On the Regular Emptiness Problem of Subzero Automata

In the fundamental paper [5] Rabin proved that the monadic second order logic (MSO) of the full binary tree is decidable using the automata method. This proof technique can be roughly described as follows: (1) an appropriate notion of tree automaton is defined; (2) every formula φ of the MSO logic is effectively associated to an automaton Aφ such that that φ is true if and only if Aφ accepts a non-empty language; (3) the emptiness problem, that is deciding if Aφ accepts a non-empty language, is proved to be decidable. The latter point is typically established using combinatorial reasoning about the graph structure of Aφ . Recently, Michalewski and Mio have investigated in [4] an extension of the MSO logic of the full binary tree, called MSO + ∀=1, capable of expressing probabilistic properties. While the full logic MSO + ∀=1 is undecidable (see Section 5 of [4]), the decidability of an interesting fragment called MSO+∀=1 π (see Section 6 of [4]), capable of expressing many probabilistic properties useful in program verification such as those definable by the logic pCTL and its variants, is an open problem. Bojańczyk proposed in [1] to use the automata method to prove the decidability of the weak fragment of MSO+ ∀=1 π , where second-order quantification is restricted to finite sets, which is still sufficiently expressive to express most useful probabilistic properties in program verification. Namely, Bojańczyk has: (1) introduced a special class of zero automata and (2) proved that for every formula φ of weakMSO+∀=1 π one can effectively associate a zero automaton Aφ such that φ is true if and only if Aφ accepts a non-empty language. Hence what is still missing from Bojańczyk’s approach is a proof of decidability of the emptiness problem of zero automata. In this paper we consider a simplified version of zero automata introduced by Bojańczyk1. We call the simplified class subzero automata and prove the following result: