M. Benedikt, L. Libkin, T. Schwentick, and L. Segoufin. A Model-Theoretic approach to regular string relations. In Proceedings of the 16th Annual IEEE Symposium on Logic in Computer Science (LICS 2001.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....automatic if the elements of the universe can be represented as words from a regular language and every relation of the structure can be recognized by a nite state automaton with several heads that proceed synchronously. Automatic structures received increasing interest during the last years [1, 3, 14, 16 18]. One of the main motivations for investigating automatic structures is the fact that every automatic structure has a decidable rst order theory. On the other hand it is known that there exist automatic structures with a nonelementary rst order theory [3] This motivates the search for ....

M. Benedikt, L. Libkin, T. Schwentick, and L. Segou n. A model-theoretic approach to regular string relations. In Proceedings of the 16th Annual IEEE Symposium on Logic in Computer Science (LICS'2001), pages 431-440. IEEE Computer Society Press, 2001.

....following are equivalent. i) PA is w automatic. ii) PA rO 9p for some (and hence all) p 2. iii) PA rO Tree(P) for some (and hence all) p 2. For the proof, see [17] There are similar characterisations for tree automatic structures [15] For further results on automatic structures, see [10, 15, 17, 71]. The model theoretic characterisations of automatic and w automatic structures in terms of interpretability suggest a general way for obtaining other domains of infinite structures that may be interesting for computational model theory: Fix a structure PA with nice (algorithmic and or ....

M. BENEDIKT, L. LIBKIN, T. SCHWENTICK, AND L. SEGOUrIN, A model-theoretic approach to regular string relations, in Proc. 16th IEEE Symp. on Logic in Computer Science, 2001, pp. 431-440.

....in particular, quantifier elimination, bounded VC dimension and expressive power. The last part contains database applications in terms of expressiveness, data complexity and safety of the corresponding query languages. Earlier presentation of this work appeared in two conference proceedings: [13, 12]. 2 Notation Throughout the paper, Sigma denotes a finite alphabet, and Sigma the set of all finite strings over Sigma. We consider a number of operations and predicates on Sigma ffl x Delta y concatenation of two strings x and y. ffl x y x is a prefix of y. ffl l a (x) a 2 ....

M. Benedikt, L. Libkin, T. Schwentick, L. Segoufin. A model-theoretic approach to regular string relations. In LICS'01, pages 431--440.

....in particular, quanti er elimination, bounded VC dimension and expressive power. The last part contains database applications in terms of expressiveness, data complexity and safety of the corresponding query languages. Earlier presentation of this work appeared in two conference proceedings: [13, 12]. 2 Notation Throughout the paper, denotes a nite alphabet, and the set of all nite strings over . We consider a number of operations and predicates on x y concatenation of two strings x and y. x y x is a pre x of y. l a (x) a 2 , is x a (adds l ast ....

M. Benedikt, L. Libkin, T. Schwentick, L. Segou n. A model-theoretic approach to regular string relations. In LICS'01, pages 431-440.

....automata. In the other setting, one considers the family of all strings # # or the family of all trees, and defines some operations on them. This gives us a first order structure M, and formulae in one free variable #(x) define sets of trees strings x #(x) This approach was studied in [3, 5, 10, 6, 20, 19]. The second approach led to the study of automatic structures, that is, structures in which every definable predicate can be represented by a finite automaton [19, 20] It was shown in [6] that there is a universal automatic structure over strings, that is, a structure S such that every other ....

....strings, that is, a structure S such that every other automatic structure can be embedded into S. That structure S also had several reducts defining regular and starfree languages, and having some nice properties that made them useful as the basis for relational calculi on databases over strings [3, 4]. Recently, automatic structures have been studied in the context of ranked trees [5] In that case, the universe is the set of all trees, and the universal treeautomatic structure T has as its definable relations precisely the relations recognized by tree automata [15] In this paper, we study ....

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M. Benedikt, L. Libkin, T. Schwentick, L. Segoufin. A model-theoretic approach to regular string relations. In LICS'01, pages 431--440.

....to , image of f a , and simply tests if two strings x and y have the same length, which we shall denote by el(x; y) Summing up, for n = 1, T is equivalent to S = h ; f a ) a2 ; eli and T p is equivalent to its reduct S p = h ; f a ) a2 ; i. These structures are well known [11, 10, 9]. 6 Figure 1 summarizes results on T; T p , and their string analogs S [11, 10] and S p [9] It turns out that modeltheoretically S and T are rather close, but S p and T p are very different. The first line of the table talks about one dimensional definable sets, that is, subsets of or ....

....denote by el(x; y) Summing up, for n = 1, T is equivalent to S = h ; f a ) a2 ; eli and T p is equivalent to its reduct S p = h ; f a ) a2 ; i. These structures are well known [11, 10, 9] 6 Figure 1 summarizes results on T; T p , and their string analogs S [11, 10] and S p [9]. It turns out that modeltheoretically S and T are rather close, but S p and T p are very different. The first line of the table talks about one dimensional definable sets, that is, subsets of or Trees n ( The second line is about arbitrary definable sets. The automaton construction for S ....

M. Benedikt, L. Libkin, T. Schwentick, L. Segoufin. A model-theoretic approach to regular string relations. In LICS'01, pages 431--440.

....algebra, and proves a quanti er elimination result for this model. The section also connects this model to the minimal model S. Section 7 gives an additional example of a regular algebra, which contains each of the previous examples. Section 8 gives conclusions. All proofs are in the full version [6]. 2 Notations Throughout the paper, denotes a nite alphabet, and the set of all nite strings over . We consider a number of operations on : x y concatenation of two strings x and y. x y x is a pre x of y. l a (x) a 2 , is x a (adds l ast character) f a ....

....(which is actually stronger than what is needed for quanti er elimination) The theorem follows from the lemma, as each type of the form atp S (t 1 (c i 1 ) t k (c i k ) is also an atomic type of S left . Hence, the atomic types determine the types. For details, see the full version [6]. 2 From the previous theorem we get the following corollaries. First, the back and forth property of 1 gives us the following normal form for FO(S left ) formulae. Corollary 4 For every FO(S left ) formula (x; y) there is an FO(S) formula 0 (x; z) and a nite set of nice S left ....

M. Benedikt, L. Libkin, T. Schwentick, L. Segoun. A model-theoretic approach to regular string relations. INRIA Technical Report, 2000. Available at http://www-rocq.inria.fr/verso/publications/.

....ordering, as well as substrings of constant length, and TRIM TRAILING, that removes all trailing occurrences of a given symbol. S len is much more powerful, and covers the SIMILAR pattern matching of the SQL3 standard [21] which is essentially grep) The properties listed below can be found in [13, 10, 8] for S len and [8, 10] for S. Note that they are properties of the underlying structures alone, without reference to a database. Properties of S. Every formula is equivalent to a formula in which quanti cation is restricted to pre xes of free variables. Moreover, S has quanti er elimination in ....

....of constant length, and TRIM TRAILING, that removes all trailing occurrences of a given symbol. S len is much more powerful, and covers the SIMILAR pattern matching of the SQL3 standard [21] which is essentially grep) The properties listed below can be found in [13, 10, 8] for S len and [8, 10] for S. Note that they are properties of the underlying structures alone, without reference to a database. Properties of S. Every formula is equivalent to a formula in which quanti cation is restricted to pre xes of free variables. Moreover, S has quanti er elimination in the signature extended ....

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M. Benedikt, L. Libkin, T. Schwentick, L. Segoun. A model-theoretic approach to regular string relations. Technical report, INRIA, 2000.

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M. Benedikt, L. Libkin, T. Schwentick, and L. Segoufin. A Model-Theoretic approach to regular string relations. In Proceedings of the 16th Annual IEEE Symposium on Logic in Computer Science (LICS 2001.

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