L. Libkin. On counting logics and local properties. In Proceedings of 13th IEEE Symposium on Logic in Computer Science, 501--512, Indianapolis, Indiana, 1998.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....extension by modulo counting quantifiers. Thus, we obtain that local properties of this modulo counting logic are decidable (Theorem 5.14) The decidability of non local properties is not answered in this paper, the missing tool is an analogue of Gaifman s locality theorem for this logic. Libkin [34 36] and Nurmonen [53] proved generalizations of a version of Gaifman s Theorem for this and other counting extensions of first order logic, but we could not make these generalizations serve our purposes. As mentioned above, the first step in our decidability proof is an application of Gaifman s ....

....proof for the first order theory of does not work for this more expressive logic; but the second step of our proof, i.e. the recognizability of the set of Note that is built using the signature of while uses the larger signature of . Libkin [34 36] and Nurmonen [53] proved locality theorems for counting logics including modulo counting, but not in the form of Theorem 5.1. We could not make them work in our situation. 27 traces satisfying some local formula in FO (cf. Theorems 5.9 and 5.7) extends to the logic FO MOD. Thus, we obtain the ....

L. Libkin. On counting logics and local properties. In Proceedings of the 13th Annual IEEE Symposium on Logic in Computer Science (LICS'98), pages 501--512. IEEE Computer Society Press, 1998.

....addition and multiplication cannot express the transitive closure of S, and therefore fails to capture LOGSPACE. The proof in the journal version of the paper relies on the fact that first order logic with counting quantifiers can only express local properties, which follows from [22] Libkin [18, 19, 20] also considered local properties of a variety of logics involving counting quantifiers. Unfortunately, in the presence of a total ordering, these proof techniques do not apply anymore, since all elements of the structure are directly connected by the ordering. The result of this paper may ....

L. Libkin. On counting logics and local properties. Proceedings LICS'98, pages 501--512, 1998.

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L. Libkin. On counting logics and local properties. In Proceedings of 13th IEEE Symposium on Logic in Computer Science, 501--512, Indianapolis, Indiana, 1998.

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L. Libkin. On counting logics and local properties. In LICS'98, pages 501-512.

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L. Libkin. On counting logics and local properties. TOCL 1(1): 33-59 (2000).

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L. Libkin. On counting logics and local properties. TOCL 1(1): 33--59 (2000).

....for other logics, e.g. FO and FO(C) 3,28] Our main goal is to study the impact of auxiliary relations, such as orderings, on the expressive power of logics with counting. Our results apply to a variety of logics, starting with FO and FO(C) and ending with a logic L 1 (C) proposed in [22]. This logic subsumes FO(C) and all other known pure counting extensions of FO. Note that when we speak of counting extensions of FO, we mean extensions that only add a counting mechanism, as opposed to those extensively studied in the literature [27] that add both counting and fixpoint. We ....

....is defined as quantifier rank. That is, it is 0 for atomic formulae; rk( W i i ) max i rk( i ) rk( rk( and rk(9x ) rk(9ix ) rk( 1 as usual. But it does not take into account quantification over N: rk(9i ) rk( Furthermore, rk(# x: rk( j xj. Definition 1 (see [22]) The logic L 1 (C) is defined to be the restriction of L1 (C) to terms and formulae of finite rank. It is known [22] that L 1 (C) formulae are closed under Boolean connectives and all quantification, and that every predicate on N Theta : Theta N is definable by a L 1 (C) ....

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L. Libkin. On counting logics and local properties. In Proc. 12th IEEE Symp. on Logic in Computer Science (LICS'97), Warsaw, Poland, June--July 1996, pages 204--215.

....isomorphic; again d is determined by k. It was shown that Hanf s theorem is strictly stronger than Gaifman s, and that both apply to a variety of logics that extend FO with counting mechanisms and limited in nitary connectives [Grohe and Schwentick 2000; Hella et al. 1999a; Hella et al. 1999b; Libkin 2000; Nurmonen 1996] These results found applications in descriptive complexity and database theory. Since the complexity class TC 0 (with the appropriate notion of uniformity) can be captured by FO with counting quanti ers [Barrington et al. 1990] locality can be used to prove lower bounds for ....

.... 1998] The above mentioned papers considered a sequence of more and more powerful logics, each of which was proved to be local, starting with FO with counting quanti ers, and ending with a logic that permits arbitrary predicates on natural numbers, a limited form of in nitary connectives [Libkin 2000] and even aggregate functions [Hella et al. 1999b] However, it was not clear how much one can add to these logics and still preserve its locality. Our goal, therefore, is to give a precise characterization of local logics. Note that the abstract notions of locality were previously characterized ....

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Libkin, L. 2000. On counting logics and local properties. ACM TOCL, 1, 1, 33-59.

....Can we find a powerful logic into which aggregate queries can be easily embedded, and whose properties can be analyzed so that bounds for query languages can be derived Our main goal is to give the positive answer to this question. To do so, we combine a powerful infinitary counting logic from [22] with an elegant framework of [11] for adding aggregation. As the numerical domain, we choose the set of rational numbers Q, although other domains (e.g. Z; R) can be chosen. The resulting logic L aggr defines every arithmetic operation and every aggregate function. We then show that it has very ....

....on Q : one, denoted by Omega Gamma of functions and predicates, and one, denoted by Theta, of aggregates. In addition we assume that there is a constant symbol c q for each q 2 Q. We now define an aggregate logic L aggr( Omega ; Theta) on twosorted structures. We do it, similarly to [22], in two steps. We first define a larger logic L aggr( Omega ; Theta) and then put a restriction on its formulae. We define terms and formulae of the two sorted logic L aggr( Omega ; Theta) over two sorted structures, by simultaneous induction. Every variable of the ith sort is a term of ....

[Article contains additional citation context not shown here]

L. Libkin. On counting logics and local properties. In LICS'98, pages 501--512.

.... see [SB98] The above mentioned papers considered a sequence of more and more powerful logics, each of which was proved to be local, starting with FO with counting quantifiers, and ending with a logic that permits arbitrary predicates on natural numbers, a limited form of infinitary connectives [Li00] and even aggregate functions [HLNW99] However, it was not clear how much one can add to these logics and still preserve its locality. Our goal, therefore, is to give a precise characterization of local logics. Note that the abstract notions of locality were previously characterized on finite ....

....is to give a precise characterization of local logics. Note that the abstract notions of locality were previously characterized on finite structures of bounded valence (e.g. for graphs of fixed maximum degree) The characterization for Hanf locality uses a logic L 1 (C) introduced in [Li00]. This logic subsumes a number of counting extensions of FO (such as FO with counting quantifiers [IL90] FO with unary generalized quantifiers [He96; KV95] FO with unary counters [BK97] and is quite easy to deal with. A result in [HLN99] states that Hanf local properties on structures of ....

[Article contains additional citation context not shown here]

L. Libkin. On counting logics and local properties. ACM TOCL, 1 (1) (2000), 33--59.

....degree. The restriction allows us to use the best possible bounds in the statements of those theorems. The proof relies on new locality based conditions that provide winning strategies for the duplicator. Concluding remarks are given in Section 8. Complete proofs are given in the full version [23]. 2 Notations Finite Structures and Logics All structures are assumed to be finite. A relational signature oe is a set of relation symbols fR 1 , R l g, with associated arities p i 0. We write oe n for oe extended with n new constant symbols. A oe structure is A = hA; R A 1 ; R ....

....0 (x) U (x) and ff k 1 (x) 9z:d k (x; z) ff k (z) This formula says that there exists an element of U at the distance at most 2 k Gamma 1 from x. It is easy to see that lr(ff k ) 2 k Gamma 1 and rk(ff k ) k; hence Loc rankL (n) 2 n Gamma 1. For the proof of the upper bound, see [23]. 2 The reason why the separation property itself could not be used to prove this theorem, is the following. It is possible to find, for any n, a formula ff n (x) such that ff n 2 SP(2 n ) but ff n 62 SP(r) for any r 2 n . In fact, the formulae we ff n (x) we introduced in the proof to show ....

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L. Libkin. On counting logics and local properties. Bell Labs, Technical Memo, 1997.

....e.g. FO and FO(C) 3, 30] Our main goal is to study the impact of auxiliary re1 lations, such as orderings, on the expressive power of counting. The primary motivation comes from complexity theory: while good expressivity bounds exists for counting logics, e.g. FO(C) over unordered structures [8, 23, 24], no nontrivial bounds are known for the ordered case. As we mentioned, FO(C) over ordered structures, captures TC 0 , the class of problems solvable by polynomial size, constant depth threshold circuits, under DLOGTIME uniformity, see [2] This is an important complexity class: problems such ....

.... to circuit lower bounds, see [1, 32] One might thus hope that the approach based on proving expressivity bounds for logics may circumvent the problems raised by [32] The results we prove apply to a variety of logics, starting with FO and FO(C) and ending with a logic L 1 (C) proposed in [24]. This logic subsumes FO(C) and all other known pure counting extensions of FO. When we speak of counting extensions of FO, we mean extensions that only add a counting mechanism, as opposed to those extensively studied in the literature, see [29] that add both counting and fixpoint. We ....

[Article contains additional citation context not shown here]

L. Libkin. On counting logics and local properties. In LICS'98, to pages 501--512.

....Can we nd a powerful logic into which aggregate queries can be easily embedded, and whose properties can be analyzed so that bounds for query languages can be derived Our main goal is to give the positive answer to this question. To do so, we combine a powerful in nitary counting logic from [22] with an elegant framework of [11] for adding aggregation. As the numerical domain, we choose the set of rational numbers Q , although other domains (e.g. Z; R ) can be chosen. The resulting logic L aggr de nes every arithmetic operation and every aggregate function. We then show that it has very ....

....that we are given two signatures on Q : one, denoted by of functions and predicates, and one, denoted by , of aggregates. In addition we assume that there is a constant symbol c q for each q 2 Q . We now de ne an aggregate logic L aggr( on twosorted structures. We do it, similarly to [22], in two steps. We rst de ne a larger logic L aggr( and then put a restriction on its formulae. We de ne terms and formulae of the two sorted logic L aggr( over two sorted structures, by simultaneous induction. Every variable of the ith sort is a term of the ith sort, i = 1; 2. ....

[Article contains additional citation context not shown here]

L. Libkin. On counting logics and local properties. In LICS'98, pages 501-512.

....logics, e.g. FO and FO(C) 3, 29] 2 Our main goal is to study the impact of auxiliary relations, such as orderings, on the expressive power of logics with counting. The results we prove apply to a variety of logics, starting with FO and FO(C) and ending with a logic L 1 (C) proposed in [23]. This logic subsumes FO(C) and all other known pure counting extensions of FO. When we speak of counting extensions of FO, we mean extensions that only add a counting mechanism, as opposed to those extensively studied in the literature, see [28] that add both counting and fixpoint. We ....

....by rk. It is defined as quantifier rank (that is, it is 0 for atomic formulae, rk( W i i ) max i rk( i ) rk( rk( rk(9x ) rk(9ix ) rk( 1) but it does not take into account quantification over N: rk(9i ) rk( Furthermore, rk(# x: rk( j xj. Definition 1 (see [23]) The logic L 1 (C) is defined to be the restriction of L1 (C) to terms and formulae of finite rank. It is known [23] that L 1 (C) formulae are closed under Boolean connectives and all quantification, and that every predicate on N Theta : Theta N is definable by a L 1 (C) formula of ....

[Article contains additional citation context not shown here]

L. Libkin. On counting logics and local properties. In LICS'98, to pages 501--512.

....of ntp(d; A) from graph queries to arbitrary ones. In this paper, the only extension of this kind was for the Gaifman graph of the output. It turns out that an analog of Theorem 3.1 can be proved for queries of arbitrary arity, with d depending on both locality rank and the arity. For details, see [31]. Another problem mentioned in [9] was to develop techniques for proving languages local. One such technique was proposed in [30] which showed that queries in any reasonable logic that satisfies an analog of Hanf s theorem [24, 16] are local. Using this, and results of [25, 38] the paper [30] ....

....local. One such technique was proposed in [30] which showed that queries in any reasonable logic that satisfies an analog of Hanf s theorem [24, 16] are local. Using this, and results of [25, 38] the paper [30] showed that first order logic extended with unary generalized quantifiers is local. In [31], a technique was presented that allows one to prove locality without a recourse to Hanf s theorem. The same paper showed a version of infinitary logic that can define every numerical property, but expresses only local queries when restricted to finite relational structures. Two problems related ....

L. Libkin. On counting logics and local properties. In Proceedings of 13th IEEE Symposium on Logic in Computer Science, 501--512, Indianapolis, Indiana, 1998.

....next step, we show that locality of SQL withstands adding numerical relations, those of type fQ Theta : Theta Q g, as long as there is no ordering on b. To prove this, we first code SQL into an infinitary logic with counting, as was done in [15] and then modify the induction argument from [17] to prove locality in the presence of extra numerical relations. Finally, a finite number, say m, of unary relations of type fbg, amounts to coloring nodes of a graph with 2 m colors. If we assume that q is definable with auxiliary unary relations, we fix a number r witnessing its locality, and ....

L. Libkin. On counting logics and local properties. In LICS'98, pages 501-512.

....Can we find a powerful logic into which aggregate queries can be easily embedded, and whose properties can be analyzed so that bounds for query languages can be derived Our main goal is to give the positive answer to this question. To do so, we combine a powerful infinitary counting logic from [22] with an elegant framework of [12] for adding aggregation. As the numerical domain, we choose the set of rational numbers Q, although any other domain can be chosen. The resulting logic L aggr defines every arithmetic operation and every aggregate function. We then show that it has very nice ....

....on Q : one, denoted by Omega Gamma of functions and predicates, and one, denoted by Theta, of aggregates. In addition we assume that there is a constant symbol c q for each q 2 Q. We now define an aggregate logic L aggr( Omega ; Theta) on two sorted structures. We do it, similarly to [22], in two steps. We first define a larger logic L aggr( Omega ; Theta) and then put a restriction on its formulae. We define terms and formulae of the two sorted logic L aggr( Omega ; Theta) over two sorted structures, by simultaneous induction. Every variable of the ith sort is a term of ....

[Article contains additional citation context not shown here]

L. Libkin. On counting logics and local properties. In LICS'98, pages 501--512.

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