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Formalized Mathematics

(a computer assisted approach)

Editor-in-Chief: Matuszewski, Roman

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Volume 22, Issue 3


Algebraic Approach to Algorithmic Logic

Grzegorz Bancerek
Published Online: 2014-03-31 | DOI: https://doi.org/10.2478/forma-2014-0025


We introduce algorithmic logic - an algebraic approach according to [25]. It is done in three stages: propositional calculus, quantifier calculus with equality, and finally proper algorithmic logic. For each stage appropriate signature and theory are defined. Propositional calculus and quantifier calculus with equality are explored according to [24]. A language is introduced with language signature including free variables, substitution, and equality. Algorithmic logic requires a bialgebra structure which is an extension of language signature and program algebra. While-if algebra of generator set and algebraic signature is bialgebra with appropriate properties and is used as basic type of algebraic logic.

Keywords : propsitional calcus; quantifier calcus; algorithmic logic


  • [1] Grzegorz Bancerek. Mizar analysis of algorithms: Preliminaries. Formalized Mathematics, 15(3):87-110, 2007. doi:10.2478/v10037-007-0011-x.CrossrefGoogle Scholar

  • [2] Grzegorz Bancerek. Program algebra over an algebra. Formalized Mathematics, 20(4): 309-341, 2012. doi:10.2478/v10037-012-0037-6.CrossrefGoogle Scholar

  • [3] Grzegorz Bancerek. Cardinal numbers. Formalized Mathematics, 1(2):377-382, 1990.Google Scholar

  • [4] Grzegorz Bancerek. K¨onig’s theorem. Formalized Mathematics, 1(3):589-593, 1990.Google Scholar

  • [5] Grzegorz Bancerek. Algebra of morphisms. Formalized Mathematics, 6(2):303-310, 1997.Google Scholar

  • [6] Grzegorz Bancerek. Institution of many sorted algebras. Part I: Signature reduct of an algebra. Formalized Mathematics, 6(2):279-287, 1997.Google Scholar

  • [7] Grzegorz Bancerek. Translations, endomorphisms, and stable equational theories. Formalized Mathematics, 5(4):553-564, 1996.Google Scholar

  • [8] Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41-46, 1990.Google Scholar

  • [9] Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91-96, 1990.Google Scholar

  • [10] Grzegorz Bancerek. Sequences of ordinal numbers. Formalized Mathematics, 1(2):281-290, 1990.Google Scholar

  • [11] Grzegorz Bancerek. Ordinal arithmetics. Formalized Mathematics, 1(3):515-519, 1990.Google Scholar

  • [12] Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107-114, 1990.Google Scholar

  • [13] Grzegorz Bancerek and Andrzej Trybulec. Miscellaneous facts about functions. Formalized Mathematics, 5(4):485-492, 1996.Google Scholar

  • [14] Ewa Burakowska. Subalgebras of the universal algebra. Lattices of subalgebras. Formalized Mathematics, 4(1):23-27, 1993.Google Scholar

  • [15] Czesław Bylinski. Binary operations. Formalized Mathematics, 1(1):175-180, 1990.Google Scholar

  • [16] Czesław Bylinski. Finite sequences and tuples of elements of a non-empty sets. Formalized Mathematics, 1(3):529-536, 1990.Google Scholar

  • [17] Czesław Bylinski. Functions and their basic properties. Formalized Mathematics, 1(1): 55-65, 1990.Google Scholar

  • [18] Czesław Bylinski. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.Google Scholar

  • [19] 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.Google Scholar

  • [20] Czesław Bylinski. Partial functions. Formalized Mathematics, 1(2):357-367, 1990.Google Scholar

  • [21] Czesław Bylinski. Some basic properties of sets. Formalized Mathematics, 1(1):47-53, 1990.Google Scholar

  • [22] Agata Darmochwał. Finite sets. Formalized Mathematics, 1(1):165-167, 1990.Google Scholar

  • [23] Jarosław Kotowicz, Beata Madras, and Małgorzata Korolkiewicz. Basic notation of universal algebra. Formalized Mathematics, 3(2):251-253, 1992.Google Scholar

  • [24] Elliott Mendelson. Introduction to Mathematical Logic. Chapman Hall/CRC, 1997. http://books.google.pl/books?id=ZO1p4QGspoYC.Google Scholar

  • [25] Grazyna Mirkowska and Andrzej Salwicki. Algorithmic Logic. PWN-Polish Scientific Publisher, 1987.Google Scholar

  • [26] Beata Perkowska. Free universal algebra construction. Formalized Mathematics, 4(1): 115-120, 1993.Google Scholar

  • [27] Beata Perkowska. Free many sorted universal algebra. Formalized Mathematics, 5(1): 67-74, 1996.Google Scholar

  • [28] Krzysztof Retel. Properties of first and second order cutting of binary relations. Formalized Mathematics, 13(3):361-365, 2005.Google Scholar

  • [29] Andrzej Trybulec. Binary operations applied to functions. Formalized Mathematics, 1 (2):329-334, 1990.Google Scholar

  • [30] Andrzej Trybulec. Tuples, projections and Cartesian products. Formalized Mathematics, 1(1):97-105, 1990.Google Scholar

  • [31] Andrzej Trybulec. Many sorted algebras. Formalized Mathematics, 5(1):37-42, 1996.Google Scholar

  • [32] Andrzej Trybulec. Many sorted sets. Formalized Mathematics, 4(1):15-22, 1993.Google Scholar

  • [33] Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.Google Scholar

  • [34] Edmund Woronowicz. Many argument relations. Formalized Mathematics, 1(4):733-737, 1990.Google Scholar

  • [35] Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1 (1):73-83, 1990. Google Scholar

About the article

Received: 2014-09-15

Published Online: 2014-03-31

Published in Print: 2014-09-01

Citation Information: Formalized Mathematics, Volume 22, Issue 3, Pages 225–255, ISSN (Online) 1898-9934, DOI: https://doi.org/10.2478/forma-2014-0025.

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© by Grzegorz Bancerek. This work is licensed under the Creative Commons Attribution-ShareAlike 3.0 License. (CC BY-SA 3.0) BY-SA 3.0

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