arXiv subjects
Publications and source records attributed to Mikołaj Bojańczyk.
We consider extensions of monadic second order logic over $ω$-words, which are obtained by adding one language that is not $ω$-regular. We show that if the added language $L$ has a neutral letter, then the resulting logic is necessarily undecidable. A corollary is that the $ω$-regular languages are the only decidable Boolean-closed full trio over $ω$-words.
We prove the undecidability of MSO on $ω$-words extended with the second-order predicate $U_1(X)$ which says that the distance between consecutive positions in a set $X \subseteq \mathbb{N}$ is unbounded. This is achieved by showing that adding $U_1$ to MSO gives a logic with the same expressive power as $MSO+U$, a logic on $ω$-words with undecidable satisfiability. As a corollary, we prove that MSO on $ω$-words becomes undecidable if allowing to quantify over sets of positions that are ultimately periodic, i.e., sets $X$ such that for some positive integer $p$, ultimately either both or none of positions $x$ and $x+p$ belong to $X$.
$ω$-clones are multi-sorted structures that naturally emerge as algebras for infinite trees, just as $ω$-semigroups are convenient algebras for infinite words. In the algebraic theory of languages, one hopes that a language is regular if and only if it is recognized by an algebra that is finite in some simple sense. We show that, for infinite trees, the situation is not so simple: there exists an $ω$-clone that is finite on every sort and finitely generated, but recognizes a non-regular language.
In this article we provide effective characterisations of regular languages of infinite trees that belong to the low levels of the Wadge hierarchy. More precisely we prove decidability for each of the finite levels of the hierarchy; for the class of the Boolean combinations of open sets $BC(Σ_1^0)$ (i.e. the union of the first $ω$ levels); and for the Borel class $Δ_2^0$ (i.e. for the union of the first $ω_1$ levels).
String-to-string MSO interpretations are like Courcelle's MSO transductions, except that a single output position can be represented using a tuple of input positions instead of just a single input position. In particular, the output length is polynomial in the input length, as opposed to MSO transductions, which have output of linear length. We show that string-to-string MSO interpretations are exactly the polyregular functions. The latter class has various characterizations, one of which is that it consists of the string-to-string functions recognized by pebble transducers. Our main result implies the surprising fact that string-to-string MSO interpretations are closed under composition.
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 \in {0, 1}^w satisfies a formula ϕ(π)''. This logic was introduced by Michalewski and Mio; we call it MSO+nabla following notation of Shelah and Lehmann. The logic MSO+nabla subsumes many qualitative probabilistic formalisms, including qualitative probabilistic CTL, probabilistic LTL, or parity tree automata with probabilistic acceptance conditions. We show that it is undecidable to check if a given sentence of MSO+nabla is true in the full binary tree.
This paper is about certain string-to-string functions, called the polyregular functions. These are like the regular string-to-string functions, except that they can have polynomial (and not just linear) growth. The class has four equivalent definitions: 1. deterministic two-way transducers with pebbles; 2 the smallest class of string-to-string functions that is closed under composition, contains all sequential functions as well as two extra functions called squaring and iterated reverse 3. a fragment of the lambda-calculus, which has a list type constructor and limited forms of iteration such as map but not fold; 4. an imperative programming language, which has for loops that range over input positions. The first definition comes from [milo2003typechecking], while the remaining three are new to the author's best knowledge. The class of polyregular functions contains known classes of string-to-string transducers, such as the sequential, rational, or regular ones, but goes beyond them because of super-linear growth. Polyregular functions have good algorithmic properties, such as: - the output can be computed in linear time (in terms of combined input and output size); - the inverse image of a regular word language is (effectively) regular. We also identify a fragment of polyregular functions, called the first-order polyregular functions, which has additional good properties, e.g. the output can be computed by an AC0 circuit.
We define a new class of languages of $ω$-words, strictly extending $ω$-regular languages. One way to present this new class is by a type of regular expressions. The new expressions are an extension of $ω$-regular expressions where two new variants of the Kleene star $L^*$ are added: $L^B$ and $L^S$. These new exponents are used to say that parts of the input word have bounded size, and that parts of the input can have arbitrarily large sizes, respectively. For instance, the expression $(a^Bb)^ω$ represents the language of infinite words over the letters $a,b$ where there is a common bound on the number of consecutive letters $a$. The expression $(a^Sb)^ω$ represents a similar language, but this time the distance between consecutive $b$'s is required to tend toward the infinite. We develop a theory for these languages, with a focus on decidability and closure. We define an equivalent automaton model, extending Büchi automata. The main technical result is a complementation lemma that works for languages where only one type of exponent---either $L^B$ or $L^S$---is used. We use the closure and decidability results to obtain partial decidability results for the logic MSOLB, a logic obtained by extending monadic second-order logic with new quantifiers that speak about the size of sets.
The following problem is shown undecidable: given regular languages L,K of finite trees, decide if there exists a deterministic tree-walking automaton which accepts all trees in L and rejects all trees in K. The proof uses a technique of Kopczyński from LICS 2016.
One of the major open problems in automata and logic is the following: is there an algorithm which inputs a regular tree language and decides if the language can be defined in first-order logic? The goal of this paper is to present this problem and similar ones using the language of universal algebra, highlighting potential connections to the structural theory of finite algebras, including Tame Congruence Theory.
Subzero automata is a class of tree automata whose acceptance condition can express probabilistic constraints. Our main result is that the problem of determining if a subzero automaton accepts some regular tree is decidable.
We prove a conjecture of Courcelle, which states that a graph property is definable in MSO with modular counting predicates on graphs of constant treewidth if, and only if it is recognizable in the following sense: constant-width tree decompositions of graphs satisfying the property can be recognized by tree automata. While the forward implication is a classic fact known as Courcelle's theorem, the converse direction remained open
We consider the logic MSO+U, which is monadic second-order logic extended with the unbounding quantifier. The unbounding quantifier is used to say that a property of finite sets holds for sets of arbitrarily large size. We prove that the logic is undecidable on infinite words, i.e. the MSO+U theory of (N,<) is undecidable. This settles an open problem about the logic, and improves a previous undecidability result, which used infinite trees and additional axioms from set theory.
The principle behind algebraic language theory for various kinds of structures, such as words or trees, is to use a compositional function from the structures into a finite set. To talk about compositionality, one needs some way of composing structures into bigger structures. It so happens that category theory has an abstract concept for this, namely a monad. The goal of this paper is to propose monads as a unifying framework for discussing existing algebras and designing new algebras.
We study languages over infinite alphabets equipped with some structure that can be tested by recognizing automata. We develop a framework for studying such alphabets and the ensuing automata theory, where the key role is played by an automorphism group of the alphabet. In the process, we generalize nominal sets due to Gabbay and Pitts.
This paper shows that over infinite trees, satisfiability is decidable for weak monadic second-order logic extended by the unbounding quantifier U and quantification over infinite paths. The proof is by reduction to emptiness for a certain automaton model, while emptiness for the automaton model is decided using profinite trees.
Call a string-to-string transducer regular if it can be realised by one of the following equivalent models: mso transductions, two-way deterministic automata with output, and streaming transducers with registers. This paper proposes to treat origin information as part of the semantics of a regular string-to-string transducer. With such semantics, the model admits a machine-independent characterisation, Angluin-style learning in polynomial time, as well as effective characterisations of natural subclasses such as one-way transducers or first-order definable transducers.
We investigate finite deterministic automata in sets with non-homogeneous atoms: integers with successor. As there are uncount- ably many deterministic finite automata in this setting, we restrict our attention to automata with semilinear transition function. The main re- sults is a minimization procedure for semilinear automata. The proof is subtle and refers to decidability of existential Presburger arithmetic with divisibility predicates. Interestingly, the minimization is not obtained by the standard partition refinement procedure, and we demonstrate that this procedure does not necessarily terminate for semilinear automata.