Bases of admissible rules for <Emphasis Type="Italic">K</Emphasis>-saturated logics

Admissible inference rules for table modal and superintuitionistic logics are investigated. K-saturated logics are defined semantically. Such logics are proved to have finite bases for admissible inference rules in finitely many variables. __________ Translated from Algebra i Logika, Vol. 47, No. 6, pp. 750–761, November–December, 2008.     

An explicit basis for admissible inference rules in table modal logics of width 2

We construct an explicit finite basis for admissible inference rules in an arbitrary modal logic of width 2 extending the logic Grz. Translated from Algebra i Logika, Vol. 48, No. 1, pp. 122–148, January–February, 2009.     

An explicit basis for the admissible inference rules of the modal logics extending <Emphasis Type="Italic">S</Emphasis>4.1 And <Emphasis Type="Italic">Grz</Emphasis>

We study bases for the admissible inference rules in a broad class of modal logics. We construct an explicit basis for all admissible rules in the logics S4.1, Grz, and their extensions whose number is at least countable. The resulting basis consists of an infinite sequence of rules in a concise and simple form. In the case of a logic of finite width a basis for all admissible rules consists of a finite sequence of rules.     

Complexity of admissible rules

We investigate the computational complexity of deciding whether a given inference rule is admissible for some modal and superintuitionistic logics. We state a broad condition under which the admissibility problem is coNEXP-hard. We also show that admissibility in several well-known systems (including GL, S4, and IPC) is in coNE, thus obtaining a sharp complexity estimate for admissibility in these systems. The research was done while the author was visiting the Department of Philosophy of the Utrecht University. Supported by grant IAA1019401 of GA AV ČR     

An explicit basis for the admissible inference rules in the Gödel-Löb logic <Emphasis Type="Italic">GL</Emphasis>

We describe an explicit basis for the admissible inference rules in the Gödel-Löb logic. The basis consists of a sequence of inference rules in infinitely many variables. Inference rules in the reduced form play an important role in this study. Alongside a basis for the admissible rules we obtain a basis for the quasi-identities of the countable rank free algebra in the Gödel-Löb logic.     

A Criterion for Admissibility of Inference Rules in Some Class of S4-Logics Without the Branching Property

We consider a finitely approximable modal S4-logic without the branching property. Although Rybakov's criterion is inapplicable, using his method we manage to obtain an algorithmic criterion for admissibility of inference rules in a given logic.     

Bases of admissible inference rules for table modal logics of depth 2

In the present article, we prove the theorem which states that every table modal logic λ of depth 2 over S4 has a finite basis of admissible inference rules. In addition, it is established that a finite algebra ℒ belongs to F_ω(λ)^Q iff there exist numbers n₁…, n_k such that (Lemma 5). Let F be a λ-frame of depth 2 and b a cluster of the second layer in F. We show that for any n₁,…,n_k, there exist no p-morphisms from (F_n1⊔…⊔F_nk)⁺ a local component K (b) such that, for any n, there is no p-morphism from any local component of F_n onto K (b) (Lemma 6). Translated fromAlgebra i Logika, Vol. 35, pp. 612–622, September–October, 1996.     

A markov chain sampler for contingency table exact inference

Summary  In the inference of contingency table, when the cell counts are not large enough for asymptotic approximation, conditioning exact method is used and often computationally impractical for large tables. Instead, various sampling methods can be used. Based on permutation, the Monte Carlo sampling may become again impractical for large tables. For this, existing the Markov chain method is to sample a few elements of the table at each iteration and is inefficient. Here we consider a Markov chain, in which a sub-table of user specified size is updated at each iteration, and it achieves high sampling efficiency. Some theoretical properties of the chain and its applications to some commonly used tables are discussed. As an illustration, this method is applied to the exact test of the Hardy-Weinberg equilibrium in the population genetics context.     

Theory of T-norms and fuzzy inference methods

In this paper, the theory of T-norm and T-conorm is reviewed and the T-norm, T-conorm and negation function are defined as a set of T-operators. Some typical T-operators and their mathematical properties are presented. Finally, the T-operators are extended to the conventional fuzzy reasoning methods which are based on the and operators. This extended fuzzy reasoning provides both a general and a flexible method for the design of fuzzy logic controllers and, more generally, for the modelling of any decision-making process.     

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. Supported by RFBR (project No. 06-01-00358) and by the Council for Grants (under RF President) and State Aid of Leading Scientific Schools (grant NSh-335.2008.1). Translated from Algebra i Logika, Vol. 48, No. 1, pp. 54-89, January-February, 2009.     

乒乓球赛制的概率分析

乒乓球比赛的每局原先是21分制现在是11分制,单打由5局3胜制改为7局4胜制。赛制的改变增加了比赛结果的偶然性。本文用概率方法对赛制的改变进行了定量分析,给出了新赛制和旧赛制下运动员取胜的概率。     

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.     

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. 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-2069.2003.1. __________ Translated from Algebra i Logika, Vol. 45, No. 1, pp. 85–113, January–February, 2006.     

Mining axiomatic fuzzy set association rules for classification problems

In this paper, we propose a novel method to mine association rules for classification problems namely AFSRC (AFS association rules for classification) realized in the framework of the axiomatic fuzzy set (AFS) theory. This model provides a simple and efficient rule generation mechanism. It can also retain meaningful rules for imbalanced classes by fuzzifying the concept of the class support of a rule. In addition, AFSRC can handle different data types occurring simultaneously. Furthermore, the new model can produce membership functions automatically by processing available data. An extensive suite of experiments are reported which offer a comprehensive comparison of the performance of the method with the performance of some other methods available in the literature. The experimental result shows that AFSRC outperforms most of other methods when being quantified in terms of accuracy and interpretability. AFSRC forms a classifier with high accuracy and more interpretable rule base of smaller size while retaining a sound balance between these two characteristics.     

Some considerations on fuzzy conditional inference

In much of human reasoning, the form of reasoning is approximate rather than exact as in ‘A red apple is ripe and this apple is more or less red. Then this apple is more or less ripe.’L.A. Zadeh and E.H. Mamdani suggested methods for such a fuzzy reasoning as an application of fuzzy set theory. The method involves an inference rule and a conditional proposition which contains fuzzy concepts.In this paper we point out that the consequence inferred by their methods does not always fit our intuitions and we suggest the improved methods which fit our intuitions under several criteria.     

Inductive inference of ultimately periodic sequences

An algorithm is presented for inductive inference of ultimately periodic sequences. It can be implemented in linear time and space.Dedicated to Peter Naur on the occasion of his 60th birthday