The completeness modulo ideal in functional systems of a program type

Abstract

In a functional system of a program type the fundamental set R consists of functions and predicates, and the operations are defined by a class of program schemes or a set of programming operations. Systems of one-place partially recursive functions and predicates are investigated. A set M C R is called J-complete where J denotes an ideal in the lattice of recursively enumerable sets, if for every element t G R and for every set X G J there exists an element t' in the closure of the set M such that t' is equal to t on X. It is established that the J-completeness problem is not simpler than the problem of ordinary completeness if the ideal J contains an infinite set, and therefore the perfectly investigated completeness modulo ideal of finite sets is unique.A functional system (F. S.) (#, Ω) is called a functional system of a program type if the set R is two-basic and consists of a set P of functions and a set Q of predicates, and the set of operations Ω is defined by a class of program schemes (or by an equivalent collection of programming operations).The system (Α,Ω) contains F.S. (P,C) and F.S. (φ,ν,Λ,-»), however in Ω besides the operation of superposition C and the logical operations V, Λ, -»there are operations defined on the whole set R. Specifically we will consider the set Ρ of all one-place partially recursive functions, the set Q of all one-place partial predicates whose truth and falsity domains are recursively enumerable sets, the set Ω which contains three more operations besides those mentioned above, i.e., the branching of two functions / and g by a predicate a (notation [a] (/ V #)), the iteration (cycle) of a function / by a predicate α ([<*]{/}) and the substitution of a function / into a predicate a(f · a). F.S. with the given set of opeiations is called by Glushkov ^1} the system of algorithmic algebras. As Ω one can take the class of Yanov's program schemes with the same sets Ρ and Q [2]. At the end of the article we show the possibility of extension of the concept of the completeness modulo ideal to many-place functions and predicates (for example Redko's program logics [3] or Zaslavskii's graph-schemes with memory [4]).Generalizing results of investigations of various functional systems Kudryavtsev noted [5] that 'the results on completeness, being negative on the whole from the point of view of their effectiveness, leads us to the necessity to consider various modifications of that problem'. There are some modifications for which criterial systems appear to be simpler than those for the ordinary completeness. For example a criterial system is countable for the ^-completeness of automata functions [6] or for linear automata [7], while its cardinality is equal to continuum for the ordinary completeness [8].Some of the most natural and practically important modifications are 'finite completeness' and 'effective completeness': it is required to find a completeness criterion

The completeness problem in the function algebra of linear integer-coefficient polynomials

“…I S / of k-valued and countably-infinite-valued functions with superposition operations (k-valued and countably-infinite-valued logics); the algebra .P BD I S; F / of boundedly determinate functions with superposition and feedback operations (finite automata); the algebra .P par:r I S; PR; M / of partial recursive (computable) functions with superposition, primitive recursion, and minimisation operations [1][2][3][4]. Also the following kinds of function algebras are studied: the algebra of partial k-valued functions [5][6][7][8][9][10], the algebras of functions whose domain is the Cartesian product of finite sets [11][12][13][14][15][16][17], the algebras of mappings whose domain and range are distinct [1,2,[18][19][20][21], the algebra of mappings of regular sets of a certain finite alphabet words [22]; the algebra of special sets, so-called bundles, of functions [23,24]; the algebras with various definitions of closure operators [1,2,[25][26][27][28][29][30]; the algebras with operations similar to algorithms and programs [31][32][33][34][35][36][37][38]; the alge...…”