Abstract
We refine and extend the known results that the set of ordinary binary relations forms a Kleene algebra, the set of up-closed multirelations forms a lazy Kleene algebra, the set of up-closed finite multirelations forms a monodic tree Kleene algebra, and the set of total up-closed finite multirelations forms a probabilistic Kleene algebra. For the refinement, we introduce a notion of type of multirelations. For each of eight classes of relaxation of Kleene algebra, we give a sufficient condition on type T so that the set of up-closed multirelations of T belongs to the class. Some of the conditions are not only sufficient, but also necessary.
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References
Kozen, D.: A Completeness Theorem for Kleene Algebras and the Algebra of Regular Events. Inf. Comput. 110(2), 366–390 (1994)
Möller, B.: Lazy Kleene Algebra. In: Kozen, D., Shankland, C. (eds.) MPC 2004. LNCS, vol. 3125, pp. 252–273. Springer, Heidelberg (2004)
Takai, T., Furusawa, H.: Monodic Tree Kleene Algebra. In: Schmidt, R.A. (ed.) RelMiCS/AKA 2006. LNCS, vol. 4136, pp. 402–416. Springer, Heidelberg (2006)
McIver, A., Weber, T.: Towards Automated Proof Support for Probabilistic Distributed Systems. In: Sutcliffe, G., Voronkov, A. (eds.) LPAR 2005. LNCS (LNAI), vol. 3835, pp. 534–548. Springer, Heidelberg (2005)
Parikh, R.: The Logic of Games. Annals of Discrete Mathematics 24, 111–140 (1985)
Pauly, M., Parikh, R.: Game Logic - An Overview. Studia Logica 75(2), 165–182 (2003)
Goranko, V.: The Basic Algebra of Game Equivalences. Studia Logica 75(2), 221–238 (2003)
Venema, Y.: Representation of Game Algebras. Studia Logica 75(2), 239–256 (2003)
Furusawa, H., Tsumagari, N., Nishizawa, K.: A Non-probabilistic Relational Model of Probabilistic Kleene Algebras. In: Berghammer, R., Möller, B., Struth, G. (eds.) RelMiCS/AKA 2008. LNCS, vol. 4988, pp. 110–122. Springer, Heidelberg (2008)
Tsumagari, N., Nishizawa, K., Furusawa, H.: Multirelational Model of Lazy Kleene Algebra. In: Berghammer, R., Möller, B., Struth, G. (eds.) Relations and Kleene Algebra in Computer Science, PhD Programme at RelMiCS10/AKA5 2008-04, Institut für Informatik, Universität Augsburg, Germany (April 2008)
Tsumagari, N., Nishizawa, K., Furusawa, H.: Reflexive Transitive Closure of Binary Multirelations. In: Proc. of 25th Conference of Japan Society for Software Science and Technology, Tokyo, Japan (September 2008) (in Japanese)
Furusawa, H., Nishizawa, K., Tsumagari, N.: Multirelational Models of Lazy, Monodic Tree, and Probabilistic Kleene Algebras. Bulletin of Informatics and Cybernetics (to appear)
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Nishizawa, K., Tsumagari, N., Furusawa, H. (2009). The Cube of Kleene Algebras and the Triangular Prism of Multirelations. In: Berghammer, R., Jaoua, A.M., Möller, B. (eds) Relations and Kleene Algebra in Computer Science. RelMiCS 2009. Lecture Notes in Computer Science, vol 5827. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-04639-1_19
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DOI: https://doi.org/10.1007/978-3-642-04639-1_19
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