Kleene Algebra with Tests and Commutativity Conditions

We give an equational proof, using Kleene algebra with tests and commutativity conditions, of the following classical result: every while program can be simulated by a while program with at most one while loop. The proof illustrates the use of Kleene algebra with extra conditions in program equivalence proofs. We also show, using a construction of Cohen, that the universal Horn theory of *-continuous Kleene algebras is not finitely axiomatizable.

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