Non-associative Kleene algebra and temporal logics
- In earlier papers we have presented algebraic semantics of CTL and CTL*. However, they were not fully satisfactory. In particular, the treatment of the iteration operators U ("Until") and W ("While") employed ad-hoc definitions with general recursion, and their interrelation in CTL* and CTL was not set up in a very algebraic way. In this paper we use a new operator with which U and W can be described by variants of the Kleene star and omega iteration. However, the operator is neither associative nor does it have a neutral element. Therefore we present a general investigation of the star and omega for such operators in what we call iteration algebras. The relation between various semantics for CTL* and CTL can then be expressed by homomorphisms between iteration algebras, which is more satisfactory than the original approach.
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