Abstract
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 \(\mathsf {KAT}\).
In honor of Rohit Parikh
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Acknowledgements
Many thanks to Jiří Adámek, Marcello Bonsangue, Helle Hvid Hansen, Raul Leal, Jan Rutten, Mehrnoosh Sadrzadeh, Luigi Santocanale, Alexandra Silva, Yde Venema, and James Worthington. An earlier version of this work (Kozen 2008) was presented at CMCS 2008 and was supported by NSF Grant CCF-0635028. The preparation of this revised version was supported by NSF grants CCF-1535952 and CCF-1637532 and by the National Security Agency.
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Kozen, D. (2017). On the Coalgebraic Theory of Kleene Algebra with Tests. In: Başkent, C., Moss, L., Ramanujam, R. (eds) Rohit Parikh on Logic, Language and Society. Outstanding Contributions to Logic, vol 11. Springer, Cham. https://doi.org/10.1007/978-3-319-47843-2_15
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