Mikołaj Bojańczyk

Two-Variable Logic on Words with Data

In a data word each position carries a label from a finite alphabet and a data value from some infinite domain. These models have been already considered in the realm of semistructured data, timed automata and extended temporal logics. It is shown that satisfiability for the two-variable first-order logic FO2(~,<,+1) is decidable over finite and over infinite data words, where ~ is a binary predicate testing the data value equality and +1,< are the usual successor and order predicates. The complexity of the problem is at least as hard as Petri net reachability. Several extensions of the logic are considered, some remain decidable while some are undecidable

Two-variable logic on data words

In a data word each position carries a label from a finite alphabet and a data value from some infinite domain. This model has been already considered in the realm of semistructured data, timed automata, and extended temporal logics. This article shows that satisfiability for the two-variable fragment FO2(∼,<,+1) of first-order logic with data equality test ∼ is decidable over finite and infinite data words. Here +1 and < are the usual successor and order predicates, respectively. The satisfiability problem is shown to be at least as hard as reachability in Petri nets. Several extensions of the logic are considered; some remain decidable while some are undecidable.

Automata with Group Actions

Our motivating question is a My hill-Nerode theorem for infinite alphabets. We consider several kinds of those: alphabets whose letters can be compared only for equality, but also ones with more structure, such as a total order or a partial order. We develop a framework for studying such alphabets, where the key role is played by the automorphism group of the alphabet. This framework builds on the idea of nominal sets of Gabbay and Pitts, nominal sets are the special case of our framework where letters can be only compared for equality. We use the framework to uniformly generalize to infinite alphabets parts of automata theory, including decidability results. In the case of letters compared for equality, we obtain automata equivalent in expressive power to finite memory automata, as defined by Francez and Kaminski.

Two-variable logic on data trees and XML reasoning

Motivated by reasoning tasks in the context of XML languages, the satisfiability problem of logics on data trees is investigated. The nodes of a data tree have a label from a finite set and a data value from a possibly infinite set. It is shown that satisfiability for two-variable first-order logic is decidable if the tree structure can be accessed only through the child and the next sibling predicates and the access to data values is restricted to equality tests. From this main result decidability of satisfiability and containment for a data-aware fragment of XPath and of the implication problem for unary key and inclusion constraints is concluded.

Bounds in w-Regularity

We consider an extension of omega-regular expressions where two new variants of the Kleene star L* are added: LB and LS . These exponents act as the standard star, but restrict the number of iterations to be bounded (for LB) or to tend toward infinity (for LS). These expressions can define languages that are not omega-regular. We develop a theory for these languages. We study the decidability and closure questions. We also define an equivalent automaton model, extending Buchi automata. This culminates with a - partial -complementation result

Automata theory in nominal sets

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.

Towards nominal computation

Nominal sets are a different kind of set theory, with a more relaxed notion of finiteness. They offer an elegant formalism for describing lambda-terms modulo alpha-conversion, or automata on data words. This paper is an attempt at defining computation in nominal sets. We present a rudimentary programming language, called Nlambda. The key idea is that it includes a native type for finite sets in the nominal sense. To illustrate the power of our language, we write short programs that process automata on data words.

Characterizing EF and EX tree logics

The expressive power of temporal branching time logics that use the modalities EX and EF is described. Forbidden pattern characterizations are given for tree languages definable in three logics: EX, EF and EX + EF. The characterizations give algorithms for the definability problem in the respective logics that are polynomial in the size of a deterministic tree automaton representing the language.

Piecewise Testable Tree Languages

This paper presents a decidable characterization of tree languages that can be defined by a boolean combination of Sigma1 formulas. This is a tree extension of the Simon theorem, which says that a string language can be defined by a boolean combination of Sigma1 formulas if and only if its syntactic monoid is J-trivial.

A Bounding Quantifier

The logic MSOL+\(\mathbb{B}\) is defined, by extending monadic second-order logic on the infinite binary tree with a new bounding quantifier\(\mathbb{B}\). In this logic, a formula \(\mathbb{B}\)X. φ(X) states that there is a finite bound on the size of sets satisfying φ(X). Satisfiability is proved decidable for two fragments of MSOL+\(\mathbb{B}\): formulas of the form \(\neg\mathbb{B}\)X.φ(X), with φ a \(\mathbb{B}\)-free formula; and formulas built from \(\mathbb{B}\)-free formulas by nesting \(\mathbb{B}\), ∃, ∨ and ∧.