paraconsistent extensions of the minimal logic

The interpolation property in extensions of Johansson’s minimal logic is investigated. The construction of a matched product of models is proposed, which allows us to prove the interpolation property in a number of known extensions of the minimal logic. It is shown that, unlike superintuitionistic, positive, and negative logics, a sum of J-logics with the interpolation property CIP may fail to possess _CIP, nor even the restricted interpolation property. Supported by RFBR grant No. 06-01-00358, by INTAS grant No. 04-77-7080, and by the Council for Grants (under RF President) and State Aid of Fundamental Science Schools, project NSh-4787.2006.1. __________ Translated from Algebra i Logika, Vol. 46, No. 5, pp. 627–648, September–October, 2007.     

Some decidable classes of formulas of pure hybrid logic

We consider the formulas of pure hybrid logic without occurrences of a satisfaction operator. We describe a couple of formula derivation tactics in sequent calculus and prove the decidability of two classes of formulas. Decidable classes are obtained by setting restrictions on nominal occurrences in the formulas. Published in Lietuvos Matematikos Rinkinys, Vol. 44, No. 4, pp. 563–572, October–December, 2007.     

Proof-theoretical investigation of temporal logic with time gaps

In the paper, the first-order intuitionistic temporal logic sequent calculus LBJ is considered. The invertibility of some of the LBJ rules, syntactic admissibility of the structural rules and the cut rule in LBJ, as well as Harrop and Craig's interpolation theorems for LBJ are proved. Gentzen's midsequent theorem is proved for the LBJ' calculus which is obtained from LBJ by removing the antecedent disjunction rule from it. Published in Lietuvos Matematikos Rinkinys, Vol. 40, No. 3, pp. 255–276, July–September, 2000.     

On a logic of involutive quantales

The logic just corresponding to (non‐commutative) involutive quantales, which was introduced by Wendy MacCaull, is reconsidered in order to obtain a cut‐free sequent calculus formulation, and the completeness theorem (with respect to the involutive quantale model ) for this logic is proved using a new admissible rule. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A modal provability logic of explicit and implicit proofs

We establish the bi-modal forgetful projection of the Logic of Proofs and Formal Provability GLA. That is to say, we present a normal bi-modal provability logic with modalities □ and whose theorems are precisely those formulas for which the implicit provability assertions represented by the modality can be realized by explicit proof terms.     

A new technique for proving realisability and consistency theorems using finite paraconsistent models of cut‐free logic

A new technique for proving realisability results is presented, and is illustrated in detail for the simple case of arithmetic minus induction. CL is a Gentzen formulation of classical logic. CPQ is CL minus the Cut Rule. The basic proof theory and model theory of CPQ and CL is developed. For the semantics presented CPQ is a paraconsistent logic, i.e. there are non‐trivial CPQ models in which some sentences are both true and false. Two systems of arithmetic minus induction are introduced, CL‐A and CPQ‐A based on CL and CPQ, respectively. The realisability theorem for CPQ‐A is proved: It is shown constructively that to each theorem A of CPQ‐A there is a formula A *, a so‐called “realised disjunctive form of A ”, such that variables bound by essentially existential quantifiers in A * can be written as recursive functions of free variables and variables bound by essentially universal quantifiers. Realisability is then applied to prove the consistency of CL‐A, making use of certain finite non‐trivial inconsistent models of CPQ‐A. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Sequent Calculi with Analytic Cut for Logics of Time and Knowledge with Perfect Recall

In this paper, we consider two logics of time and knowledge. These logics involve the discrete time linear temporal logic operators ``next' and ``until'. In addition, they contain an indexed set of unary epistemic modalities ``agent $i$ knows'. In these logics, the temporal and epistemic dimensions may interact. The particular interactions we consider capture perfect recall. We consider perfect recall in synchronously distributed systems and in systems without any assumptions. For these logics, we present sequent calculi with an analytic cut rule. Thus, we get proof systems where proof-search becomes decidable. The soundness and completeness of these calculi are proved.     

PSPACE complexity of modal logic <Emphasis Type="Italic">K D</Emphasis>45<Subscript>n</Subscript>

We study sequent calculus for multi-modal logic K D45_n and its complexity. We introduce a loop-check free sequent calculus. Loop-check is eliminated by using the marked modal operator □_i^•, which is used as an alternative to sequents with histories ([8], [3], [5]). All inference rules are invertible or semi-invertible. To get this, we use or branches beside common and branches. We prove the equivalence between known sequent calculus and our newly introduced efficient sequent calculus. We concentrate on the complexity analysis of the introduced sequent calculus for multi-modal logic K D45_n. We prove that the space complexity of the given calculus is polynomial (O(l ³)). We show the maximum height of the constructed derivation tree that leads to the reduction of the time and space complexity. We present a decision algorithm for multi-modal logic K D45_n and some nontrivial examples to improve the introduced loop-check free sequent calculus.     

扰动模糊命题逻辑的代数结构及其广义重言式性质

着眼于扰动模糊命题逻辑的代数结构,为研究二维扰动模糊命题逻辑最大子代数I2R及其广义重言式提供了一些代数理论基础,最后研究了子代数间广义重言式的关系.     

Finite problems and the logic of the weak law of excluded middle

A new characteristic of propositional formulas as operations on finite problems, the cardinality of a sufficient solution set, is defined. It is proved that if a formula is deducible in the logic of the weak law of excluded middle, then the cardinality of a sufficient solution set is bounded by a constant depending only on the number of variables; otherwise, the accessible cardinality of a sufficient solution set is close to (greater than the nth root of) its trivial upper bound. This statement is an analog of the authors result about the algorithmic complexity of sets obtained as values of propositional formulas, which was published previously. Also, we introduce the notion of Kolmogorov complexity of finite problems and obtain similar results.Translated from Matematicheskie Zametki, vol. 77, no. 2, 2005, pp. 291–302.Original Russian Text Copyright © 2005 by A. V. Chernov.This revised version was published online in April 2005 with a corrected issue number.     

Modal Aggregation and the Theory of Paraconsistent Filters

This paper articulates the structure of a two species of weakly aggregative necessity in a common idiom, neighbourhood semantics, using the notion of a k-filter of propositions. A k-filter on a non-empty set I is a collection of subsets of I which (i) contains I, (ii) is closed under supersets on I, and (iii) contains ∪{X_i ≤ X_j : 0 ≤ i < j ≤ k} whenever it contains the subsets X₀,…, X_k. The mathematical content of the proof that weakly aggregative modal logic is complete relative to k-ary frame theory, the standard semantic idiom for weakly aggregative modal logic (see [1]) is presented in language-independent terms as a representation theorem for k-filters: every non-trivial k-filter is included in the union of ≤ k non-trivial filters. The elementary theory of k-filters is developed and then applied in the form of an ultrafilter extension result for k-ary frame theory. Mathematics Subject Classification: 03B45.     

Nonmatricial version of the Arveson-Wittstock extension principle,and its generalization

We consider the algebra ? = ?(H) of bounded operators in a Hilbert space H, ?-bimodules, and morphisms of these bimodules into the algebra ?(L ? H), where L is a Hilbert space. We study the problem of extension of a morphism defined on a sub-?-bimodule Y ? Z to Z. This problem is solved for Ruan bimodules.     

An Edgeworth expansion for functionals of Gaussian fields and its applications

This paper is concerned with the rate of convergence in the normal approximation of the sequence $\left\{F_n\right\}$, where each $F_n$ is a functional of an infinite-dimensional Gaussian field. We develop new and powerful techniques for computing the exact rate of convergence in distribution with respect to the Kolmogorov distance. As a tool for our works, the Edgeworth expansion of general orders, with an explicitly expressed remainder, will be obtained, and this remainder term will be controlled to find upper and lower bounds of the Kolmogorov distance in the case of an arbitrary sequence $\left\{F_n\right\}$. As applications, we provide the optimal fourth moment theorem of the sequence $\left\{F_n\right\}$ in the case when $\left\{F_n\right\}$ is a sequence of random variables living in a fixed Wiener chaos or a finite sum of Wiener chaoses. In the former case, our results show that the conditions given in this paper seem more natural and minimal than ones appeared in the previous works.     

A matricial extension of the Helson-Szegö theorem and its application in multivariate prediction

It is shown that the positivity of the angle between the past and future of a multivariate stationary process is sufficient for the existence of a mean-convergent autoregressive series representation of its linear predictor. A large class of multivariate processes whose past and future are at positive angle is characterized, thus providing a matricial extension of the Helson-Szegö theorem.     

小议数学发展的哲学问题及其在数学与计算机科学关系中的表现

本文从哲学角度讨论数学与实践的联系 ,并以数学与计算机科学的关系为例说明这种联系 .