J.-E. Pin. Syntactic semigroups. In G. Rozenberg and A. Salomaa, editors, Handbook of Formal Languages, volume 1, chapter 10. Springer, 1997.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....(DFG) grant Wa 847 4 1. the theory of finite automata, the theory of finite semigroups, first order theory and other fields of research, due to the various characterizations of starfree languages. For an overview on these rich theories and their surprising relations we refer to the articles [Brz76, Pin96a, Pin96b, Tho96] and to the textbooks [MP71, Pin86] A fragment TL[ of TL is a subset of TL where only the use of the temporal operators specified in brackets is allowed, e.g. TL[ TL[F] TL[X;F] TL[X;F;U] TL. For a class of formulas Phi TL we denote by L( Phi) the class of Phi definable languages. ....

J. E. Pin. Syntactic semigroups. In G.Rozenberg and A.Salomaa, editors, Handbook of formal languages, volume I, pages 679--746. Springer, 1996.

....exactly those languages with an L trivial syntactic monoid. The intersection of both is known to equal level 1 of the Straubing Th erien Hierarchy (STH) of star free regular languages, known to correspond to so called J trivial monoids. For background on these classes, we refer the reader to [Pin97], see also [Pin86] These classes have interesting combinatorial characterizations: Consider an alphabet A. A left deterministic product over A is a concatenation of the form k , where a i 2 A, A i A, and a i 62 A i 1 . Now a language is recognized by a p. o. FA i it is a nite disjoint ....

J. E. Pin. Syntactic semigroups. In G. Rozenberg and A. Salomaa, editors, Handbook of Formal Languages, volume I, pages 679-746. Springer Verlag, Berlin Heidelberg, 1997.

....then both a and b are permutations. For an automaton to be solvable, it must have at least one constant event. One way to generalize the notion of decidability would be to extend it to constant composition of events, hinting at the possibility that properties of the syntactic monoid (see [9] and [7]) are related to the existence of vector algorithms. ....

J.-E. Pin, Syntactic Semigroups, in Handbook of Formal Languages, Vol. 1, Springer, (1997) 679-738.

....Utt Sigma is a very simple regular language, a finite union of languages of the form Sigma a 1 Sigma a 2 Sigma Delta Delta Delta a k Sigma for some letters a i 2 Sigma . This family of languages corresponds exactly to level 1=2 in the concatenation hierarchy of Straubing Th erien [10]) Finally, Sigma a 1 Sigma a 2 Sigma Delta Delta Delta a k Sigma ] a i 1 Delta Delta Deltaa i k 2[a1 Delta Delta Deltaa k ] Sigma a i 1 Sigma Delta Delta Delta a i k Sigma . 6 Conclusion In this paper we presented specification and verification methods for MSCs, which ....

J.-E. Pin. Syntactic semigroups. In G. Rozenberg and A. Salomaa, editors, Handbook of Formal Languages, volume 1, pages 679--738. Springer, Berlin-HeidelbergNew York, 1997.

....1 . L 2 Sigma . The following problem has been widely investigated: find a function Psi . such that the language L 1 . L 2 is recognized by Psi . M 1 ; M 2 ) For more details on this problem, as well as for a large bibliography, the reader is referred to [1] 4] or more recently, [5]. 5 In the sequel we solve this problem for the operation T , where T is an arbitrary set of trajectories. The solution offers a uniform method to find monoids that recognize a large number of operations with languages. Also, we compare our solution with other well known constructions, mainly ....

J.E. Pin, "Syntactic Semigroups", in Handbook of Formal Languages, eds. G. Rozenberg and A. Salomaa, Vol. 1, Springer, 1997, 679-746.

....exactly those languages with an L trivial syntactic monoid. The intersection of both is known to equal level 1 of the Straubing Th erien Hierarchy (STH) of star free regular languages, known to correspond to so called J trivial monoids. For background on these classes, we refer the reader to [Pin97], see also [Pin86] These classes have interesting combinatorial characterizations: Consider an alphabet A. A left deterministic product over A is a concatenation of the form A 0 a 1 A 1 : a k A k , where a i 2 A, A i A, and a i 62 A i 1 . Now a language is recognized by a ....

J. E. Pin. Syntactic semigroups. In G. Rozenberg and A. Salomaa, editors, Handbook of Formal Languages, volume I, pages 679-746. Springer Verlag, Berlin Heidelberg, 1997.

....together with concatenation. To determine for a given language the minimal number of alternations between these two kinds of operations is known as the dot depth problem, recently considered as one of the most important open questions on regular languages [9] For an overview we refer to [8]. We deal with the dot depth hierarchy [3] and the Straubing Th erien hierarchy [12, 15, 13] which both formalize the dot depth problem in terms of the membership problems of their hierarchy classes. Fix some nite alphabet A with jAj 2. For a class C of languages let Pol(C) be its polynomial ....

J.-E. Pin. Syntactic semigroups. In G.Rozenberg and A.Salomaa, editors, Handbook of formal languages, volume I, pages 679-746. Springer, 1996.

....have a close connection to complexity theory (via the leaf languages approach) It is known that these hierarchies are strict, and that they exhaust the class of starfree languages. The decidability of their levels is an open problem. For a comprehensive introduction to this area have a look at [Brz76, Pin96a, Pin96b, Tho96, Sch98]. To understand complicated objects in a better way, it is useful to develop normalforms for them. In this paper we will provide such normalforms for the levels n 1=2 of both hierarchies: L n 1=2 = Pol (coL n 1=2 ) B n 1=2 = Pol (co B n 1=2 ) Notice that coL n 1=2 ( L n and co ....

J. E. Pin. Syntactic semigroups. In G. Rozenberg and A. Salomaa, editors, Handbook of formal languages, volume I, pages 679-746. Springer, 1996.

....F , for short) is known as the dot depth problem. Although many researchers believe the answer should be yes, some suspect the contrary. To our knowledge, only the classes B 0 , B 1=2 and B 1 were known to be decidable [Kna83,PW97] Especially, the case of dot depth 3=2 was mentioned open in [Pin96,PW97]. This can be seen in contrast to the Straubing Th erien hierarchy, for which beside levels 0, 1=2 and 1 also level 3=2 is known to be decidable [Arf91,PW97] Some partial results are known for level 2 of this hierarchy, e.g. its decidability in case of a 2 letter alphabet [Str88] In this ....

....corollary. Corollary 1. Given a regular language L, it is decidable whether L is definable by a Sigma 2 formula of the logic FO[ min; max; S; P ] An algebraic interpretation of our Theorem 6 can also be given. For an introduction to the algebraic theory of finite automata we refer to [Pin96]. Let L be a regular language of A and let FL = A; S; ffi; s 0 ; S 0 ) be its unique minimal dfa. We define the syntactic semigroup of L via the transition semigroup of FL , i.e. as S L = def fffi w : w 2 A g where the composition is defined as ffi u Delta ffi v = def ffi uv . ....

J. E. Pin. Syntactic semigroups. In G.Rozenberg and A.Salomaa, editors, Handbook of formal languages, volume I, pages 679--746. Springer, 1996.

....generalize the lexicographical normal form, as well as the priority normal form used in [18] Moreover, right factor resp. left factor normal forms suce in order to obtain an analogue of Prop. 16. As a byproduct we obtain an alternative characterization of the priority normal form. De nition 18 ([21]) A monoid S is called ordered if there exists some total order such that s t implies xsy xty for all x; y 2 S. For a homomorphism h : S, a monoid element s 2 S and a subalphabet A , we denote by h 1 A (s) the set h 1 (s) fw j alph(w) Ag of words with alphabet A which ....

J.-E. Pin. Syntactic semigroups. In G. Rozenberg and A. Salomaa, editors, Handbook of Formal Languages, volume 1, pages 679-738. Springer, Berlin-HeidelbergNew York, 1997.

.... closely related Straubing Therien hierarchy [Str81, The81, Str85] Their investigation is of major interest in many research areas since surprisingly close connections have been exposed, e.g. to finite model theory, theory of finite semigroups, complexity theory and others (for an overview, see [Pin96]) Let A be some finite alphabet with jAj 2. For a class C of languages let Pol(C) be its polynomial closure, i.e. the closure under finite union and concatenation. Denote by BC(C) its Boolean closure (taking complements with respect to A ) Then the classes B n=2 of the dot depth ....

....Since most results in the field were obtained via this theory, the following definitions of concatenation hierarchies are widely used. We denote the Boolean closure of a class D of languages of A by BC (D) taking complements with respect to A ) Definition 2. 10 (DDH due to [Pin96]) Let B 1=2 be the class of all languages of A which can be written as finite unions of languages of the form u 0 A u 1 A um where m 0 and u i 2 A . For n 0 let B n 1 = def BC(B n 1=2 ) and B n 3=2 = def Pol B (B n 1 ) Definition 2.11 (STH due ....

J.-E. Pin. Syntactic semigroups. In G.Rozenberg and A.Salomaa, editors, Handbook of formal languages, volume I, pages 679--746. Springer, 1996.

....in different research areas since surprisingly close connections to finite model theory, theory of finite semigroups, topology, Boolean circuits, complexity theory and others have been exposed. For an overview or as a good starting point to this rich field of research see e.g. the articles [Brz76, Pin96a, Pin96b]. Here we deal with the dot depth hierarchy. Let A be some finite alphabet with jAj 2. For a class C of languages over A let POL(C) be its polynomial closure, i.e. the class of languages L that can be written as a finite union of languages u 0 L 1 u 1 L 2 u 2 Delta Delta Delta L n u n , ....

....that these classes are also closed under intersection (see [Arf87, Arf91] To our knowledge, only the classes of levels 0, 1=2 and 1 of the dot depth hierarchy were known to be decidable. Especially, the case of dot depth 3=2 was mentioned as an open question in the Handbook of Formal Languages [Pin96b] and in [PW97] This can be seen in contrast to the Straubing Therien hierarchy, for which beside levels 0, 1=2 and 1 also level 3=2 is known to be decidable. The latter was first proved in [Arf87, Arf91] see also [PW97] Due to a fundamental result in [Str85] it is known that level n of the ....

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J. E. Pin. Syntactic semigroups. In G.Rozenberg and A.Salomaa, editors, Handbook of formal languages, volume I, pages 679--746. Springer, 1996. 49

....it is closely related to other research areas, e.g. connections to finite model theory, theory of finite semigroups, topology, Boolean circuits, complexity theory and others have been exposed. For an overview or as a good starting point to this rich field of research see e.g. Brz76, Pin96a, Pin96b] Let A be some finite alphabet with jAj 2 and let B 1=2 denote the class of languages having dot depth 1=2 (we use notations from [PW97] i.e. the class of languages of A that can we written as finite unions of languages u 0 A u 1 A Delta Delta Delta u n Gamma1 A u n , ....

J. E. Pin. Syntactic semigroups. In G.Rozenberg and A.Salomaa, editors, Handbook of formal languages, volume I, pages 679--746. Springer, 1996.

....decidability of its levels is still an open problem, although a lot of e ort via di erent approaches has been invested. It is only known that the levels 1=2 and 1 are decidable (cf. Kna83, Ste85b, PW97] For an overview of concatenation hierarchies see for example the articles [Brz76, Pin96a, Pin96b, Tho96, Sch98] For a class C of languages of A , let BC (C) be the Boolean closure of C, i.e. BC (C) is the smallest class containing C and being closed under union, intersection and complementation w.r.t. A . For a class C of languages of A , let Pol (C) be its polynomial closure, ....

J. E. Pin. Syntactic semigroups. In G. Rozenberg and A. Salomaa, editors, Handbook of formal languages, volume I, pages 679-746. Springer, 1996.

....are of major interest in different research areas since there are close connections to finite model theory, theory of finite semigroups, topology, boolean circuits and others. For an overview or as a good starting point to this rich field of research see e.g. the articles [Brz76, Pin96a, Pin96b, Tho96] In this paper we deal with the so called Straubing Therien hierarchy. Let A be some finite alphabet with jAj 2. For a class C of languages over A let POL(C) be its polynomial closure, i.e. the class of languages L that can be written as a finite union of languages L 0 a 1 L 1 a 2 ....

J. E. Pin. Syntactic semigroups. In G.Rozenberg and A.Salomaa, editors, Handbook of formal languages, volume I, pages 679--746. Springer, 1996.

....form level one of the dot depth hierarchy, first studied in [CB71] This hierarchy is known to be strict [BK78] and exhausts the class of regular starfree languages. For an overview or as a good starting point to the study of concatenation hierarchies and related problems see e.g. the articles [Brz76, Pin96a, Pin96b, Tho96]. We consider languages over a finite alphabet A with jAj 2. All definitions will be made for arbitrary but fixed alphabet A. For a class C of languages of A , let BC(C) be the boolean closure of C, i.e. BC(C) is the smallest class containing C and being closed under union, intersection ....

....by a deterministic finite automaton, decide whether or not L has dot depth n=2. To our knowledge, only levels 0; 1=2 and 1 of the dot depth hierarchy are known to be decidable (cf. PW97] We deal in this paper with level one, i.e. with the class B 1 . The definitions given here are adopted from [Pin96b] and differ a little bit from the ones given in earlier literature, e.g. CB71, BK78] On one hand the levels n 1=2 of the dot depth hierarchy extent the corresponding classes defined in [CB71] but levels n coincide. On the other hand languages having dot depth one have been studied in earlier ....

[Article contains additional citation context not shown here]

J. E. Pin. Syntactic semigroups. In G.Rozenberg and A.Salomaa, editors, Handbook of formal languages, volume I, pages 679--746. Springer, 1996.

....(DFG) grant Wa 847 4 1. the theory of finite automata, the theory of finite semigroups, first order theory and other fields of research, due to the various characterizations of starfree languages. For an overview on these rich theories and their surprising relations we refer to the articles [Brz76, Pin96a, Pin96b, Tho96] and to the textbooks [MP71, Pin86] A fragment TL[ of TL is a subset of TL where only the use of the temporal operators specified in brackets is allowed, e.g. TL[ TL[F] TL[X;F] TL[X;F;U] TL. For a class of formulas Phi TL we denote by L( Phi) the class of Phi definable languages. ....

J. E. Pin. Syntactic semigroups. In G.Rozenberg and A.Salomaa, editors, Handbook of formal languages, volume I, pages 679--746. Springer, 1996.

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J.-  E. Pin, Syntactic semigroups, in Handbook of formal languages, G. Rozenberg and A. Salomaa (ed.), Springer Verlag, 1997.

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J.-E. Pin, Syntactic semigroups, in Handbook of formal languages, G. Rozenberg et A. Salomaa eds., vol. 1, ch. 10, pp. 679-746, Springer (1997).

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J.-E. Pin, 1997, Syntactic semigroups, Chapter 10 in Handbook of formal languages, G. Rozenberg and A. Salomaa eds., Springer.

....eSe is an idempotent and commutative semigroup, and ese e for each idempotent e and for each element s. It is known that a language is recognized by an element of LJ 1 if and only if it is a positive Boolean combination of languages of the form fxg, xA , A x and A xA (x 2 A ) [14]. By the same reasoning as in the previous example, one can show that the pseudovariety of finite ordered semigroups associated with the variety of quasi ecarts defined above is LJ 1 . In particular, the quasi order LJ 1 is trivial. ....

J.-E. Pin. Syntactic semigroups, in Handbook of language theory , G. Rozenberg and A. Salomaa eds., Springer (1997).

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J.-E. Pin. Syntactic semigroups. In G. Rozenberg and A. Salomaa, editors, Handbook of Formal Languages, volume 1, chapter 10. Springer, 1997.

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J.-E. Pin. Syntactic semigroups. In G. Rozenberg and A. Salomaa, editors, Handbook of Formal Languages, Vol. 1, pp. 679--746. Springer-Verlag, 1997.

No context found.

J. E. Pin. Syntactic semigroups. In G.Rozenberg and A.Salomaa, editors, Handbook of formal languages, volume I, pages 679--746. Springer, 1996.

No context found.

J. E. Pin. Syntactic semigroups. In G.Rozenberg and A.Salomaa, editors, Handbook of formal languages, volume I, pages 679--746. Springer, 1996.

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