By Vladimir Uspensky, A.L. Semenov

Today the idea of the set of rules is well-known not just to mathematicians. It kinds a conceptual base for info processing; the lifestyles of a corresponding set of rules makes computerized info processing attainable. the idea of algorithms (together with mathematical common sense ) types the the­ oretical foundation for contemporary laptop technological know-how (see [Sem Us 86]; this text is named "Mathematical common sense in desktop technology and Computing perform" and in its identify mathematical good judgment is known in a wide feel together with the speculation of algorithms). in spite of the fact that, no longer everybody realizes that the note "algorithm" contains a remodeled toponym Khorezm. Algorithms have been named after a superb sci­ entist of medieval East, is al-Khwarizmi (where al-Khwarizmi potential "from Khorezm"). He lived among c. 783 and 850 B.C. and the yr 1983 used to be selected to have fun his 1200th birthday. a quick biography of al-Khwarizmi compiled within the 10th century begins as follows: "al-Khwarizmi. His identify is Muhammad ibn Musa, he's from Khoresm" (cited in line with [Bul Rozen Ah eighty three, p.8]).

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Mark 57]) should be regarded only as an explanation and not as a definition. However, explanations of this kind are sufficient to establish certain meaningful facts. For example, we can conclude that not all natural-valued functions of a natural argument can be computed by an algorithm (because the set of all possible prescriptions is countable). lo1], [Us 70], [Us 77]). As Kolmogorov writes: "We start with the following obvious ideas concerning algorithms: 1) An algorithm r being applied to any "input" (= "initial state") A which belongs to some set (the "domain" of the algorithm) gives a "solution" (="final state") B.

Kolmogorov machines 23 process terminates and the state A"+1 is called the result of a computation performed by the machine on input AO. 3]. The definition of Kolmogorov machines given above can be easily modified for the case of an algorithm working with (B,k)-complexes. To do so we must replace (in the definition of the immediate transformation operator) Kolmogorov actions by actions in the aggregate of (B,k)-complexes. From the informal viewpoint the main difference here from Kolmogorov algorithms described above is that we use directed graphs (where the indegree of a vertex is not bounded).

Bya quasi-identity ofa given signature (J we mean any expression of the form "CE~ C ::} D where t is a finite set of atomic formulas and D is an atomic formula. A special case of a quasi-identity is defined as a result of substitution of closed terms instead of variables in a quasi-identity. Let us mention that any identity, Le. any atomic formula D (in particular, any equality) is equivalent to a quasi-identity x = x ::} D. We say that a quasi-identity is true in an algebraic system if all its special cases are true in this system.

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