Certifying Inexpressibility

Abstract

Different classes of automata on infinite words have different expressive power. Deciding whether a given language $$L \subseteq \varSigma ^\omega $$ L ⊆ Σ ω can be expressed by an automaton of a desired class can be reduced to deciding a game between Prover and Refuter: in each turn of the game, Refuter provides a letter in $$\varSigma $$ Σ , and Prover responds with an annotation of the current state of the run (for example, in the case of Büchi automata, whether the state is accepting or rejecting, and in the case of parity automata, what the color of the state is). Prover wins if the sequence of annotations she generates is correct: it is an accepting run iff the word generated by Refuter is in L. We show how a winning strategy for Refuter can serve as a simple and easy-to-understand certificate to inexpressibility, and how it induces additional forms of certificates. Our framework handles all classes of deterministic automata, including ones with structural restrictions like weak automata. In addition, it can be used for refuting separation of two languages by an automaton of the desired class, and for finding automata that approximateL and belong to the desired class.

Certifying DFA Bounds for Recognition and Separation

“…Our setting considers automata on finite words, and it focuses on the number of states required for recognizing a regular language. In [12], we used a similar methodology for refuting the recognizability of ω-regular languages by automata with limited expressive power. For example, deterministic Büchi automata (DBAs) are less expressive than their non-deterministic counterpart, and a DBA-refuter generates certificates that a given language cannot be recognized by a DBA.…”

“…For example, deterministic Büchi automata (DBAs) are less expressive than their non-deterministic counterpart, and a DBA-refuter generates certificates that a given language cannot be recognized by a DBA. Thus, the setting in [12] is of automata on infinite words, and it focuses on expressive power.…”

“…Unlike DFAs, which allow polynomial minimization, minimization of DBAs is NP-complete [23]. Combining our setting here with the one in [12] would enable the certification and refutation of k-DBA-recognizability, namely recognizability by a DBA with k states. The NP-hardness of DBA minimization makes this combination very interesting.…”

Using the past for resolving the future

“…Proving that NBWs are strictly more expressive than DBWs, Landweber showed that the language L 1 = (0 + 1) * • 1 ω (only finitely many 0's) is in NBW\DBW. The proof is simple and can be stated in a few lines or using a two-state expressiveness refuter (Kupferman and Sickert, 2021). Much harder is the proof that NRTs are strictly more expressive than NBTs.…”