Rational subsets of polycyclic monoids and valence automata

Render, Elaine; Kambites, Mark

doi:10.1016/j.ic.2009.02.012

Cited by 16 publications

(18 citation statements)

References 14 publications

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“…Furthermore, we prove a Kleene-style theorem relating T-automata and the corresponding expressions, thus generalizing previous work in Silva et al 2011]. This generic correspondence instantiates to three large classes of machines actively studied in the literature: -state machines over various types of store, as classically studied in formal language theory [Rozenberg and Salomaa 1997]; here we elaborate in detail push-down stores and their combination with one another and with nondeterminism, as well as Turing tapes; -valence automata [Render and Kambites 2009;Kambites 2009;Zetzsche 2016], capturing nondeterministic computations over a store modeled by various classes of monoids; -weighted automata [Droste et al 2009;Sakarovitch 2009].…”

Section: Introductionmentioning

confidence: 64%

“…In this form it is rather similar to valence automata, another example of a machine previously studied in the literature (e.g. [Render and Kambites 2009;Kambites 2009]). We present the corresponding algebraic theories side by side and explain how valence automata can be formalised as T-automata.…”

Section: Context-free Languages and Valence Automatamentioning

confidence: 80%

“…Recall e.g. from [Render and Kambites 2009;Kambites 2009] that a valence automaton over M is a tuple A = (X, M, A, δ, q 0 , F ) where X is a finite set of states, δ is finite subset of X × A * × M × X of transitions, q 0 ∈ X is an initial state and F ⊆ X a set of final states. This induces a transition relation ⇒ on X ×A * ×M as usual by defining (p, w, m) ⇒ (q, wu, mn) if there exists (p, u, n, q) ∈ δ.…”

Section: Context-free Languages and Valence Automatamentioning

confidence: 99%

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Toward a Uniform Theory of Effectful State Machines

Goncharov

Milius

Silva

2020

ACM Trans. Comput. Logic

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We use recent developments on coalgebraic and monad-based semantics to obtain a generic notion of a T-automaton, where T is a monad. This enables a uniform study of various notions of machines: e.g. finite state machines, multi-stack machines, Turing machines, valence automata, and weighted automata. We use the generalized powerset construction to define a generic language semantics for T-automata, and we show by numerous examples that it correctly instantiates for some known classes of machines/languages, including regular, context-free, recursively-enumerable and various subclasses of context free languages (e.g. deterministic and real-time ones). Moreover, our approach provides new generic techniques for studying expressivity power of various machine-based models.where the left-hand map, called the observation map, represents an output function (e.g. an acceptance predicate if B = 2; here and elsewehre we identify 2 with {0, 1}) and the right-hand maps, called a-derivatives, are the next state functions indexed by input actions from A. Finite L-coalgebras are hence precisely classical Moore automata. It is straightforward to extend aderivatives to w-derivatives with w ∈ A * by induction: ∂ ǫ (x) = x; ∂ aw (x) = ∂ a (∂ w (x)) where ǫ ∈ A * is the empty word.The final L-coalgebra νL always exists and is carried by the set of all formal power series B A * . The transition structure on B A * is given byand ∂ a (σ) = λw. σ(aw), (a ∈ A)for every formal power series σ : A * → B. The unique homomorphism from an L-coalgebra X to the final one B A * assigns to every state x 0 ∈ X a formal power series that we regard as the (language) semantics of X with x 0 as an initial state. Specifically, if B = 2 then finite L-coalgebras are deterministic automata and B A * ∼ = P(A * ) is the set of all formal languages over A and the language semantics assigns to every state of a given finite deterministic automaton the language accepted by that state. The transition structure on P(A * ) is given by the predicate o distinguishing languages containing the empty word and by the maps ∂ a assigning to a language their left derivatives: o(L) = ⊤ ⇐⇒ ǫ ∈ L and ∂ a (L) = {w | aw ∈ L} (a ∈ A).

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Section: Introductionmentioning

confidence: 64%

Section: Context-free Languages and Valence Automatamentioning

confidence: 80%

Section: Context-free Languages and Valence Automatamentioning

confidence: 99%

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Toward a Uniform Theory of Effectful State Machines

Goncharov

Milius

Silva

2020

ACM Trans. Comput. Logic

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“…In this form it is rather similar to another example of a machine previously studied in the literature (e.g. [28,16]) and which we also can formalize as a T-automaton. We present the corresponding algebraic theories side by side.…”

Section: Context-free Languages and Algebraic Expressionsmentioning

confidence: 82%

Towards a Coalgebraic Chomsky Hierarchy

Goncharov¹,

Milius²,

Silva³

2014

Lecture Notes in Computer Science

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Abstract. 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 monadbased semantics to obtain a generic notion of a T-automaton, where T is a monad, which allows the uniform study of various notions of machines (e.g. finite state machines, multi-stack machines, Turing machines, valence automata, weighted automata). We use the generalized powerset construction to define a generic (trace) semantics for 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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“…Fourth, we develop a normal form result for rational subsets of monoids in C (see section 6). Such normal form results have been available for monoids described by monadic rewriting systems (see, for example, [1]), which was applied by Render and Kambites to monoids representing pushdown storages [22]. Under different terms, this normal form trick has been used by Bouajjani, Esparza, and Maler [2] and by Caucal [3] to describe rational sets of pushdown operations.…”

Section: Resultsmentioning

confidence: 99%

Silent Transitions in Automata with Storage

Zetzsche

2013

Automata, Languages, and Programming

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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 λ-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.

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Rational subsets of polycyclic monoids and valence automata

Cited by 16 publications

References 14 publications

Toward a Uniform Theory of Effectful State Machines

Toward a Uniform Theory of Effectful State Machines

Towards a Coalgebraic Chomsky Hierarchy

Silent Transitions in Automata with Storage

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