Büchi Complementation Made Tighter

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(n log n) 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 n2n, 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 of Kupferman and Vardi, 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 (0.96n)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 the construction of Kupferman and Vardi, and results in much smaller automata.

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.

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.

Checking whether one formal language is included in another is vital to many verification tasks. In this paper, we provide solutions for checking the inclusion of the languages given by visibly pushdown automata over both finite and infinite words. Visibly pushdown automata are a richer automaton model than the classical finite-state automata, which allows one, e.g., to reason about the nesting of procedure calls in the executions of recursive imperative programs. The highlight of our solutions is that they do not comprise automata constructions for determinization and complementation. Instead, our solutions are more direct and generalize the so-called Ramsey-based inclusion-checking algorithms, which apply to classical finite-state automata and proved effective there, to visibly pushdown automata. We also experimentally evaluate our algorithms thereby demonstrating the virtues of avoiding determinization and complementation constructions.

In general, a nondeterministic automaton or machine (for example a finite automaton, pushdown automaton or Turing machine) is called unambiguous if each input is accepted by at most one run or computation. Each deterministic automaton is obviously unambiguous. However, in many settings, unambiguous automata are more expressive or admit more succinct automata than deterministic models, while preserving some good algorithmic properties. The aim of this talk is to survey some classical and some more recent results on unambiguous finite automata over different kind of input structures, namely finite words, infinite words, finite trees, and infinite trees.

Complementation of finite automata on infinite words is not only a fundamental problem in automata theory, but also serves as a cornerstone for solving numerous decision problems in mathematical logic, model-checking, program analysis and verification. For Streett complementation, a significant gap exists between the current lower bound 2^{Omega(n*log(n*k))} and upper bound 2^{O(n*k*log(n*k))}, where n is the state size, k is the number of Streett pairs, and k can be as large as 2^{n}. Determining the complexity of Streett complementation has been an open question since the late 80's. In this paper we show a complementation construction with upper bound 2^{O(n*log(n)+n*k*log(k))} for k=O(n) and 2^{O(n^{2}*log(n))} for k=Omega(n), which matches well the lower bound obtained in the paper arXiv:1102.2963. We also obtain a tight upper bound 2^{O(n*log(n))} for parity complementation.

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.

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.

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

We show that the separation property fails for the classes Sigma_n of the Rabin-Mostowski index hierarchy of alternating automata on infinite trees. This extends our previous result (obtained with Szczepan Hummel) on the failure of the separation property for the class Sigma_2 (i.e., for co-Buchi sets). It remains open whether the separation property does hold for the classes Pi_n of the index hierarchy. To prove our result, we first consider the Rabin-Mostowski index hierarchy of deterministic automata on infinite words, for which we give a complete answer (generalizing previous results of Selivanov): the separation property holds for Pi_n and fails for Sigma_n-classes. The construction invented for words turns out to be useful for trees via a suitable game.

In the automata-theoretic approach to verification, we model programs and specifications by automata on infinite words. Correctness of a program with respect to a specification can then be reduced to the language-containment problem. In a concurrent setting, the program is typically a parallel composition of many coordinating processes, and the language-containment problem that corresponds to verification is $$(\dag )\mathcal{L}(P_1 ) \cap \mathcal{L}(P_2 ) \cap ... \cap \mathcal{L}(P_n ) \subseteq \mathcal{L}(T),$$ where P 1, P 2, ..., P n are automata that model the underlying coordinating processes, and T is the task they should perform. In 1994, Kurshan suggested the heuristic of Reduction Hierarchies for circumventing the exponential blow-up introduced by conventional methods that solve the problem (†). In the reduction-hierarchy heuristic, we solve the problem (†) by solving a sequence of easier problems, which involve only automata of tractable sizes. Complexity-theoretic conjectures (NP ≠ PSPACE) imply that there are settings in which the heuristic cannot circumvent the exponential blow-up. In this paper, we demonstrate the strength of the heuristic, study its properties, characterize settings in which it performs effectively, and suggest a method for searching for reduction hierarchies. In particular, we prove, independently of the NP ≠ PSPACE question, that reduction hierarchies of tractable sizes do not always exist.KeywordsProof SystemAtomic PropositionAcceptance ConditionTree HierarchyDeterministic AutomatonThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.