On the Coalgebraic Theory of Kleene Algebra with Tests

We develop a coalgebraic theory of Kleene algebra with tests (\(\mathsf {KAT}\)) along the lines of Rutten (1998) for Kleene algebra (\(\mathsf {KA}\)) and Chen and Pucella (Electron Notes Theor Comput Sci 82(1), 2003) for a limited version of \(\mathsf {KAT}\), resolving some technical issues raised by Chen and Pucella. Our treatment includes a simple definition of the Brzozowski derivative for \(\mathsf {KAT}\) expressions and an automata-theoretic interpretation involving automata on guarded strings. We also give a complexity analysis, showing that an efficient implementation of coinductive equivalence proofs in this setting is tantamount to a standard automata-theoretic construction. It follows that coinductive equivalence proofs can be generated automatically in PSPACE. This matches the bound of Worthington (2008) for the automatic generation of equational proofs in \(\mathsf {KAT}\).