By Vladimir Uspensky, A.L. Semenov

Today the proposal of the set of rules is general not just to mathematicians. It varieties a conceptual base for info processing; the lifestyles of a corresponding set of rules makes automated details processing attainable. the speculation of algorithms (together with mathematical common sense ) kinds the the oretical foundation for contemporary computing device technology (see [Sem Us 86]; this text is named "Mathematical good judgment in desktop technology and Computing perform" and in its name mathematical common sense is known in a large feel together with the idea of algorithms). even if, no longer each person realizes that the observe "algorithm" features a reworked toponym Khorezm. Algorithms have been named after an exceptional sci entist of medieval East, is al-Khwarizmi (where al-Khwarizmi ability "from Khorezm"). He lived among c. 783 and 850 B.C. and the yr 1983 used to be selected to have a good time his 1200th birthday. a brief 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 accordance with [Bul Rozen Ah eighty three, p.8]).

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Here the workspace is the aggregate of B-words; all states are basic and the output procedure is trivial (as in all algebraic examples we shall consider) - it does not change an object. The calculuses defined in this way are now called associative calculuses (though Thue did not use this term); the equivalence relation defined as above is called the equivalence relation in a given associative calculus (see [Nag 77a]). e. is preserved by left or right multiplication in the semigroup. A quotient semigroup with respect to this relation is call a semigroup with the set of generators B determined by the set of equalities Aj = Bj, j = 1, ...

Assume that any element of the underlying set is a value of some closed term; in this case B is called an algebraic system generated by a signature u. For example, any group with generators al, ... , a, is (according to our terminology) an algebraic system generated by the signature {al,. ,a"o, -I} or {e,al, . ,az,o, -I}. Any algebraic system generated by a signature is a homomorphic image of the free algebraic system generated by this signature. 3: Algebraic examples Equalities. Assume that a signature u is fixed.

A"o, -I} or {e,al, . ,az,o, -I}. Any algebraic system generated by a signature is a homomorphic image of the free algebraic system generated by this signature. 3: Algebraic examples Equalities. Assume that a signature u is fixed. By an equality of the signature u we mean an expression of the form t=s where t and s are closed terms of the signature u. All equalities and all expressions of the form p(tt, ... , tn) where p is a n-ary predicate symbol and tt, ... , tn are closed terms, are called closed atomic formulas.