The automation of decision procedures makes certification essential. We suggest to use determinacy of turn-based two-player games with regular winning conditions in order to generate certificates for the number of states that a deterministic finite automaton (DFA) needs in order to recognize a given language. Given a language L and a bound k, recognizability of L by a DFA with k states is reduced to a game between Prover and Refuter. The interaction along the game then serves as a certificate. A certificate generated by Prover consists of a DFA with at most k states. A certificate generated by Refuter consists of a faulty attempt to define the required DFA. We compare the length of offline certificates, which are generated with no interaction between Prover and Refuter, and online certificates, which are based on such an interaction, and are thus shorter. We show that our approach is useful also for certification of separability of regular languages by a DFA of a given size. Unlike DFA minimization, which can be solved in polynomial time, separation is NP-complete, and thus the certification approach is essential. In addition, we prove NP-completeness of a strict version of separation.
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