Weighted Automata and Logics for Infinite Nested Words

Traditionally, data that has both linear and hierarchical structure, such as annotated linguistic data, is modeled using ordered trees and queried using tree automata. In this paper, we argue that nested words and automata over nested words offer a better way to capture and process the dual structure. Nested words generalize both words and ordered trees, and allow both word and tree operations. We study various classes of automata over nested words, and show that while they enjoy expressiveness and succinctness benefits over word and tree automata, their analysis complexity and closure properties are analogous to the corresponding word and tree special cases. In particular, we show that finite-state nested word automata can be exponentially more succinct than tree automata, and pushdown nested word automata include the two incomparable classes of context-free word languages and context-free tree languages.

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 \(\mathcal{A}\).

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.

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

Decidability and complexity of the fragments of the modal logic of Allen's relations over the rationals

In this article we investigate the family of independence-friendly (IF) logics in the equality-free setting, concentrating on questions related to expressive power. Various natural equality-free fragments of logics in this family translate into existential second-order logic with prenex quantification of function symbols only and with the first-order parts of formulae equality-free. We study this fragment of existential second-order logic. Our principal technical result is that over finite models with a vocabulary consisting of unary relation symbols only, this fragment of second-order logic is weaker in expressive power than first-order logic (with equality). Results about the fragment could turn out useful for example in the study of independence-friendly modal logics. In addition to proving results of a technical nature, we address issues related to a perspective from which IF logic is regarded as a specification framework for games, and also discuss the general significance of understanding fragments of second-order logic in investigations related to non-classical logics.

Weighted automata and multi-valued logics over arbitrary bounded lattices

Fragments of Monadic Second-Order Logics Over Word Structures

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.

We investigate weighted asynchronous cellular automata (wACAs) with weights in valuation monoids. These automata form a distributed extension of weighted finite automata (wFAs) and allow us to model concurrency. Valuation monoids are abstract weight structures that include semirings and (non-distributive) bounded lattices but also offer the possibility to model average behaviours. We prove that wACAs and wFAs which satisfy an I-diamond property are equally expressive. The main result of this paper gives a characterization of this expressiveness by weighted monadic second-order logic.

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.

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