On Shuffling of Infinite Square-Free Words

G. Fici proved that a finite word has a minimal suffix automaton if and only if all its left special factors occur as prefixes. He called LSP all finite and infinite words having this latter property. We characterize here infinite LSP words in terms of S-adicity. More precisely we provide a finite set of morphisms S and an automaton \(\mathcal{A}\) such that an infinite word is LSP if and only if it is S-adic and all its directive words are recognizable by \(\mathcal{A}\).

On square-free arithmetic progressions in infinite words

We consider two-variable first-order logic FO2 and its quantifier alternation hierarchies over both finite and infinite words. Our main results are forbidden patterns for deterministic automata (finite words) and for Carton-Michel automata (infinite words). In order to give concise patterns, we allow the use of subwords on paths in finite graphs. This concept is formalized as subword-patterns. For certain types of subword-patterns there exists a non-deterministic logspace algorithm to decide their presence or absence in a given automaton. In particular, this leads to NL algorithms for deciding the levels of the FO2 quantifier alternation hierarchies. This applies to both full and half levels, each over finite and infinite words. Moreover, we show that these problems are NL-hard and, hence, NL-complete.

This paper is concerned with the language inclusion problem for timed automata: given timed automata ${\mathcal A}$ and ${\mathcal B}$, is every word accepted by ${\mathcal B}$ also accepted by ${\mathcal A}$? Alur and Dill [3] showed that the language inclusion problem is decidable if ${\mathcal A}$ has no clocks and undecidable if ${\mathcal A}$ has two clocks (with no restriction on ${\mathcal B}$). However, the status of the problem when ${\mathcal A}$ has one clock is not determined by [3]. In this paper we close this gap for timed automata over infinite words by showing that the one-clock language inclusion problem is undecidable. For timed automata over finite words, building on our earlier paper [20], we show that the one-clock language inclusion problem is decidable with non-primitive recursive complexity. This reveals a surprising divergence between the theory of timed automata over finite words and over infinite words. Finally, we show that if e-transitions or non-singular postconditions are allowed, then the one-clock language inclusion problem is undecidable over both finite and infinite words.

Over finite words, languages of dot-depth one are expressively complete for alternation-free first-order logic. This fragment is also known as the Boolean closure of existential first-order logic. Here, the atomic formulas comprise order, successor, minimum, and maximum predicates. Knast (1983) has shown that it is decidable whether a language has dot-depth one. We extend Knast's result to infinite words. In particular, we describe the class of languages definable in alternation-free first-order logic over infinite words, and we give an effective characterization of this fragment. This characterization has two components. The first component is identical to Knast's algebraic property for finite words and the second component is a topological property, namely being a Boolean combination of Cantor sets. As an intermediate step we consider finite and infinite words simultaneously. We then obtain the results for infinite words as well as for finite words as special cases. In particular, we give a new proof of Knast's Theorem on languages of dot-depth one over finite words.

In 2007, Grytczuk conjectured that for any sequence ( ℓ i ) i ≥ 1 (\ell _i)_{i\ge 1} of alphabets of size 3 3 there exists a square-free infinite word w w such that for all i i , the i i -th letter of w w belongs to ℓ i \ell _i . The result of Thue from 1906 implies that there is an infinite square-free word if all the ℓ i \ell _i are identical. On the other hand, Grytczuk, Przybyło and Zhu showed in 2011 that it also holds if the ℓ i \ell _i are of size 4 4 instead of 3 3 . In this article, we first show that if the lists are of size 4 4 , the number of square-free words is at least 2.45 n 2.45^n (the previous similar bound was 2 n 2^n ). We then show our main result: we can construct such a square-free word if the lists are subsets of size 3 3 of the same alphabet of size 4 4 . Our proof also implies that there are at least 1.25 n 1.25^n square-free words of length n n for any such list assignment. This proof relies on the existence of a set of coefficients verified with a computer. We suspect that the full conjecture could be resolved by this method with a much more powerful computer (but we might need to wait a few decades for such a computer to be available).

An infinite word is an infinite Lyndon word if it is smaller, with respect to the lexicographic order, than all its proper suffixes, or equivalently if it has infinitely many finite Lyndon words as prefixes. A characterization of binary endomorphisms generating Lyndon infinite words is provided.

formula omitted]-Regular languages are testable with a constant number of queries

We construct geometric realizations for the infimax family of substitutions by generalizing the Rauzy-Canterini-Siegel method for a single substitution to the S-adic case. The composition of each countably infinite subcollection of substitutions from the family has an asymptotic fixed sequence whose shift orbit closure is an infimax minimal set $Δ^+$. The subcollection of substitutions also generates an infinite Bratteli-Vershik diagram with prefix-suffix labeled edges. Paths in the diagram give the Dumont-Thomas expansion of sequences in $Δ^+$ which in turn gives a projection onto the asymptotic stable direction of the infinite product of the Abelianization matrices. The projections of all sequences from $Δ^+$ is the generalized Rauzy fractal which has subpieces corresponding to the images of symbolic cylinder sets. The intervals containing these subpieces are shown to be disjoint except at endpoints, and thus the induced map derived from the symbolic shift translates them. Therefore the process yields an Interval Translation Map, and the Rauzy fractal is proved to be its attractor.

Avoiding square-free words on free groups

Automata over infinite words provide a powerful framework to solve various decision problems. However, the mechanized reasoning with restricted classes of automata over infinite words is often simpler and more efficient. For instance, weak deterministic Büchi automata (wdbas) can be handled algorithmically almost as efficient as deterministic automata over finite words. In this paper, we show how and when the standard powerset construction for automata over finite words can be used to determinize automata over infinite words. An instance is the class of automata that accept wdba-recognizable languages. Furthermore, we present applications of this new determinization construction. Namely, we apply it to improve the automata-based approach for the mixed first-order linear arithmetic over the reals and the integers, and we utilize it to accelerate finite state model checking. We report on experimental results for these two applications.

Weighted automata and weighted MSO logics for average and long-time behaviors

The complementation problem for nondeterministic word automata has numerous applications in formal verification. In particular, the language-containment problem, to which many verification problems is reduced, involves complementation. For automata on finite words, which correspond to safety properties, complementation involves determinization. The 2n blow-up that is caused by the subset construction is justified by a tight lower bound. For Büchi automata on infinite words, which are required for the modeling of liveness properties, optimal complementation constructions are quite complicated, as the subset construction is not sufficient. From a theoretical point of view, the problem is considered solved since 1988, when Safra came up with a determinization construction for Büchi automata, leading to a 2O(nlogn) complementation construction, and Michel came up with a matching lower bound. A careful analysis, however, of the exact blow-up in Safra’s and Michel’s bounds reveals an exponential gap in the constants hiding in the O() notations: while the upper bound on the number of states in Safra’s complementary automaton is n 2n, Michel’s lower bound involves only an n! blow up, which is roughly (n/e)n. The exponential gap exists also in more recent complementation constructions. In particular, the upper bound on the number of states in the complementation construction in [KV01], which avoids determinization, is (6n)n. This is in contrast with the case of automata on finite words, where the upper and lower bounds coincides. In this work we describe an improved complementation construction for nondeterministic Büchi automata and analyze its complexity. We show that the new construction results in an automaton with at most (1.06n)n states. While this leaves the problem about the exact blow up open, the gap is now exponentially smaller. From a practical point of view, our solution enjoys the simplicity of [KV01], and results in much smaller automata.

On generating binary words palindromically

Metric Temporal Logic (MTL) is a prominent specification formalism for real-time systems. In this paper, we show that the satisfiability problem for MTL over finite timed words is decidable, with non-primitive recursive complexity. We also consider the model-checking problem for MTL: whether all words accepted by a given Alur-Dill timed automaton satisfy a given MTL formula. We show that this problem is decidable over finite words. Over infinite words, we show that model checking the safety fragment of MTL--which includes invariance and time-bounded response properties--is also decidable. These results are quite surprising in that they contradict various claims to the contrary that have appeared in the literature.