Provability, Computability and Reflection.

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. (shrink)

Post algebras of order + as a semantic foundation for +-valued predicate calculi were examined in [5]. In this paper Post spaces of order + being a modification of Post spaces of order n2 (cf. Traczyk [8], Dwinger [1], Rasiowa [6]) are introduced and Post fields of order + are defined. A representation theorem for Post algebras of order + 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 +-valued predicate calculi in order to obtain a topological characterization of open theories. (shrink)

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.

This paper presents a monotonic system of Post algebras of order +* whose chain of Post constans is isomorphic with 012 ... -3-2-1. Besides monotonic operations, other unary operations are considered; namely, disjoint operations, the quasi-complement, succesor, and predecessor operations. The successor and predecessor operations are basic for number theory.

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. (shrink)

The aim of this paper is to give a geometric interpretation of quantifiers in the intutionistic predicate calculus. We obtain it treating formulae withn free variables as functions withn arguments which run over an abstract set whereas the values of functions are open subsets of a suitable topological space.

Extended algorithmic logic (EAL) as introduced in [18] is a modified version of extended +-valued algorithmic logic. Only two-valued predicates and two-valued propositional variables occur in EAL. The role of the +-valued logic is restricted to construct control systems (stacks) of pushdown algorithms whereas their actions are described by means of the two-valued logic. Thus EAL formalizes a programming theory with recursive procedures but without the instruction CASE.The aim of this paper is to discuss EAL and prove the completeness theorem. (...) A complete formalization of EAL was announced in [20] but no proof of the completeness theorem was given. (shrink)

An algebraic approach to programs called recursive coroutines — due to Janicki [3] — is based on the idea to consider certain complex algorithms as algebraics models of those programs. Complex algorithms are generalizations of pushdown algorithms being algebraic models of recursive procedures (see Mazurkiewicz [4]). LCA — logic of complex algorithms — was formulated in [11]. It formalizes algorithmic properties of a class of deterministic programs called here complex recursive ones or interacting stacks-programs, for which complex algorithms constitute mathematical (...) models. LCA is in a sense an extension of algorithmic logic as initiated by Salwicki [14] and of extended algorithmic logic EAL as formulated and examined by the present author in [8], [9], [10]. In LCA — similarly as in EAL-ω + -valued logic is applied as a tool to construct control systems (stacks) occurring in corresponding algorithms. The aim of this paper is to give a complete axiomatization. of LCA and to prove a completeness theorem. (shrink)