## Algebraic recognizability of languages (2004)

Venue: | In Proc. 29th Int. Symp. Math. Found. of Comp. Sci. (MFCS’04 |

Citations: | 11 - 3 self |

### BibTeX

@INPROCEEDINGS{Weil04algebraicrecognizability,

author = {Pascal Weil},

title = {Algebraic recognizability of languages},

booktitle = {In Proc. 29th Int. Symp. Math. Found. of Comp. Sci. (MFCS’04},

year = {2004},

pages = {149--175},

publisher = {Springer}

}

### OpenURL

### Abstract

Abstract. Recognizable languages of finite words are part of every computer science cursus, and they are routinely described as a cornerstone for applications and for theory. We would like to briefly explore why that is, and how this word-related notion extends to more complex models, such as those developed for modeling distributed or timed behaviors. In the beginning was the Word... Recognizable languages of finite words are part of every computer science cursus, and they are routinely described as a cornerstone for applications and for theory. We would like to briefly explore why that is, and how this wordrelated notion extends to more complex models, such as those developed for modeling distributed or timed behaviors. The notion of recognizable languages is a familiar one, associated with classical theorems by Kleene, Myhill, Nerode, Elgot, Büchi, Schützenberger, etc. It can be approached from several angles: recognizability by automata, recognizability by finite monoids or finite-index congruences, rational expressions, monadic second

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Citation Context ...the natural language for logical definability of recognizable sets would seem to be MSO, monadic second order logic. It is actually more natural to use CMSO, counting monadic second order logic. CMSO =-=[16,19]-=- is monadic second order logic, enriched with the modulo quantifiers ∃ mod qx introduced in Sect. 1. In the case of words, CMSO is equivalent to MSO. In fact, this holds for any relational structure t... |

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Citation Context ...the natural language for logical definability of recognizable sets would seem to be MSO, monadic second order logic. It is actually more natural to use CMSO, counting monadic second order logic. CMSO =-=[16,19]-=- is monadic second order logic, enriched with the modulo quantifiers ∃ mod qx introduced in Sect. 1. In the case of words, CMSO is equivalent to MSO. In fact, this holds for any relational structure t... |

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Citation Context ...e LDs subject to a boundedness condition. Henriksen, Kumar, Mukund, Sohoni, Thiagarajan [43,44,62] considered the class of bounded finite MSC languages, defined by so-called bounded (Alur, Yannakakis =-=[3]-=-) or locally synchronised (Muscholl, Peled [63]) MSC-graphs. For bounded MSC languages, MSO-definability is equivalent to rationality of the language of all linearizations, and to recognizability by d... |

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Citation Context ... sets are defined in terms of morphisms into finite algebras of the same type (or in terms of finite index congruences), and equational sets in terms of systems of polynomial equations (Mezei, Wright =-=[60]-=-, Courcelle [17]). With those definitions, recognizable sets form a Boolean algebra, equational sets are closed under union, recognizable sets are always equational, finite sets are equational (even t... |

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Citation Context ... of Languages. Many more examples can be found in the literature, where natural algebraic properties of finite monoids match natural combinatorial or logical properties of languages (see for instance =-=[1,33,67,69]-=-). The scope of this matching is described in Eilenberg’s variety theorem; the latter identifies the closure properties on classes of recognizable languages and on classes of finite monoids, that char... |

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Citation Context ...everal attempts to develop appropriate notions ofsrational expressions, that would be equivalent to the expressive power of timed automata, see Henzinger, Raskin, Schobbens [45], Asarin, Maler, Caspi =-=[4]-=-, Dima [28], Maler, Pnueli [58] among others. At the same time, timed automata and timed languages may exhibit paradoxical behaviors, due to the continuous nature of time, so the central ideas and tec... |

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Citation Context ...]. A poset is series-parallel if it can be obtained from singletons by using the operations of sequential and parallel product. There is a combinatorial characterization of these posets (Nfree posets =-=[41,87]-=-), but the definition above naturally leads to the consideration of the so-called series-parallel algebras [55], that is, sets equipped with two binary associative operations, one of which is commutat... |

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Citation Context ...iksen, Kumar, Mukund, Sohoni, Thiagarajan [43,44,62] considered the class of bounded finite MSC languages, defined by so-called bounded (Alur, Yannakakis [3]) or locally synchronised (Muscholl, Peled =-=[63]-=-) MSC-graphs. For bounded MSC languages, MSO-definability is equivalent to rationality of the language of all linearizations, and to recognizability by deterministic (resp. nondeterministic) message-p... |

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Citation Context ...tc). There have also been several attempts to develop appropriate notions ofsrational expressions, that would be equivalent to the expressive power of timed automata, see Henzinger, Raskin, Schobbens =-=[45]-=-, Asarin, Maler, Caspi [4], Dima [28], Maler, Pnueli [58] among others. At the same time, timed automata and timed languages may exhibit paradoxical behaviors, due to the continuous nature of time, so... |

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Citation Context ... arity. The operations of arity 2 are the parallel product, the sequential product and the clique product: they are defined by the graphs with 2 vertices and no edge, 1 edge and 2 edges, respectively =-=[18,88]-=-. The theory of modular decomposition of graphs [57,61] shows that the class of all finite graphs is generated by the singleton graph and F∞, and describes the relations between the operations in F∞. ... |

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Citation Context ...ificantly better algorithm is possible [85]. Elegant results generalize this discussion to transfinite words, that is, Alabeled ordinals longer than ω, see the work of Bedon, Bruyère, Carton, Choueka =-=[15,9,10,13]-=-. For infinite trees, we know models of automata that are equivalent to MSOdecidability (Rabin, see [85]), but the extension of the algebraic ideas sketched above remains to be done. The finiteness re... |

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Citation Context ...An abstraction of this model is given by Lamport Diagrams (LDs) and by Layered Lamport Diagrams (LLDs), which are LDs subject to a boundedness condition. Henriksen, Kumar, Mukund, Sohoni, Thiagarajan =-=[43,44,62]-=- considered the class of bounded finite MSC languages, defined by so-called bounded (Alur, Yannakakis [3]) or locally synchronised (Muscholl, Peled [63]) MSC-graphs. For bounded MSC languages, MSO-def... |

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Citation Context ...An abstraction of this model is given by Lamport Diagrams (LDs) and by Layered Lamport Diagrams (LLDs), which are LDs subject to a boundedness condition. Henriksen, Kumar, Mukund, Sohoni, Thiagarajan =-=[43,44,62]-=- considered the class of bounded finite MSC languages, defined by so-called bounded (Alur, Yannakakis [3]) or locally synchronised (Muscholl, Peled [63]) MSC-graphs. For bounded MSC languages, MSO-def... |

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Citation Context ...-1990s, Pin introduced the usage of ordered semigroups to refine the classification of recognizable languages [69]. The same idea can as naturally be used in any algebra (and has been for instance in =-=[55]-=-), but we will keep it outside the discussion in this paper to avoid increased complexity. 3.3 Definability vs. Recognizability Based on the examples of words and trees, the natural language for logic... |

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Citation Context ...class of finite monoids in which every regular element is idempotent (if xyx = x for some y, then x 2 = x). Then a combination of results of Etessami, Pin, Schützenberger, Thérien, Vardi, Weil, Wilke =-=[74,70,37,80]-=- shows that a language L is in FO2, if and only if it is defined by a Σ2- and by a Π2-formula, if and only if its syntactic monoid is in DA, if and only if L is defined by a PTL formula which does not... |

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Citation Context ...An abstraction of this model is given by Lamport Diagrams (LDs) and by Layered Lamport Diagrams (LLDs), which are LDs subject to a boundedness condition. Henriksen, Kumar, Mukund, Sohoni, Thiagarajan =-=[43,44,62]-=- considered the class of bounded finite MSC languages, defined by so-called bounded (Alur, Yannakakis [3]) or locally synchronised (Muscholl, Peled [63]) MSC-graphs. For bounded MSC languages, MSO-def... |

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Citation Context ...s with bounded tree-width, then CMSO[inc]-definability is equivalent to HR-recognizability [54]. Moreover, if L is uniformly sparse, then CMSO[E]- and CMSO[inc]-definability are equivalent (Courcelle =-=[20]-=-). In view of the equivalence result between HR- and V R-recognizability mentioned above, it would be interesting to find out whether both definabilities are also equivalent if L is without Kn,n for s... |

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Citation Context ...of Σ-trees is FO-definable is still open, and various attempts to use the structure of Σ-algebras described in Sect. 2 to solve it in the spirit of Schützenberger’s theorem (Theorem 1.2), have failed =-=[46,71,72]-=-. Recently, Ésik and Weil introduced a new algebraic framework to investigate this particular problem on tree languages [36]. The point was to enrich the algebraic framework, without modifying the not... |

21 |
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Citation Context ... all linearizations, and to recognizability by deterministic (resp. nondeterministic) message-passing automata. Kuske extended these results to FOdefinable MSC languages, and to infinite bounded MSCs =-=[53]-=-.sThe restriction to classes of posets with a rational language of linearizations is rather severe, but little work so far has discussed definability or recognizability outside this hypothesis. Meenak... |

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Citation Context ...nteresting notion of algebraic recognizability is introduced, that is at least as powerful as timed automata. languages. Moreover, several logics have been introduced (see for instance Demri, D’Souza =-=[24]-=-), but none is completely satisfactory with respect to the motivation of formulating and solving the controller synthesis problem, and none is connected in a robust way to an algebraic approach of rec... |

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Citation Context ...so traces form a monoid); or as a so-called dependence graph, that is, an A-labeled poset where the order is constrained by the distributed structure of the alphabet, see Diekert and Rozenberg’s book =-=[27]-=-. The latter is the more significant model, from the point of view of the original motivation of traces as a model of distributed computation. The power of MSO-definability – interpreted on the depend... |

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Citation Context ...d in terms of morphisms into finite algebras of the same type (or in terms of finite index congruences), and equational sets in terms of systems of polynomial equations (Mezei, Wright [60], Courcelle =-=[17]-=-). With those definitions, recognizable sets form a Boolean algebra, equational sets are closed under union, recognizable sets are always equational, finite sets are equational (even though they may f... |

16 | Pure future local temporal logics are expressively complete for Mazurkiewicz traces
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Citation Context ...f their syntactic monoid (Guaiana, Restivo, Salemi [42]), and important temporal logics with the same expressive power have been developed (see Thiagarajan and Walukiewicz [82] and Diekert and Gastin =-=[26]-=-).sThere is a large body of literature on recognizable trace languages, and the results summarized above point to a rather well understood situation. Some questions however are not solved in a complet... |

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Citation Context ...r additional boundedness hypothesis. For general graph languages, Thomas introduced the notion of graph acceptor [84,86], generalizing the tiling systems introduced earlier by Giammarresi and Restivo =-=[40]-=- for pictures (A-labeled rectangular grids). Recognizability by a graph acceptor was shown to be equivalent to EMSO-definability, where EMSO is the extension of FO by existential quantification of mon... |

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Citation Context ... in the spirit of Schützenberger’s theorem (Theorem 1.2), have failed [46,71,72]. Recently, Ésik and Weil introduced a new algebraic framework to investigate this particular problem on tree languages =-=[36]-=-. The point was to enrich the algebraic framework, without modifying the notion of a recognizable subset, but introducing additional algebraic structure. Ésik and Weil’s algebras, called preclones, ar... |

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Citation Context ...[inc]-definability implies HR-recognizability, see Courcelle [19]. Lapoire showed that if L is a set of graphs with bounded tree-width, then CMSO[inc]-definability is equivalent to HR-recognizability =-=[54]-=-. Moreover, if L is uniformly sparse, then CMSO[E]- and CMSO[inc]-definability are equivalent (Courcelle [20]). In view of the equivalence result between HR- and V R-recognizability mentioned above, i... |

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Citation Context ...-trees, where each child of the root is in fact a smaller sΣ-term. Lodaya and Weil introduced a model of automata and a notion of rational expressions, both of which are equivalent to recognizability =-=[56]-=- – a result which generalizes the characterization of recognizability by finite automata for both words and trees. They showed that their result could be adapted if some of the operations in Σ were as... |

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Citation Context ...e renaming of sources, forgetting sources, and the fusion of two sources [19]. A number of variants can be considered, which do not affect the class of HR-recognizable subsets, see Courcelle and Weil =-=[21]-=-: the disjoint union can be replaced with parallel composition (source name sets need not be disjoint, and sources with the same name get identified), the sources may be assumed to be pairwise distinc... |

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Citation Context ...nd of a logical formalism is also still in its infancy. One should mention however the recent work of Maler and Pnueli [58], and the results of Francez and Kaminski [38] and Bouyer, Petit and Thérien =-=[12]-=- on generalizations of timed languages and automata, to automata on infinite alphabets and to data languages, respectively. In both cases, an interesting notion of algebraic recognizability is introdu... |

11 |
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Citation Context ...lopment of an algebraic apparatus and of a logical formalism is also still in its infancy. One should mention however the recent work of Maler and Pnueli [58], and the results of Francez and Kaminski =-=[38]-=- and Bouyer, Petit and Thérien [12] on generalizations of timed languages and automata, to automata on infinite alphabets and to data languages, respectively. In both cases, an interesting notion of a... |

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Citation Context ...aic approach of (a subclass of LDs) that would match the power of MSO-definability. Series-Parallel Pomsets. Sets of series-parallel pomsets (or sp-languages) were investigated by Lodaya, Weil, Kuske =-=[52,55]-=-. A poset is series-parallel if it can be obtained from singletons by using the operations of sequential and parallel product. There is a combinatorial characterization of these posets (Nfree posets [... |

9 | On recognizable timed languages
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Citation Context ...ropriate notions ofsrational expressions, that would be equivalent to the expressive power of timed automata, see Henzinger, Raskin, Schobbens [45], Asarin, Maler, Caspi [4], Dima [28], Maler, Pnueli =-=[58]-=- among others. At the same time, timed automata and timed languages may exhibit paradoxical behaviors, due to the continuous nature of time, so the central ideas and techniques from the classical theo... |

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Citation Context ...empts to develop appropriate notions ofsrational expressions, that would be equivalent to the expressive power of timed automata, see Henzinger, Raskin, Schobbens [45], Asarin, Maler, Caspi [4], Dima =-=[28]-=-, Maler, Pnueli [58] among others. At the same time, timed automata and timed languages may exhibit paradoxical behaviors, due to the continuous nature of time, so the central ideas and techniques fro... |

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Citation Context ...ts. An A-labeled text is a finite A-labeled set, equipped with 2 linear orders. Texts form a particular class of the 2-structures studied by Ehrenfeucht, Engelfriet, Harju, Proskurowski and Rozenberg =-=[31,32,34]-=-. Hoogeboom and ten Pas introduced an algebraic structure on the set of all texts [47]. This algebra has an infinite signature, but within any finitely generated subalgebra (generated by A and any fin... |

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Citation Context ...ificantly better algorithm is possible [85]. Elegant results generalize this discussion to transfinite words, that is, Alabeled ordinals longer than ω, see the work of Bedon, Bruyère, Carton, Choueka =-=[15,9,10,13]-=-. For infinite trees, we know models of automata that are equivalent to MSOdecidability (Rabin, see [85]), but the extension of the algebraic ideas sketched above remains to be done. The finiteness re... |

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Citation Context ...ty. Good results are also known for FO-definable trace languages: they are characterized by star-free rational expressions, and by the aperiodicity of their syntactic monoid (Guaiana, Restivo, Salemi =-=[42]-=-), and important temporal logics with the same expressive power have been developed (see Thiagarajan and Walukiewicz [82] and Diekert and Gastin [26]).sThere is a large body of literature on recogniza... |