The class of rational subsets of a group G is the smallest class that contains all finite subsets of G and that is closed with respect to union, product and taking the monoid generated by a set. The rational subset membership problem for a finitely generated group G is the decision problem, where for a given rational subset of G and a group element g it is asked whether g ∈ G. This paper presents a survey on known decidability and undecidability results for the rational subset membership problem for groups. The membership problems for finitely generated submonoids and finitely generated subgroups will be discussed as well.

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The Rational Subset Membership Problem for Groups: A Survey

The program of the 30th Symposium on Logic in Computer Science held in 2015 in Kyoto included two contributions on the computational complexity of the reachability problem for vector addition systems: Blondin, Finkel, Goller, Haase, and McKenzie [2015] attacked the problem by providing the first tight complexity bounds in the case of dimension 2 systems with states, while Leroux and Schmitz [2015] proved the first complexity upper bound in the general case. The purpose of this column is to present the main ideas behind these two results, and more generally survey the current state of affairs.

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The complexity of reachability in vector addition systems

The Chomsky hierarchy plays a prominent role in the foundations of theoretical computer science relating classes of formal languages of primary importance. In this paper we use recent developments on coalgebraic and monad-based semantics to obtain a generic notion of a $\mathbb{T}$-automaton, where $\mathbb{T}$ is a monad, which allows the uniform study of various notions of machines (e.g. finite state machines, multi-stack machines, Turing machines, weighted automata). We use the generalized powerset construction to define a generic (trace) semantics for $\mathbb{T}$-automata, and we show by numerous examples that it correctly instantiates for some known classes of machines/languages captured by the Chomsky hierarchy. Moreover, our approach provides new generic techniques for studying expressivity power of various machine-based models.

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Towards a Coalgebraic Chomsky Hierarchy

We consider the computational power of silent transitions in one-way automata with storage. Specifically, we ask which storage mechanisms admit a transformation of a given automaton into one that accepts the same language and reads at least one input symbol in each step. We study this question using the model of valence automata. Here, a finite automaton is equipped with a storage mechanism that is given by a monoid. This work presents generalizations of known results on silent transitions. For two classes of monoids, it provides characterizations of those monoids that allow the removal of silent transitions. Both classes are defined by graph products of copies of the bicyclic monoid and the group of integers. The first class contains pushdown storages as well as the blind counters while the second class contains the blind and the partially blind counters.

Silent transitions in automata with storage

A monoid is special if it admits a presentation in which all the defining relations are of the form $w = 1$. In this article, we study the word problem, in the sense of Duncan & Gilman, for special monoids, and relate the language-theoretic properties of this set to the word problem for the group of units of the monoid, by developing the theory of the words representing invertible words. We show that a special monoid has regular word problem if and only if it is a finite group. When $\mathcal{C}$ is a class of languages satisfying some restrictions modelled on the context-free languages, we show that a special monoid has word problem in $\mathcal{C}$ if and only if its group of units has word problem in $\mathcal{C}$. As a corollary, we generalise the Muller-Schupp theorem to special monoids: a finitely presented special monoid has context-free word problem if and only if its group of units is virtually free. This completely answers a question from 1993 of Book & Otto, and for the class of special monoids answers a question from 2004 of Duncan & Gilman. We show that it is decidable whether a one-relator special monoid has context-free word problem. Finally, by proving that any context-free monoid has decidable rational subset membership problem, we also obtain a large class of special monoids for which this problem is decidable.

On the Word Problem for Special Monoids.

We study the classes of languages defined by valence automata with rational target sets (or equivalently, regular valence grammars with rational target sets), where the valence monoid is drawn from the important class of polycyclic monoids. We show that for polycyclic monoids of rank 2 or more, such automata accept exactly the context-free languages. For the polycyclic monoid of rank 1 (that is, the bicyclic monoid), they accept a class of languages strictly including the partially blind one-counter languages. Key to the proof is a description of the rational subsets of polycyclic and bicyclic monoids, other consequences of which include the decidability of the rational subset membership problem, and the closure of the class of rational subsets under intersection and complement.

Rational subsets of polycyclic monoids and valence automata

Semigroup automata with rational initial and terminal sets

We study the classes of languages defined by valence automata with rational target sets (or equivalently, regular valence grammars with rational target sets), where the valence monoid is drawn from the important class of polycyclic monoids. We show that for polycyclic monoids of rank 2 or more, such automata accept exactly the context-free languages. For the polycyclic monoid of rank 1 (that is, the bicyclic monoid), they accept a class of languages strictly including the partially blind one-counter languages. Key to the proofs is a description of the rational subsets of polycyclic and bicyclic monoids, other consequences of which include the decidability of the rational subset membership problem for these monoids, and the closure of the class of rational subsets under intersection and complement.

Polycyclic and Bicyclic Valence Automata

We study hybrid systems with strong resets from the perspective of formal language theory. We define a notion of hybrid regular expression and prove a Kleene-like theorem for hybrid systems. We also prove the closure of these systems under determinisation and complementation. Finally, we prove that the reachability problem is undecidable for synchronized products of hybrid systems.

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Formal language properties of hybrid systems with strong resets

We study the classes of languages defined by valence automata with rational target sets (or equivalently, regular valence grammars with rational target sets), where the valence monoid is drawn from the important class of polycyclic monoids. We show that for polycyclic monoids of rank 2 or more, such automata accept exactly the context-free languages. For the polycyclic monoid of rank 1 (that is, the bicyclic monoid), they accept a class of languages strictly including the partially blind one-counter languages. Key to the proof is a description of the rational subsets of polycyclic and bicyclic monoids, other consequences of which include the decidability of the rational subset membership problem for these monoids, and the closure of the class of rational subsets under intersection and complement.

Elaine Render

Papers

Rational subsets of polycyclic monoids and valence automata

Semigroup automata with rational initial and terminal sets

Polycyclic and Bicyclic Valence Automata

Formal language properties of hybrid systems with strong resets

Rational subsets of polycyclic monoids and valence automata