Helena Rasiowa

Logic of Approximation Reasoning

An algebraic and set-theoretical approach to approximation reasoning as proposed in [10] and [5] leads to a formulation of a class of first order logics. They are certain intermediate logics equipped with approximation operators dt for t e T — where (T, ≤) is a poset establishing a type of logic under consideration — and with modal connectives Ct of possibility and It of necessity, t e T and possibly with CT and IT. Their semantics is based on the idea that a set of objects to be recognized in a process of an approximation reasoning is approximated by means of a family of sets covering this set and by their intersection. Approximating sets with equivalence classes of equivalence relations, as connected with Pawlaks rough sets methods (see [8], [7], [12], [13]) is an additional tool. The main task of this paper is to formulate and prove the completeness theorem for the logics under consideration. For that purpose a theory of plain semi-Post algebras as introduced and developed in [3] has been applied. These algebras replace more complicated semi-Post algebras occurring in [10].

Plain Semi-Post algebras as a poset-based generalization of post algebras and their representability

Semi-Post algebras of any type T being a poset have been introduced and investigated in [CR87a], [CR87b]. Plain Semi-Post algebras are in this paper singled out among semi-Post algebras because of their simplicity, greatest similarity with Post algebras as well as their importance in logics for approximation reasoning ([Ra87a], [Ra87b], [RaEp87]). They are pseudo-Boolean algebras generated in a sense by corresponding Boolean algebras and a poset T. Every element has a unique descending representation by means of elements in a corresponding Boolean algebra and primitive Post constants which form a poset T. An axiomatization and another characterization, subalgebras, homomorphisms, congruences determined by special filters and a representability theory of these algebras, connected with that for Boolean algebras, are the subject of this paper.

Approximating sets with equivalence relations

Abstract In various considerations of computer science (for instance in image processing and databases) one faces the following situation: given a set (of points or of documents), one considers a descending sequence of equivalence relations (‘approximation spaces of order ξ’). These equivalence relations determine a sequence of closure operations Cl i . Given a set X , the approximation sequence of X is simply 〈Cl i ( X )〉 i ξ . We characterize here those sets X which satisfy the conditions: X = ∩{Cl i ( X ): i ξ }.

Semi-Post algebras

In this paper, semi-Post algebras are introduced and investigated. The generalized Post algebras are subcases of semi-Post algebras. The so called primitive Post constants constitute an arbitrary partially ordered set, not necessarily connected as in the case of the generalized Post algebras examined in [3]. By this generalization, semi-Post products can be defined. It is also shown that the class of all semi-Post algebras is closed under these products and that every semi-Post algebra is a semi-Post product of some generalized Post algebras.

Axiomatization and completeness of uncountably valued approximation logic

A first order uncountably valued logicLQ(0,1) for management of uncertainty is considered. It is obtained from approximation logicsLT of any poset type (T, ⩽) (see Rasiowa [17], [18], [19]) by assuming (T, ⩽)=(Q(0, 1), ⩽) — whereQ(0, 1) is the set of all rational numbersq such that 0<q<1 and ⩽ is the arithmetic ordering — by eliminating modal connectives and adopting a semantics based onLT-fuzzy sets (see Rasiowa and Cat Ho [20], [21]). LogicLQ(0,1) can be treated as an important case ofLT-fuzzy logics (introduced in Rasiowa and Cat Ho [21]) for (T, ⩽)=(Q(0, 1), ⩽), i.e. asLQ(0, 1)-fuzzy logic announced in [21] but first examined in this paper.LQ(0,1) deals with vague concepts represented by predicate formulas and applies approximate truth-values being certain subsets ofQ(0, 1). The set of all approximate truth-values consists of the empty set ø and all non-empty subsetss ofQ(0, 1) such that ifq∈s andq′⩽q, thenq′∈s. The setLQ(0, 1) of all approximate truth-values is uncountable and covers up to monomorphism the closed interval [0, 1] of the real line.LQ(0, 1) is a complete set lattice and therefore a pseudo-Boolean (Heyting) algebra. Equipped with some additional operations it is a basic plain semi-Post algebra of typeQ(0, 1) (see Rasiowa and Cat Ho [20]) and is taken as a truth-table forLQ(0,1) logic.LQ(0,1) can be considered as a modification of Zadehs fuzzy logic (see Bellman and Zadeh [2] and Zadeh and Kacprzyk, eds. [29]). The aim of this paper is an axiomatization of logicLQ(0,1) and proofs of the completeness theorem and of the theorem on the existence ofLQ(0, 1)-models (i.e. models under the semantics introduced) for consistent theories based on any denumerable set of specific axioms. Proofs apply the theory of plain semi-Post algebras investigated in Cat Ho and Rasiowa [4].

Topological representations of Post algebras of order ω+ and open theories based on ω+-valued Post logic

Post algebras of order ω+ as a semantic foundation for ω+-valued predicate calculi were examined in [5]. In this paper Post spaces of order ω+ being a modification of Post spaces of order n≥2 (cf. Traczyk [8], Dwinger [1], Rasiowa [6]) are introduced and Post fields of order ω+ are defined. A representation theorem for Post algebras of order ω+ as Post fields of sets is proved. Moreover necessary and sufficient conditions for the existence of representations preserving a given set of infinite joins and infinite meets are established and applied to Lindenbaum-Tarski algebras of elementary theories based on ω+-valued predicate calculi in order to obtain a topological characterization of open theories.

Many-Valued Algorithmic Logic as a Tool to Investigate Programs

The attempt to develop research in mathematical programming theory has led to various ideas, methods and approaches. Progress has recently been made in a logical approach to programming theory on the basis of algorithmic logic. Formalized languages of algorithmic logic are extensions of first order predicate languages, and formulas describe properties of these programs. Extensions to m-valued branchings and logics lead to ω+-valued logic, which may be considered as an attempt to formalize programming theory which includes recursive procedures. Formulations of extended ω+-valued algorithmic logics, including the syntax and the semantics of their formalized languages, are given.

Subalgebras and homomorphisms of semi-Post algebras

Semi-Post algebras have been introduced and investigated in [6]. This paper is devoted to semi-Post subalgebras and homomorphisms. Characterization of semi-Post subalgebras and homomorphisms, relationships between subalgebras and homomorphisms of semi-Post algebras and of generalized Post algebras are examined.

LT -fuzzy sets

The problem of axiomatization of fuzzy algebra is one of the most interesting topics in fuzzy sets theory. There is not as yet a uniform point of view concerning the question what a fuzzy algebra is (cf. [1, 6, 7, 20, 4]) and there are different fuzzy algebras useful in different applications. In this paper a poset-based concept of LT-fuzzy sets is proposed, being a simple modification of L-fuzzy sets (cf. [5]). This new approach permits the development of an axiomatic fuzzy sets algebra based on the theory of plain semi-Post algebras as investigated in [3] and applied to logic for approximation reasoning [14, 15, 16].

A proof of Herbrand's theorem

Publisher Summary This chapter gives a proof of the Herbrands theorem by means of the algebraic method that has been successfully applied to logical problems by various writers in recent years. The first algebraic proof of the Herbrands theorem was found in 1951 independently by Los and Mostowski. Their proof was afterward simplified considerably by Rasiowa and is presented in a simplified form in this chapter.