Complexity of weak acceptance conditions in tree automata

Weak acceptance conditions for automata on infinite words or trees are defined in terms of the set of states that appear in the run. This is in contrast with, more usual, strong conditions that are defined in terms of states appearing infinitely often on the run. Weak conditions appear in the context of model-checking and translations of logical formalisms to automata. We study the complexity of the emptiness problem for tree automata with weak conditions. We also study the translations between automata with weak and strong conditions.     

A note on partially ordered tree automata

A recent paper by Bouajjani, Muscholl and Touili shows that the class of languages accepted by partially ordered word automata (or equivalently accepted by Σ₂-formulae) is closed under semi-commutation and it suggested the following open question: can we extend this result to tree languages? This problem can be addressed by proving (1) that the class of tree regular languages accepted by Σ₂ formulae is strictly included in the class of languages accepted by partially ordered automata, and (2) that Bouajjani and the others results cannot be extended to tree.     

A characterization of Büchi tree automata

We show that a language of infinite binary trees is definable by a Σ₂-formula of the monadic second order logic of two successors (with no additional symbols) iff it can be accepted by a Büchi automaton. The same result has been obtained by G. Lenzi, but our proof is simpler.     

Weighted tree automata and weighted logics

We define a weighted monadic second order logic for trees where the weights are taken from a commutative semiring. We prove that a restricted version of this logic characterizes the class of formal tree series which are accepted by weighted bottom-up finite state tree automata. The restriction on the logic can be dropped if additionally the semiring is locally finite. This generalizes corresponding classical results of Thatcher, Wright, and Doner for tree languages and it extends recent results of Droste and Gastin [Weighted automata and weighted logics, in: Automata, Languages and Programming—32nd International Colloquium, ICALP 2005, Lisbon, Portugal, 2005, Proceedings, Lecture Notes in Computer Science, Vol. 3580, Springer, Berlin, 2005, pp. 513–525, full version in Theoretical Computer Science, to appear.] from formal power series on words to formal tree series.     

Weighted automata with discounting

We investigate weighted automata with discounting and their behaviors over semirings and finitely generated graded monoids. We characterize the discounted behaviors of weighted automata precisely as rational formal power series with a discounted form of the Cauchy product. This extends a classical result of Kleene-Schützenberger. Here we show that the very special case of Schützenberger's result for free monoids over singleton alphabets suffices to deduce our generalization.     

Folk theorems on the determinization and minimization of timed automata

Timed automata are known not to be complementable or determinizable. Natural questions are, then, could we check whether a given TA enjoys these properties? These problems are not algorithmically solvable. Minimizing the “resources” of a TA (number of clocks or size of constants) are also unsolvable problems. In this paper we provide simple undecidability proofs using a “constructive” version of the problems where we require not just a yes/no answer, but also a “witness”. Proofs are then simple reductions from the universality problem. Recent work of Finkel shows that the corresponding decision problems are also undecidable [O. Finkel, On decision problems for timed automata, Bulletin of the European Association for Theoretical Computer Science 87 (2005) 185-190].     

Parameterized complexity of basic decision problems for tree automata

There are many decision problems in automata theory (including membership, emptiness, inclusion and universality problems) that are NP-hard for some classes of tree automata (TA). The study of their parameterized complexity allows us to find new bounds of their nonpolynomial time algorithmic behaviours. We present results of such a study for classical TA, rigid tree automata, TA with global equality and disequality and t-DAG automata. As parameters we consider the number of states, the cardinality of the signature, the size of the term or the t-dag and the size of the automaton.     

Flip-pushdown automata with k pushdown reversals and E0L systems are incomparable

We prove that any propagating E0L system cannot generate the language {w#w|w∈{0,1}^?}$\{w\mathrm{\#}w|w\in {\{0,1\}}^{?}\}$. This result, together with some known ones, enables us to conclude that the flip-pushdown automata with k pushdown reversals, i.e., the pushdown automata with the ability to flip the pushdown, and E0L systems are incomparable. This result solves an open problem stated by Holzer and Kutrib in 2003.     

Two-way deterministic automata with two reversals are exponentially more succinct than with one reversal

We prove that limiting the number of reversals from two to one can cause an exponential blow-up in the size of two-way deterministic automaton.     

Compositional synthesis of asynchronous automata

Asynchronous automata are a model of communication processes with a control structure distributed on a set P of processes, global initializations and global accepting conditions. The well-known theorem of Zielonka states that they recognize exactly the class of regular Mazurkiewicz trace languages. The corresponding synthesis problem is, given a global specification A of any regular trace language L, to build an asynchronous automaton that recognizes L, automatically. Yet, all such existing constructions are quite involved and yield an explosion of the number of states in each process, which is exponential in both the sizes of A and P. In this paper, we introduce the particular case of distributed asynchronous automata, which require that the initializations and the accepting conditions are distributed as well. We present an original technique based on simple compositions/decompositions of these distributed asynchronous automata that results in the construction of smaller non-deterministic asynchronous automata: now, the number of states in each process is only polynomial in the size of A, but is still exponential in the size of P.     

On the efficient construction of quasi-reversible automata for reversible languages

Quasi-reversible automata is a suitable representation for reversible languages. In this work a method is proposed to obtain such an automaton for any given reversible language represented by its minimal DFA. Our method runs in polynomial time respect to the size of the minimal DFA and improves a previous exponential method. Previous bound for the size of quasi-reversible automata is also reduced.     

Pattern expressions and pattern automata

We define the pattern expressions as an extension of both regular expressions and patterns. We prove several properties of the new family of languages, similar to those of extended regex languages [Câmpeanu et al., Int. J. Found. Comput. Sci. 14 (6) (2003) 1007-1018]. We also define an automata system that recognizes these languages. Differences between regex and pattern expressions are also discussed.     

The complexity of tree automata and XPath on grammar-compressed trees

The complexity of various membership problems for tree automata on compressed trees is analyzed. Two compressed representations are considered: dags, which allow to share identical subtrees in a tree, and straight-line context-free tree grammars, which moreover allow to share identical intermediate parts in a tree. Several completeness results for the classes NL, P, and PSPACE are obtained. Finally, the complexity of the evaluation problem for (structural) XPath queries on trees that are compressed via straight-line context-free tree grammars is investigated.     

Cellular automata with limited inter-cell bandwidth

A d-dimensional cellular automaton is a d-dimensional grid of interconnected interacting finite automata. There are models with parallel and sequential input modes. In the latter case, the distinguished automaton at the origin, the communication cell, is connected to the outside world and fetches the input sequentially. Often in the literature this model is referred to as an iterative array. In this paper, d-dimensional iterative arrays and one-dimensional cellular automata are investigated which operate in real and linear time and whose inter-cell communication bandwidth is restricted to some constant number of different messages independent of the number of states. It is known that even one-dimensional two-message iterative arrays accept rather complicated languages such as {a^p∣p prime} or {a²ⁿ∣n∈N} (H. Umeo, N. Kamikawa, Real-time generation of primes by a 1-bit-communication cellular automaton, Fund. Inform. 58 (2003) 421-435). Here, the computational capacity of d-dimensional iterative arrays with restricted communication is investigated and an infinite two-dimensional hierarchy with respect to dimensions and messages is shown. Furthermore, the computational capacity of the one-dimensional devices in question is compared with the power of two-way and one-way cellular automata with restricted communication. It turns out that the relations between iterative arrays and cellular automata are quite different from the relations in the unrestricted case. Additionally, an infinite strict message hierarchy for real-time two-way cellular automata is obtained as well as a very dense time hierarchy for k-message two-way cellular automata. Finally, the closure properties of one-dimensional iterative arrays with restricted communication are investigated and differences to the unrestricted case are shown as well.     

Recognizability of the support of recognizable series over the semiring of the integers is undecidable

A recognizable series over the semiring of the integers is a function that maps each word over an alphabet to an integer. The support of such a series consists of all those words which are not mapped to zero. It is long known that there are recognizable series whose support is not a recognizable, but a context-free language. However, the problem of deciding whether the support of a given recognizable series is recognizable was never considered. Here we show that this problem is undecidable. The proof relies on an encoding of an instance of Post?s correspondence problem.