By Vladimir Uspensky, A.L. Semenov

Today the proposal of the set of rules is well-known not just to mathematicians. It varieties a conceptual base for info processing; the life of a corresponding set of rules makes automated details processing attainable. the speculation 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 termed "Mathematical common sense in laptop technology and Computing perform" and in its name mathematical good judgment is known in a extensive feel together with the speculation of algorithms). even if, now not every body realizes that the note "algorithm" encompasses a reworked toponym Khorezm. Algorithms have been named after an exceptional sci entist of medieval East, is al-Khwarizmi (where al-Khwarizmi skill "from Khorezm"). He lived among c. 783 and 850 B.C. and the yr 1983 was once selected to have fun his 1200th birthday. a quick biography of al-Khwarizmi compiled within the 10th century starts off as follows: "al-Khwarizmi. His identify is Muhammad ibn Musa, he's from Khoresm" (cited in keeping with [Bul Rozen Ah eighty three, p.8]).

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Model specifies in its own way the notions used in Kolmogorov's description of an algorithmic process. In [Kol 53] the most general. way to specify these notions is suggested. This scheme may be considered as an adequate formalization of the notion of 20 Ch. 1. Algorithm algorithm (for model with non-local transformations of information we need to divide non-local steps into local steps, see above). The corresponding computational model is called Kolmogorov machines. The exact definition of it will be given in chap.

In this case the set of all semantic consequences of S (and, therefore, the corresponding congruence relation) can be generated by a calculus easily obtained from S. This calculus has the following rules: (1) t = t is admissible for any tj (2) ift = s is admissible and u = v or v = u belongs to S then any equality obtained from t = s by replacing some occurrence of u by v is admissible. 3: Algebraic examples 42 To specify this calculus completely we must specify the workspace, the rule selecting basic states and the output procedure.

0. The general notion of a calculus The general concept of a calculus, or a deductive system (see [Mas 86]), is as fundamental as the concept of an algorithm and can be regarded separately from any formal definitions. The notion of a calculus reflects and generalizes an intuitive idea of inductive generation of a set (see [Mas 67], [Eb 70], [Mas 79]). Mathematical roots of the concept of a calculus go back to antiquity (see [Janovs 62]). Games with strict rules - such as chess, dominoes, majong - are probably the earliest examples of calculuses in a real world.