Michał Skrzypczak

Measure Properties of Game Tree Languages

We introduce a general method for proving measurability of topologically complex sets by establishing a correspondence between the notion of game tree languages from automata theory and the σ-algebra of \(\mathcal{R}\)-sets, introduced by A. Kolmogorov as a foundation for measure theory. We apply the method to answer positively to an open problem regarding the game interpretation of the probabilistic μ-calculus.

The Topological Complexity of MSO+U and Related Automata Models

This work shows that for each i ∈ ω there exists a

Rabin-Mostowski Index Problem: A Step beyond Deterministic Automata

\Sigma ^1_i

On the Decidability of MSO+U on Infinite Trees

-hard ω-word language definable in Monadic Second Order Logic extended with the unbounding quantifier (MSO+U). This quantifier was introduced by Bojanczyk to express some asymptotic properties. Since it is not hard to see that each language expressible in MSO+U is projective, our finding solves the topological complexity of MSO+U. The result can immediately be transferred from ω-words to infinite labelled trees. As a consequence of the topological hardness we note that no alternating automaton with a Borel acceptance condition — or even with an acceptance condition of a bounded projective complexity — can capture all of MSO+U. The same holds for deterministic and nondeterministic automata since they are special cases of alternating ones. We also give exact topological complexities of related classes of languages recognized by nondeterministic ωB-, ωS- and ωBS-automata studied by Bojanczyk and Colcombet. Furthermore, we show that corresponding alternating automata have higher topological complexity than nondeterministic ones — they inhabit all finite levels of the Borel hierarchy. The paper is an extended journal version of [8]. The main theorem of that article is strengthened here.

On the topological complexity of MSO+U and related automata models

For a given regular language of infinite trees, one can ask about the minimal number of priorities needed to recognise this language with a non-deterministic or alternating parity automaton. These questions are known as, respectively, the non-deterministic and the alternating Rabin-Mostowski index problems. Whether they can be answered effectively is a long-standing open problem, solved so far only for languages recognisable by deterministic automata (the alternating variant trivialises). We investigate a wider class of regular languages, recognisable by so-called game automata, which can be seen as the closure of deterministic ones under complementation and composition. Game automata are known to recognise languages arbitrarily high in the alternating Rabin-Mostowski index hierarchy, i.e., the alternating index problem does not trivialise any more. Our main contribution is that both index problems are decidable for languages recognisable by game automata. Additionally, we show that it is decidable whether a given regular language can be recognised by a game automaton.

Nondeterminism in the presence of a diverse or unknown future

This paper is about MSO+U, an extension of monadic second-order logic, which has a quantifier that can express that a property of sets is true for arbitrarily large sets. We conjecture that the MSO+U theory of the complete binary tree is undecidable. We prove a weaker statement: there is no algorithm which decides this theory and has a correctness proof in zfc. This is because the theory is undecidable, under a set-theoretic assumption consistent with zfc, namely that there exists of projective well-ordering of 2 ω of type ω 1. We use Shelah’s undecidability proof of the MSO theory of the real numbers.

Unambiguity and uniformization problems on infinite trees

This work shows that for each i ∈ ω there exists a

On the Borel Complexity of MSO Definable Sets of Branches

\Sigma ^1_i

On Determinisation of Good-for-Games Automata

-hard ω-word language definable in Monadic Second Order Logic extended with the unbounding quantifier (MSO+U). This quantifier was introduced by Bojanczyk to express some asymptotic properties. Since it is not hard to see that each language expressible in MSO+U is projective, our finding solves the topological complexity of MSO+U. The result can immediately be transferred from ω-words to infinite labelled trees. As a consequence of the topological hardness we note that no alternating automaton with a Borel acceptance condition — or even with an acceptance condition of a bounded projective complexity — can capture all of MSO+U. The same holds for deterministic and nondeterministic automata since they are special cases of alternating ones. We also give exact topological complexities of related classes of languages recognized by nondeterministic ωB-, ωS- and ωBS-automata studied by Bojanczyk and Colcombet. Furthermore, we show that corresponding alternating automata have higher topological complexity than nondeterministic ones — they inhabit all finite levels of the Borel hierarchy. The paper is an extended journal version of [8]. The main theorem of that article is strengthened here.

Measure properties of regular sets of trees

Choices made by nondeterministic word automata depend on both the past (the prefix of the word read so far) and the future (the suffix yet to be read). In several applications, most notably synthesis, the future is diverse or unknown, leading to algorithms that are based on deterministic automata. Hoping to retain some of the advantages of nondeterministic automata, researchers have studied restricted classes of nondeterministic automata. Three such classes are nondeterministic automata that are good for trees (GFT; i.e., ones that can be expanded to tree automata accepting the derived tree languages, thus whose choices should satisfy diverse futures), good for games (GFG; i.e., ones whose choices depend only on the past), and determinizable by pruning (DBP; i.e., ones that embody equivalent deterministic automata). The theoretical properties and relative merits of the different classes are still open, having vagueness on whether they really differ from deterministic automata. In particular, while DBP ⊆ GFG ⊆ GFT, it is not known whether every GFT automaton is GFG and whether every GFG automaton is DBP. Also open is the possible succinctness of GFG and GFT automata compared to deterministic automata. We study these problems for ω-regular automata with all common acceptance conditions. We show that GFT=GFG⊃DBP, and describe a determinization construction for GFG automata.