A New Technique for Reachability of States in Concatenation Automata

We present a new technique for demonstrating the reachability of states in deterministic finite automata representing the concatenation of two languages. Such demonstrations are a necessary step in establishing the state complexity of the concatenation of two languages, and thus in establishing the state complexity of concatenation as an operation. Typically, ad-hoc induction arguments are used to show particular states are reachable in concatenation automata. We prove some results that seem to capture the essence of many of these induction arguments. Using these results, reachability proofs in concatenation automata can often be done more simply and without using induction directly.Comment: 23 pages, 1 table. Added missing affiliation/funding informatio

Automata and rational expressions

This text is an extended version of the chapter 'Automata and rational expressions' in the AutoMathA Handbook that will appear soon, published by the European Science Foundation and edited by JeanEricPin

Complexity of Equivalence and Learning for Multiplicity Tree Automata

We consider the complexity of equivalence and learning for multiplicity tree automata, i.e., weighted tree automata over a field. We first show that the equivalence problem is logspace equivalent to polynomial identity testing, the complexity of which is a longstanding open problem. Secondly, we derive lower bounds on the number of queries needed to learn multiplicity tree automata in Angluin's exact learning model, over both arbitrary and fixed fields. Habrard and Oncina (2006) give an exact learning algorithm for multiplicity tree automata, in which the number of queries is proportional to the size of the target automaton and the size of a largest counterexample, represented as a tree, that is returned by the Teacher. However, the smallest tree-counterexample may be exponential in the size of the target automaton. Thus the above algorithm does not run in time polynomial in the size of the target automaton, and has query complexity exponential in the lower bound. Assuming a Teacher that returns minimal DAG representations of counterexamples, we give a new exact learning algorithm whose query complexity is quadratic in the target automaton size, almost matching the lower bound, and improving the best previously-known algorithm by an exponential factor

Equivalence checking for weak bi-Kleene algebra

Pomset automata are an operational model of weak bi-Kleene algebra, which describes programs that can fork an execution into parallel threads, upon completion of which execution can join to resume as a single thread. We characterize a fragment of pomset automata that admits a decision procedure for language equivalence. Furthermore, we prove that this fragment corresponds precisely to series-rational expressions, i.e., rational expressions with an additional operator for bounded parallelism. As a consequence, we obtain a new proof that equivalence of series-rational expressions is decidable

Minimisation of Multiplicity Tree Automata

We consider the problem of minimising the number of states in a multiplicity tree automaton over the field of rational numbers. We give a minimisation algorithm that runs in polynomial time assuming unit-cost arithmetic. We also show that a polynomial bound in the standard Turing model would require a breakthrough in the complexity of polynomial identity testing by proving that the latter problem is logspace equivalent to the decision version of minimisation. The developed techniques also improve the state of the art in multiplicity word automata: we give an NC algorithm for minimising multiplicity word automata. Finally, we consider the minimal consistency problem: does there exist an automaton with $n$ states that is consistent with a given finite sample of weight-labelled words or trees? We show that this decision problem is complete for the existential theory of the rationals, both for words and for trees of a fixed alphabet rank.Comment: Paper to be published in Logical Methods in Computer Science. Minor editing changes from previous versio

The complexity of linear-time temporal logic over the class of ordinals

We consider the temporal logic with since and until modalities. This temporal logic is expressively equivalent over the class of ordinals to first-order logic by Kamp's theorem. We show that it has a PSPACE-complete satisfiability problem over the class of ordinals. Among the consequences of our proof, we show that given the code of some countable ordinal alpha and a formula, we can decide in PSPACE whether the formula has a model over alpha. In order to show these results, we introduce a class of simple ordinal automata, as expressive as B\"uchi ordinal automata. The PSPACE upper bound for the satisfiability problem of the temporal logic is obtained through a reduction to the nonemptiness problem for the simple ordinal automata.Comment: Accepted for publication in LMC

The Bottom-Up Position Tree Automaton, the Father Automaton and their Compact Versions

The conversion of a given regular tree expression into a tree automaton has been widely studied. However, classical interpretations are based upon a Top-Down interpretation of tree automata. In this paper, we propose new constructions based on the Gluskov's one and on the one of Ilie and Yu one using a Bottom-Up interpretation. One of the main goals of this technique is to consider as a next step the links with deterministic recognizers, consideration that cannot be performed with classical Top-Down approaches. Furthermore, we exhibit a method to factorize transitions of tree automata and show that this technique is particularly interesting for these constructions, by considering natural factorizations due to the structure of regular expression.Comment: extended version of a paper accepted at CIAA 201

Exact and Approximate Determinization of Discounted-Sum Automata

A discounted-sum automaton (NDA) is a nondeterministic finite automaton with edge weights, valuing a run by the discounted sum of visited edge weights. More precisely, the weight in the i-th position of the run is divided by $\lambda^i$, where the discount factor $\lambda$ is a fixed rational number greater than 1. The value of a word is the minimal value of the automaton runs on it. Discounted summation is a common and useful measuring scheme, especially for infinite sequences, reflecting the assumption that earlier weights are more important than later weights. Unfortunately, determinization of NDAs, which is often essential in formal verification, is, in general, not possible. We provide positive news, showing that every NDA with an integral discount factor is determinizable. We complete the picture by proving that the integers characterize exactly the discount factors that guarantee determinizability: for every nonintegral rational discount factor $\lambda$, there is a nondeterminizable $\lambda$-NDA. We also prove that the class of NDAs with integral discount factors enjoys closure under the algebraic operations min, max, addition, and subtraction, which is not the case for general NDAs nor for deterministic NDAs. For general NDAs, we look into approximate determinization, which is always possible as the influence of a word's suffix decays. We show that the naive approach, of unfolding the automaton computations up to a sufficient level, is doubly exponential in the discount factor. We provide an alternative construction for approximate determinization, which is singly exponential in the discount factor, in the precision, and in the number of states. We also prove matching lower bounds, showing that the exponential dependency on each of these three parameters cannot be avoided. All our results hold equally for automata over finite words and for automata over infinite words