We show that it is decidable in time complexity \(2^{2^{2^{O{(n)}}}}\) whether the language accepted by an n-state non-deterministic automaton is of star height one, which is the first ever complexity result for the star height one problem. To achieve this, we introduce distance desert automata as a joint generalization of distance automata and desert automata, and show the decidability of its limitedness problem by solving the underlying Burnside problem.

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Distance desert automata and the star height problem

Derivatives of rational expressions with multiplicity

Determinism of devices is a key aspect throughout all of computer science, simply because of considerations of efficiency of the implementation. One possible way (among others) to relax this notion is to consider unambiguous machines: non-deterministic machines that have at most one accepting run on each input.

Unambiguity in Automata Theory

Max-plus algebra in the history of discrete event systems

This paper is a survey on the universal automaton, which is an automaton canonically associated with every language. In the last forty years, many objects have been defined or studied, that are indeed closely related to the universal automaton. We first show that every automaton that accepts a given language has a morphic image which is a subautomaton of the universal automaton of this language. This property justifies the name “universal” that we have coined for this automaton. The universal automaton of a regular language is finite and can be effectively computed in the syntactic monoid or, more efficiently, from the minimal automaton of the language. We describe the construction that leads to tight bounds on the size of the universal automaton. Another outcome of the effective construction of the universal automaton is the computation of a minimal NFA accepting a given language, or approximations of such a minimal NFA. From another point of view, the universal automaton of a language is based on the factorisations of this language, and is thus involved in the problems of factorisations and approximations of languages. Last, but not least, we show how the universal automaton gives an elegant solution to the star height problem for some classes of languages (pure-group or reversible languages). With every language is canonically associated an automaton, called the universal automaton of the language, which is finite whenever the language is regular. It is large, it is complex, it is complicated to compute, but it contains, hopefully, many interesting informations on the language. In the last forty years, it has been described a number of times, more or less explicitly,

The universal automaton.

This paper addresses a decision problem highlighted by Grigorchuk, Nekrashevich, and Sushchanskiĭ, namely the finiteness problem for automaton (semi)groups. For semigroups, we give an effective sufficient but not necessary condition for finiteness and, for groups, an effective necessary but not sufficient condition. The efficiency of the new criteria is demonstrated by testing all Mealy automata with small stateset and alphabet. Finally, for groups, we provide a necessary and sufficient condition that does not directly lead to a decision procedure.

On the finiteness problem for automaton (semi)groups

Consider partial maps from the free monoid into the field of real numbers with a rational domain. We show that two families of such series are actually the same: the unambiguous rational series on the one hand, and the max-plus and min-plus rational series on the other hand. The decidability of equality was known to hold in both families with different proofs, so the above unifies the picture. We give an effective procedure to build an unambiguous automaton from a max-plus automaton and a min-plus one that recognize the same series.

https://hal.archives-ouvertes.fr/hal-00017220/document

Series which are both max-plus and min-plus rational are unambiguous

This invited talk presents the work conducted on the problems that arise when dealing with weighted automata containing \(\varepsilon \)-transitions: how to define the behaviour of such automata in which the presence of \(\varepsilon \)-circuits results in infinite summations, and second how to eliminate the \(\varepsilon \)-transitions in an automaton whose behaviour has been recognised to be well-defined. The origin of this work is the implementation, in the Awali platform [19], of an \(\varepsilon \)-transition removal algorithm for automata with weight in \(\mathbb {Q}\) or \(\mathbb {R}\), a case that had never been treated before in the rich literature on the subject of \(\varepsilon \)-transition removal algorithms (cf. [16] for a survey). The results of this work have been published in [14].

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The validity of weighted automata

In this paper we investigate how it is possible to recover an automaton from a rational expression that has been computed from that automaton. The notion of derived term of an expression, introduced by Antimirov, appears to be instrumental in this problem. The second important ingredient is the co-minimization of an automaton, a dual and generalized Moore algorithm on non-deterministic automata. If an automaton is then sufficiently decorated, the combination of these two algorithms gives the desired result. Reducing the amount of decoration is still the object of ongoing investigation.

Sylvain Lombardy

Papers

On the finiteness problem for automaton (semi)groups

Series which are both max-plus and min-plus rational are unambiguous

The validity of weighted automata

How expressions can code for automata

On the star height of rational languages a new presentation for two old results