Program Algebra over an Algebra

Grzegorz Bancerek

doi:10.2478/v10037-012-0037-6

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Program Algebra over an Algebra

Formalized Mathematics

Volume 20 (2012): Issue 4 (December 2012)

By: Grzegorz Bancerek

Open Access

|Feb 2013

Abstract

We introduce an algebra with free variables, an algebra with undefined values, a program algebra over a term algebra, an algebra with integers, and an algebra with arrays. Program algebra is defined as universal algebra with assignments. Programs depend on the set of generators with supporting variables and supporting terms which determine the value of free variables in the next state. The execution of a program is changing state according to successor function using supporting terms.

References

[1] Grzegorz Bancerek. Cardinal numbers. Formalized Mathematics, 1(2):377-382, 1990.
Search in Google Scholar
[2] Grzegorz Bancerek. Introduction to trees. Formalized Mathematics, 1(2):421-427, 1990.
Search in Google Scholar
[3] Grzegorz Bancerek. K¨onig’s theorem. Formalized Mathematics, 1(3):589-593, 1990.
Search in Google Scholar
[4] Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91-96, 1990.
Search in Google Scholar
[5] Grzegorz Bancerek. Cartesian product of functions. Formalized Mathematics, 2(4):547-552, 1991.
Search in Google Scholar
[6] Grzegorz Bancerek. K¨onig’s lemma. Formalized Mathematics, 2(3):397-402, 1991.
Search in Google Scholar
[7] Grzegorz Bancerek. Algebra of morphisms. Formalized Mathematics, 6(2):303-310, 1997.
Search in Google Scholar
[8] Grzegorz Bancerek. Institution of many sorted algebras. Part I: Signature reduct of an algebra. Formalized Mathematics, 6(2):279-287, 1997.
Search in Google Scholar
[9] Grzegorz Bancerek. Mizar analysis of algorithms: Preliminaries. Formalized Mathematics, 15(3):87-110, 2007, doi:10.2478/v10037-007-0011-x.10.2478/v10037-007-0011-x
Search in Google Scholar
[10] Grzegorz Bancerek. Sorting by exchanging. Formalized Mathematics, 19(2):93-102, 2011, doi: 10.2478/v10037-011-0015-4.10.2478/v10037-011-0015-4
Search in Google Scholar
[11] Grzegorz Bancerek. Free term algebras. Formalized Mathematics, 20(3):239-256, 2012, doi: 10.2478/v10037-012-0029-6.10.2478/v10037-012-0029-6
Search in Google Scholar
[12] Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107-114, 1990.
Search in Google Scholar
[13] Grzegorz Bancerek and Piotr Rudnicki. The set of primitive recursive functions. FormalizedMathematics, 9(4):705-720, 2001.
Search in Google Scholar
[14] Grzegorz Bancerek and Andrzej Trybulec. Miscellaneous facts about functions. FormalizedMathematics, 5(4):485-492, 1996.
Search in Google Scholar
[15] Ewa Burakowska. Subalgebras of the universal algebra. Lattices of subalgebras. FormalizedMathematics, 4(1):23-27, 1993.
Search in Google Scholar
[16] Czesław Bylinski. Binary operations. Formalized Mathematics, 1(1):175-180, 1990.
Search in Google Scholar
[17] Czesław Bylinski. Finite sequences and tuples of elements of a non-empty sets. FormalizedMathematics, 1(3):529-536, 1990.
Search in Google Scholar
[18] Czesław Bylinski. Functions and their basic properties. Formalized Mathematics, 1(1):55-65, 1990.
Search in Google Scholar
[19] Czesław Bylinski. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.
Search in Google Scholar
[20] Czesław Bylinski. The modification of a function by a function and the iteration of the composition of a function. Formalized Mathematics, 1(3):521-527, 1990.
Search in Google Scholar
[21] Czesław Bylinski. Partial functions. Formalized Mathematics, 1(2):357-367, 1990.
Search in Google Scholar
[22] Czesław Bylinski. Some basic properties of sets. Formalized Mathematics, 1(1):47-53, 1990.
Search in Google Scholar
[23] Agata Darmochwał. Finite sets. Formalized Mathematics, 1(1):165-167, 1990.
Search in Google Scholar
[24] Artur Korniłowicz. On the group of automorphisms of universal algebra & many sorted algebra. Formalized Mathematics, 5(2):221-226, 1996.
Search in Google Scholar
[25] Artur Korniłowicz and Marco Riccardi. The Borsuk-Ulam theorem. Formalized Mathematics, 20(2):105-112, 2012, doi: 10.2478/v10037-012-0014-0.10.2478/v10037-012-0014-0
Search in Google Scholar
[26] Małgorzata Korolkiewicz. Homomorphisms of many sorted algebras. Formalized Mathematics, 5(1):61-65, 1996.
Search in Google Scholar
[27] Jarosław Kotowicz, Beata Madras, and Małgorzata Korolkiewicz. Basic notation of universal algebra. Formalized Mathematics, 3(2):251-253, 1992.
Search in Google Scholar
[28] Yatsuka Nakamura and Grzegorz Bancerek. Combining of circuits. Formalized Mathematics, 5(2):283-295, 1996.
Search in Google Scholar
[29] Andrzej Nedzusiak. Probability. Formalized Mathematics, 1(4):745-749, 1990.
Search in Google Scholar
[30] Beata Perkowska. Free many sorted universal algebra. Formalized Mathematics, 5(1):67-74, 1996.
Search in Google Scholar
[31] Andrzej Trybulec. Binary operations applied to functions. Formalized Mathematics, 1(2):329-334, 1990.
Search in Google Scholar
[32] Andrzej Trybulec. Enumerated sets. Formalized Mathematics, 1(1):25-34, 1990.
Search in Google Scholar
[33] Andrzej Trybulec. Many sorted sets. Formalized Mathematics, 4(1):15-22, 1993.
Search in Google Scholar
[34] Andrzej Trybulec. Many sorted algebras. Formalized Mathematics, 5(1):37-42, 1996.
Search in Google Scholar
[35] Andrzej Trybulec. A scheme for extensions of homomorphisms of many sorted algebras. Formalized Mathematics, 5(2):205-209, 1996.
Search in Google Scholar
[36] Andrzej Trybulec. On the sets inhabited by numbers. Formalized Mathematics, 11(4):341-347, 2003.
Search in Google Scholar
[37] Michał J. Trybulec. Integers. Formalized Mathematics, 1(3):501-505, 1990.
Search in Google Scholar
[38] Wojciech A. Trybulec. Pigeon hole principle. Formalized Mathematics, 1(3):575-579, 1990.
Search in Google Scholar
[39] Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.
Search in Google Scholar
[40] Tetsuya Tsunetou, Grzegorz Bancerek, and Yatsuka Nakamura. Zero-based finite sequences. Formalized Mathematics, 9(4):825-829, 2001.
Search in Google Scholar
[41] Edmund Woronowicz. Many argument relations. Formalized Mathematics, 1(4):733-737, 1990.
Search in Google Scholar
[42] Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1(1):73-83, 1990.
Search in Google Scholar