On the Expressive Power of Non-deterministic and Unambiguous Petri Nets over Infinite Words

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.

The Maximal Complexity of Quasiperiodic Infinite Words

A quasiperiod of a finite or infinite string is a word whose occurrences cover every part of the string. An infinite string is referred to as quasiperiodic if it has a quasiperiod. We present a characterisation of the set of infinite strings having a certain word q as quasiperiod via a finite language Pq consisting of prefixes of the quasiperiod q. It turns out its star root Pq* is a suffix code having a bounded delay of decipherability. This allows us to calculate the maximal subword (or factor) complexity of quasiperiodic infinite strings having quasiperiod q and further to derive that maximally complex quasiperiodic infinite strings have quasiperiods aba or aabaa. It is shown that, for every length l≥3, a word of the form anban (or anbban if l is even) generates the most complex infinite string having this word as quasiperiod. We give the exact ordering of the lengths l with respect to the achievable complexity among all words of length l.

The Overlap Gap Between Left-Infinite and Right-Infinite Words

We study ultimate periodicity properties related to overlaps between the suffixes of a left-infinite word [Formula: see text] and the prefixes of a right-infinite word [Formula: see text]. The main theorem states that the set of minimum lengths of words [Formula: see text] and [Formula: see text] such that [Formula: see text] or [Formula: see text] is finite, where [Formula: see text] runs over positive integers and [Formula: see text] and [Formula: see text] are respectively the suffix of [Formula: see text] and the prefix of [Formula: see text] of length [Formula: see text], if and only if [Formula: see text] and [Formula: see text] are ultimately periodic words of the form [Formula: see text] and [Formula: see text] for some finite words [Formula: see text], [Formula: see text] and [Formula: see text].

Two Effective Properties of ω-Rational Functions

We prove two new effective properties of rational functions over infinite words which are realized by finite state Büchi transducers. Firstly, for each such function [Formula: see text], one can construct a deterministic Büchi automaton [Formula: see text] accepting a dense [Formula: see text]-subset of [Formula: see text] such that the restriction of [Formula: see text] to [Formula: see text] is continuous. Secondly, we give a new proof of the decidability of the first Baire class for synchronous [Formula: see text]-rational functions from which we get an extension of this result involving the notion of Wadge classes of regular [Formula: see text]-languages.

Minimization of Limit-Average Automata

LimAvg-automata are weighted automata over infinite words that aggregate weights along runs with the limit-average value function. In this paper, we study the minimization problem for (deterministic) LimAvg-automata. Our main contribution is an equivalence relation on words characterizing LimAvg-automata, i.e., the equivalence classes of this relation correspond to states of an equivalent LimAvg-automaton. In contrast to relations characterizing DFA, our relation depends not only on the function defined by the target automaton, but also on its structure. We show two applications of this relation. First, we present a minimization algorithm for LimAvg-automata, which returns a minimal LimAvg-automaton among those equivalent and structurally similar to the input one. Second, we present an extension of Angluin's L^*-algorithm with syntactic queries, which learns in polynomial time a LimAvg-automaton equivalent to the target one.

Synthesizing Good-Enough Strategies for LTLf Specifications

We consider the problem of synthesizing good-enough (GE)-strategies for linear temporal logic (LTL) over finite traces or LTLf for short. The problem of synthesizing GE-strategies for an LTL formula φ over infinite traces reduces to the problem of synthesizing winning strategies for the formula (∃Oφ)⇒φ where O is the set of propositions controlled by the system. We first prove that this reduction does not work for LTLf formulas. Then we show how to synthesize GE-strategies for LTLf formulas via the Good-Enough (GE)-synthesis of LTL formulas. Unfortunately, this requires to construct deterministic parity automata on infinite words, which is computationally expensive. We then show how to synthesize GE-strategies for LTLf formulas by a reduction to solving games played on deterministic Büchi automata, based on an easier construction of deterministic automata on finite words. We show empirically that our specialized synthesis algorithm for GE-strategies outperforms the algorithms going through GE-synthesis of LTL formulas by orders of magnitude.

Separating the Words of a Language by Counting Factors

For a given language L, we study the languages X such that for all distinct words u, v ∈ L, there exists a word x ∈ X that appears a different number of times as a factor in u and in v. In particular, we are interested in the following question: For which languages L does there exist a finite language X satisfying the above condition? We answer this question for all regular languages and for all sets of factors of infinite words.

An Inequality for the Number of Periods in a Word

We prove an inequality for the number of periods in a word [Formula: see text] in terms of the length of [Formula: see text] and its initial critical exponent. Next, we characterize all periods of the length-[Formula: see text] prefix of a characteristic Sturmian word in terms of the lazy Ostrowski representation of [Formula: see text], and use this result to show that our inequality is tight for infinitely many words [Formula: see text]. We propose two related measures of periodicity for infinite words. Finally, we also consider special cases where [Formula: see text] is overlap-free or squarefree.