Aiming to harmonise finite and infinite model reasoning, we initiate the study of partially finite models, where the reasoning task comes with a formula that specifies a part of the model that must be finite. We focus on the problem of partially finite query entailment in description logics (DLs): given a knowledge base (KB), a query, and a distinguished concept, decide whether the query holds in all models of the KB that interpret the distinguished concept as a finite set. To break the ground, we work with the DL S, an extension of the basic DL ALC with transitive roles, which is one of the simplest cases where finite and infinite query entailment diverge. Generalising previous results on the finite and infinite cases, we show that also partially finite entailment of conjunctive queries is in 2-ExpTime for S. The solution involves sophisticated infinite model surgery and goes far beyond combining the arguments for the two special cases. As a direct application, we show how the problem of query containment in the presence of closed predicates can be solved by reduction to partially finite query entailment.

In this work we introduce new generalised quantifiers which allow us to express the Rabin-Mostowski index of automata. Our main results study expressive power and decidability of the monadic second-order (MSO) logic extended with these quantifiers. We study these problems in the realm of both ω-words and infinite trees. As it turns out, the pictures in these two cases are very different. In the case of ω-words the new quantifiers can be effectively expressed in pure MSO logic. In contrast, in the case of infinite trees, addition of these quantifiers leads to an undecidable formalism. To realise index-quantifier elimination, we consider the extension of MSO by game quantifiers. As a tool, we provide a specific quantifier-elimination procedure for them. Moreover, we introduce a novel construction of transducers realising strategies in ω-regular games with monadic parameters.

We study the existence of positional strategies for the protagonist in infinite duration games over arbitrary game graphs. We prove that prefix-independent objectives in $Σ_0^2$ which are positional and admit a (strongly) neutral letter are exactly those that are recognised by history-deterministic monotone co-Bchi automata over countable ordinals. This generalises a criterion proposed by [Kopczyński, ICALP 2006] and gives an alternative proof of closure under union for these objectives, which was known from [Ohlmann, TheoretiCS 2023]. We then give two applications of our result. First, we prove that the mean-payoff objective is positional over arbitrary game graphs. Second, we establish the following completeness result: for any objective $W$ which is prefix-independent, admits a (weakly) neutral letter, and is positional over finite game graphs, there is an objective $W'$ which is equivalent to $W$ over finite game graphs and positional over arbitrary game graphs.

We focus on formulae ∃X.φ(Y, X) of monadic second-order logic over the full binary tree, such that the witness X is a well-founded set. The ordinal rank rank(X) < ω₁ of such a set X measures its depth and branching structure. We search for the least upper bound for these ranks, and discover the following dichotomy depending on the formula φ. Let η_φ be the minimal ordinal such that, whenever an instance Y satisfies the formula, there is a witness X with rank(X) ≤ η_φ. Then η_φ is either strictly smaller than ω² or it reaches the maximal possible value ω₁. Moreover, it is decidable which of the cases holds. The result has potential for applications in a variety of ordinal-related problems, in particular it entails a result about the closure ordinal of a fixed-point formula.

A uniformisation of a binary relation is a functional relation contained in it, with the same domain. The uniformisation problem asks whether such a uniformisation can be defined in a given formalism.

This short note establishes positionality of mean-payoff games over infinite game graphs by constructing a well-founded monotone universal graph.

This paper studies the complexity of operations on finite automata and the complexity of their decision problems when the alphabet is unary. Let $n$ denote the maximum of the number of states of the input finite automata considered in the corresponding results. The following main results are obtained: (1) Given two unary NFAs recognising $L$ and $H$, respectively, one can decide whether $L \subseteq H$ as well as whether $L = H$ in time $2^{O((n \log n)^{1/3})}$. The previous upper bound on time was $2^{O((n \log n)^{1/2})}$ as given by Chrobak (1986), and this bound was not significantly improved since then. (2) Given two unary UFAs (unambiguous finite automata) recognising $L$ and $H$, respectively, one can determine a UFA recognising $L \cup H$ and a UFA recognising complement of $L$, where these output UFAs have the number of states bounded by a quasipolynomial in $n$. However, in the worst case, a UFA for recognising concatenation of languages recognised by two $n$-state UFAs, uses $2^{Θ((n \log^2 n)^{1/3})}$ states. (3) Given a unary language $L$, if $L$ contains the word of length $k$, then let $L(k)=1$ else let $L(k)=0$. Let $ω_L$ be the $ω$-word $L(0)L(1)\ldots$ and let $\cal L$ be a fixed $ω$-regular language. The last section studies how difficult it is to decide, given an $n$-state UFA or NFA

The Rabin tree theorem yields an algorithm to solve the satisfiability problem for monadic second-order logic over infinite trees. Here we solve the probabilistic variant of this problem. Namely, we show how to compute the probability that a randomly chosen tree satisfies a given formula. We additionally show that this probability is an algebraic number. This closes a line of research where similar results were shown for formalisms weaker than the full monadic second-order logic.

We show that it is decidable whether two regular languages of infinite trees are separable by a deterministic language, resp., a game language. We consider two variants of separability, depending on whether the set of priorities of the separator is fixed, or not. In each case, we show that separability can be decided in EXPTIME, and that separating automata of exponential size suffice. We obtain our results by reducing to infinite duration games with ω-regular winning conditions and applying the finite-memory determinacy theorem of Büchi and Landweber.

We study guidable parity automata over infinite trees introduced by Colcombet and Löding, which form an expressively complete subclass of all non-deterministic tree automata. We show that, for any non-deterministic automaton, an equivalent guidable automaton with the smallest possible index can be effectively found. Moreover, if an input automaton is of a special kind, i.e. it is deterministic or game automaton then a guidable automaton with an optimal index can be deterministic (respectively game) automaton as well. Recall that the problem whether an equivalent non-deterministic automaton with the smallest possible index can be effectively found is open, and a positive answer is known only in the case when an input automaton is a deterministic, or more generally, a game automaton.

We prove that ω-languages of (non-deterministic) Petri nets and ω-languages of (nondeterministic) Turing machines have the same topological complexity: the Borel and Wadge hierarchies of the class of ω-languages of (non-deterministic) Petri nets are equal to the Borel and Wadge hierarchies of the class of ω-languages of (non-deterministic) Turing machines. We also show that it is highly undecidable to determine the topological complexity of a Petri net ω-language. Moreover, we infer from the proofs of the above results that the equivalence and the inclusion problems for ω-languages of Petri nets are ∏21-complete, hence also highly undecidable. Additionally, we show that the situation is quite the opposite when considering unambiguous Petri nets, which have the semantic property that at most one accepting run exists on every input. We provide a procedure of determinising them into deterministic Muller counter machines with counter copying. As a consequence, we entail that the ω-languages recognisable by unambiguous Petri nets are △30 sets.

We study alternating parity good-for-games (GFG) automata, i.e., alternating parity automata where both conjunctive and disjunctive choices can be resolved in an online manner, without knowledge of the suffix of the input word still to be read. We show that they can be exponentially more succinct than both their nondeterministic and universal counterparts. Furthermore, we present a single exponential determinisation procedure and an Exptime upper bound to the problem of recognising whether an alternating automaton is GFG. We also study the complexity of deciding "half-GFGness", a property specific to alternating automata that only requires nondeterministic choices to be resolved in an online manner. We show that this problem is PSpace-hard already for alternating automata on finite words.

We consider the problem of deciding whether a given mso-definable relation over bi-infinite words contains an mso-definable function with the same domain. We prove that this problem is decidable. There are two obstacles to the existence of such uniformisations: the first is related to the existence of non-trivial automorphisms of bi-infinite words, whereas the second, more subtle obstacle, is related to the existence of finite, discrete dynamical systems, where no trajectory can be selected by an mso formula.

We view languages of words over a product alphabet A x B as relations between words over A and words over B. This leads to the notion of regular relations - relations given by a regular language. We ask when it is possible to find regular uniformisations of regular relations. The answer depends on the structure or shape of the underlying model: it is true e.g. for ω-words, while false for words over ℤ or for infinite trees. In this paper we focus on countable orders. Our main result characterises, which countable linear orders D have the property that every regular relation between words over D has a regular uniformisation. As it turns out, the only obstacle for uniformisability is the one displayed in the case of ℤ - non-trivial automorphisms of the given structure. Thus, we show that either all regular relations over D have regular uniformisations, or there is a non-trivial automorphism of D and even the simple relation of choice cannot be uniformised. Moreover, this dichotomy is effective.

This paper is about an extension of monadic second-order logic over the full binary tree, which has a quantifier saying “almost surely a branch π ∈ {0,1}ωsatisfies a formula φ(π)”, This logic was introduced by Michalewski and Mio; we call it MSO+∇ following notation of Shelah and Lehmann. The logic Mso+∇ subsumes many qualitative probabilistic formalisms, including qualitative probabilistic check, probabilistic LTL, or parity tree automata with probabilistic acceptance conditions. We show that it is undecidable to check if a given sentence of MSO+∇ is true in the full binary tree11Independently and in parallel another proof of this result was given employing different techniques in [3]..

Given R a binary relation between words (which we treat as a language over a product alphabet AxB), a uniformisation of it is another relation L included in R which chooses a single word over B, for each word over A whenever there exists one. It is known that MSO, the full class of regular languages, is strong enough to define a uniformisation for each of its relations. The quest of this work is to see which other formalisms, weaker than MSO, also have this property. In this paper, we solve this problem for pseudo-varieties of semigroups: we show that no nonempty pseudo-variety weaker than MSO can provide uniformisations for its relations.

We study the strength of axioms needed to prove various results related to automata on infinite words and B\"uchi's theorem on the decidability of the MSO theory of $(N, {\le})$. We prove that the following are equivalent over the weak second-order arithmetic theory $RCA_0$: (1) the induction scheme for $\Sigma^0_2$ formulae of arithmetic, (2) a variant of Ramsey's Theorem for pairs restricted to so-called additive colourings, (3) B\"uchi's complementation theorem for nondeterministic automata on infinite words, (4) the decidability of the depth-$n$ fragment of the MSO theory of $(N, {\le})$, for each $n \ge 5$. Moreover, each of (1)-(4) implies McNaughton's determinisation theorem for automata on infinite words, as well as the "bounded-width" version of K\"onig's Lemma, often used in proofs of McNaughton's theorem.

This work is a study of the expressive power of unambiguity in the case of automata over infinite trees. An automaton is called unambiguous if it has at most one accepting run on every input, the language of such an automaton is called an unambiguous language. It is known that not every regular language of infinite trees is unambiguous. Except that, very little is known about which regular tree languages are unambiguous. This paper answers the question whether unambiguous languages are of bounded complexity among all regular tree languages. The notion of complexity is the canonical one, called the (parity or Rabin-Mostowski) index hierarchy. The answer is negative, as exhibited by a family of examples of unambiguous languages that cannot be recognised by any alternating parity tree automata of bounded range of priorities. Hardness of the given examples is based on the theory of signatures in parity games, previously studied by Walukiewicz. This theory is further developed here to construct canonical signatures. The technical core of the article is a parity game that compares signatures of a given pair of parity games (without an increase in the index).

We investigate the extension of Monadic Second Order logic, interpreted over infinite words and trees, with generalized "for almost all" quantifiers interpreted using the notions of Baire category and Lebesgue measure.

In GFG automata, it is possible to resolve nondeterminism in a way that only depends on the past and still accepts all the words in the language. The motivation for GFG automata comes from their adequacy for games and synthesis, wherein general nondeterminism is inappropriate. We continue the ongoing effort of studying the power of nondeterminism in GFG automata. Initial indications have hinted that every GFG automaton embodies a deterministic one. Today we know that this is not the case, and in fact GFG automata may be exponentially more succinct than deterministic ones. We focus on the typeness question, namely the question of whether a GFG automaton with a certain acceptance condition has an equivalent GFG automaton with a weaker acceptance condition on the same structure. Beyond the theoretical interest in studying typeness, its existence implies efficient translations among different acceptance conditions. This practical issue is of special interest in the context of games, where the Buchi and co-Buchi conditions admit memoryless strategies for both players. Typeness is known to hold for deterministic automata and not to hold for general nondeterministic automata. We show that GFG automata enjoy the benefits of typeness, similarly to the case of deterministic automata. In particular, when Rabin or Streett GFG automata have equivalent Buchi or co-Buchi GFG automata, respectively, then such equivalent automata can be defined on a substructure of the original automata. Using our typeness results, we further study the place of GFG automata in between deterministic and nondeterministic ones. Specifically, considering automata complementation, we show that GFG automata lean toward nondeterministic ones, admitting an exponential state blow-up in the complementation of a Streett automaton into a Rabin automaton, as opposed to the constant blow-up in the deterministic case.

We show an algorithm that for a given regular tree language L decides if L is in Pi^0_2, that is if L belongs to the second level of Borel Hierarchy. Moreover, if L is in Pi^0_2, then we construct a weak alternating automaton of index (0, 2) which recognises L. We also prove that for a given language L, L is recognisable by a weak alternating (1, 3)-automaton if and only if it is recognisable by a weak non-deterministic (1, 3)-automaton.

We study the topological complexity of languages of Büchi automata on infinite binary trees. We show that such a language is either Borel and WMSO-definable, or Sigma_1^1-complete and not WMSO-definable; moreover it can be algorithmically decided which of the two cases holds. The proof relies on a direct reduction to deciding the winner in a finite game with a regular winning condition.

Infinite duration games with regular conditions are one of the crucial tools in the areas of verification and synthesis. In this paper we consider a branching variant of such games - the game contains branching vertices that split the play into two independent sub-games. Thus, a play has the form of~an~infinite tree. The winner of the play is determined by a winning condition specified as a set of infinite trees. Games of this kind were used by Mio to provide a game semantics for the probabilistic mu-calculus. He used winning conditions defined in terms of parity games on trees. In this work we consider a more general class of winning conditions, namely those definable by finite automata on infinite trees. Our games can be seen as a branching-time variant of the stochastic games on graphs. We address the question of determinacy of a branching game and the problem of computing the optimal game value for each of the players. We consider both the stochastic and non-stochastic variants of the games. The questions under consideration are parametrised by the family of strategies we allow: either mixed, behavioural, or pure. We prove that in general, branching games are not determined under mixed strategies. This holds even for topologically simple winning conditions (differences of two open sets) and non-stochastic arenas. Nevertheless, we show that the games become determined under mixed strategies if we restrict the winning conditions to open sets of trees. We prove that the problem of comparing the game value to a rational threshold is undecidable for branching games with regular conditions in all non-trivial stochastic cases. In the non-stochastic cases we provide exact bounds on the complexity of the problem. The only case left open is the 0-player stochastic case, i.e. the problem of computing the measure of a given regular language of infinite trees.

An infinite tree is called thin if it contains only countably many infinite branches. Thin trees can be seen as intermediate structures between infinite words and infinite trees. In this work we investigate properties of regular languages of thin trees. Our main tool is an algebra suitable for thin trees. Using this framework we characterize various classes of regular languages: commutative, open in the standard topology, and definable in weak MSO logic among all trees. We also show that in various meanings thin trees are not as rich as all infinite trees. In particular we observe a collapse of the parity index to the level (1, 3) and a collapse of the topological complexity to co-analytic sets. Moreover, a gap property is shown: a regular language of thin trees is either weak MSO-definable among all trees or co-analytic-complete.

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

A nondeterministic automaton is called unambiguous if it has at most one accepting run on every input. A regular language is called unambiguous if there exists an unambiguous automaton recognizing this language. Currently, the class of unambiguous languages of infinite trees is not well-understood. In particular, there is no known decision procedure verifying if a given regular tree language is unambiguous. In this work we study the self-dual class of bi-unambiguous languages - languages that are unambiguous and their complement is also unambiguous. It turns out that thin trees (trees with only countably many branches) emerge naturally in this context. We propose a procedure P designed to decide if a given tree automaton recognizes a bi-unambiguous language. The procedure is sound for every input. It is also complete for languages recognisable by deterministic automata. We conjecture that P is complete for all inputs but this depends on a new conjecture stating that there is no MSO-definable choice function on thin trees. This would extend a result by Gurevich and Shelah on the undefinability of choice on the binary tree. We provide a couple of equivalent statements to our conjecture, we also give several related results about uniformizability on thin trees. In particular, we provide a new example of a language that is not unambiguous, namely the language of all thin trees. The main tool in our studies are algebras that can be seen as an adaptation of Wilke algebras to the case of infinite trees.

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.

An infinite tree is called thin if it contains only countably many infinite branches. Thin trees can be seen as intermediate structures between infinite words and infinite trees. In this work we investigate properties of regular languages of thin trees. Our main tool is an algebra suitable for thin trees. Using this framework we characterize various classes of regular languages: commutative, open in the standard topology, closed under two variants of bisimulational equivalence, and definable in WMSO logic among all trees. We also show that in various meanings thin trees are not as rich as all infinite trees. In particular we observe a parity index collapse to level (1,3) and a topological complexity collapse to co-analytic sets. Moreover, a gap property is shown: a regular language of thin trees is either WMSO-definable among all trees or co-analytic-complete.

This work shows that for each i ∈ ω there exists a $\Sigma ^1_i$-hard ω-word language definable in Monadic Second Order Logic extended with the unbounding quantifier (MSO+U). This quantifier was introduced by Bojańczyk to express some asymptotic prop

An infinite binaryword can be identified with a branch in the full binary tree. We consider sets of branches definable in monadic second-order logic over the tree, where we allow some extra monadic predicates on the nodes. We show that this class equals to the Boolean combinations of sets in the Borel class Σ $^0_2$ over the Cantor discontinuum. Note that the last coincides with the Borel complexity of ω-regular languages.