Polinomial Church thesis for refal-5-functions, normal, algorithms and their generalizations

Abstract

It is important for multiple-used programs that their run be sufficiently quick. Du D.Z. and Ko K.I. have formulated a extended Church thesis: «A function computable in polynomial time in any reasonable computational model using a reasonable time complexity measure is computable by a deterministic Nuring machine in polynomial time» (i.e. it belongs to the class FP. It is more natural to call this extended Church thesis as a polynomial Church thesis for Turing machine. Different researchers differently understand the term reasonable in consequence of which at the last time many «proofs» (containing mistakes) of the famous problem whether P = FP appear. In such «proofs» a step of different algorithm models treats intuitively. In the present work it is proved that reasonability of a recursive function step may be consisted in the use of such refal-5-sentenses from FP, for which the length of result notation (including the lenth of all stack) increases not more than by additive constant unique for all arguments. In such a sence a usual definition of a recursive function is not a reasonable calculation model. The count of recursive algorithm steps is rather difficult. Nesessary and sufficient conditions of time polynomiality of a function, realized by means of programming language refal-5, are proved in the present paper. This language is a practicaly realized generalization of such mathematical models of algorithm notion as normal algoritm and here introduced Markov-Post algorithm and recursive Markov-Post algorithm. That is why conditions of time polynomiality of these mathematical models (used in theoretical reseaches) are proved as corollaries of the main theorem. The question whether it is possible to prove that a refal-5 program realizes a polynomiall time function (i.e. function belonging to FP) is very actual. A notion of a double polynomial recursive refal-5 function call, which provides the solution of this question, is introduced in the paper.