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Journal of Logic and Computation Advance Access originally published online on October 28, 2006
Journal of Logic and Computation 2007 17(1):199-217; doi:10.1093/logcom/exl036
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© The Author, 2006. Published by Oxford University Press. All rights reserved. for Permissions, please email: journals.permissions@oxfordjournals.org

Original Articles

On Action Logic: Equational Theories of Action Algebras

Wojciech Buszkowski

Faculty of Mathematics and Computer Science, Adam Mickiewicz University, Poznan, Poland. E-mail: buszko{at}amu.edu.pl

Received 12 December 2005.


  Abstract

Pratt (1991, Proceedings of JELIA’90, Volume 478, pp. 97–120) defines action algebras as Kleene algebras with residuals and action logic as the equational theory of action algebras. In contrast to Kleene algebras, action algebras form a (finitely based) variety. Jipsen (2004, Studia Logica, 76, 291–303) proposes a Gentzen-style sequent system for action logic but leaves it as an open question if this system admits cut-elimination and if action logic is decidable. We show that Jipsen's system does not admit cut-elimination. We prove that the equational theory of *-continuous action algebras and the simple Horn theory of *-continuous Kleene algebras are not recursively enumerable and they possess FMP, but action logic does not possess FMP.

Keywords: Kleene algebra; action algebra; action lattice; undecidability; finite model property


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