R. Fagin, L. Stockmeyer, M. Vardi, On monadic NP vs monadic co-NP, Information and Computation, 120 (1994), 78--92.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....4. AGGREGATE LOGIC: EXPRESSIVE POWER In this section we deal with the expressive power of the aggregate logic. Our main goal is to show that it satisfies a very strong locality property. Locality properties were introduced in model theory by Hanf [19] and Gaifman [16] and recently, following [14], they were a subject of renewed attention (see, e.g. 11; 30; 31; 33; 37] and references therein) Intuitively, those properties say that the behavior of logical formulae depends on the structure of small neighborhoods. They imply strong expressivity bounds for queries definable by logical ....

....called Hanf local if there exist a number d 0 such that for all finite one sorted structures A and B, A; a) d (B; b) implies A j= a) iff B b) The definition for open formulae is from [21] most previous papers [19; 14; 30; 37] considered its restriction to sentences. It is known [14] that A d B implies A r B for r d. It is also known that every (one sorted) first order sentence Phi is Hanf local and d can be taken to be 3 [14] This was generalized to various counting logics [37; 21] and the bound was improved to Gamma 1 [23; 31] Definition 4.6. cf. 30; 31] A ....

[Article contains additional citation context not shown here]

R. Fagin, L. Stockmeyer and M. Vardi, On monadic NP vs monadic co-NP, Information and Computation, 120 (1995), 78--92.

....of SQL queries What kind of general statement can one provide that would give us strong evidence that SQL cannot express recursive queries For that purpose, we shall use the locality of queries. Locality was the basis of a number of tools for proving expressivity bounds of rst order logic [15,13,11], and it was recently studied on its own and applied to more expressive logics [17,23] The general idea of this notion is that a query can only look at a small portion of its input. If the input is a graph, small means a neighborhood of a xed radius. For example, Fig. 1 shows that ....

R. Fagin, L. Stockmeyer and M. Vardi, On monadic NP vs monadic co-NP, Information and Computation, 120 (1995), 78-92.

....there exists some assignment of predicate names to heap objects such that the given local referencing constraints are satisfied. We call constraints defined in this way role constraints. The existential quantification over predicate names can be expressed in existential monadic second order logic [12]. Role constraints explicitly specify constraints on incoming and outgoing fields of objects as well as inverse reference and acyclicity constraints. Role constraints encode may reachability properties implicitly, through the reachability between summary nodes. The Entailment Problem. One of the ....

Ronald Fagin, Larry J. Stockmeyer, and Moshe Y. Vardi. On monadic NP vs monadic co-NP. Information and Computation, 120(1), 1995.

....these properties. We therefore say that a heap satis es a set of properties if there exists some choice of predicates that satisfy the mutually recursive de nitions. The existential quanti cation over predicates leads to constraints that have the form of existential monadic second order logic [7]. For a presentation of role analysis and related systems from the perspective of monadic second order logic, see [15] In this paper we present a very simple form of constraints that we call regular graph constraints. A set of regular graph constraints can be speci ed by a single graph G. A ....

Ronald Fagin, Larry J. Stockmeyer, and Moshe Y. Vardi. On monadic NP vs monadic co-NP. Information and Computation, 120(1), 1995.

....of D. For example, relational calculus and plain SQL are such languages. 4 Next we de ne the concept of local queries and local languages. Given a schema SC in and D 2 Inst(SC in ) its active domain, adom(D) is the set of all elements from D that occur in relations from D. The Gaifman graph [8, 11, 10] of D, G(D) is de ned as a graph hA; Ei, where A = adom(D) and (a; b) is in E i there is a tuple t 2 R D i for some i such that both a and b are in t. The distance d(a; b) is de ned as the length of the shortest path from a to b in G(D) we assume d(a; a) 0. If a = a 1 ; a n ) ....

R. Fagin, L. Stockmeyer, M. Vardi, On monadic NP vs monadic co-NP, Information and Computation, 120 (1994), 78-92.

....4. AGGREGATE LOGIC: EXPRESSIVE POWER In this section we deal with the expressive power of the aggregate logic. Our main goal is to show that it satisfies a very strong locality property. Locality properties were introduced in model theory by Hanf [19] and Gaifman [16] and recently, following [14], they were a subject of renewed attention (see, e.g. 11; 30; 31; 33; 37] and references therein) Intuitively, those properties say that the behavior of logical formulae depends on the structure of small neighborhoods. They imply strong expressivity bounds for queries definable by logical ....

....for all finite one sorted structures A and B, A; a) d (B; b) implies A j= a) iff B j= b) The definition for open formulae is from [21] most previous papers [19; 14; 30; 37] considered its 14 Delta L. Hella, L. Libkin, J. Nurmonen, L. Wong restriction to sentences. It is known [14] that A d B implies A r B for r d. It is also known that every (one sorted) first order sentence Phi is Hanf local and d can be taken to be 3 qr( Phi) Gamma1 [14] This was generalized to various counting logics [37; 21] and the bound was improved to 2 qr( Phi) Gamma1 Gamma 1 [23; 31] ....

[Article contains additional citation context not shown here]

R. Fagin, L. Stockmeyer and M. Vardi, On monadic NP vs monadic co-NP, Information and Computation, 120 (1995), 78--92.

....proving expressivity bounds in the presence of an order relation, one has to at least be able to lift the results from constant degrees to those that depend on the size of the input. A result in this direction was proved in [21] using a definition of moderate degree by Fagin, Stockmeyer and Vardi [9]. We say that a class C of graphs (more generally, relational structures) is of moderate degree, if degmax C (n) the maximal in or out degree of an n element graph from C, is at most log o(1) n. That is, for some function ffi(n) such that lim n 1 ffi(n) 0, we have degmax C (n) log ffi(n) ....

....graph from C, is at most log o(1) n. That is, for some function ffi(n) such that lim n 1 ffi(n) 0, we have degmax C (n) log ffi(n) n. Then [21] proved that there is a DLOGSPACEcomplete problem which is not definable in FO(C) in the presence of auxiliary relations of moderate degree. In [9], auxiliary relations of moderate degree were shown to be of no help for expressing connectivity of graphs in monadic Sigma 1 1 . Starting from their result, Schwentick extended it to degrees n o(1) 31] and to a linear order [32] So one may wonder if a similar program can be carried out for ....

[Article contains additional citation context not shown here]

R. Fagin, L. Stockmeyer and M. Vardi, On monadic NP vs monadic co-NP, Information and Computation, 120 (1995), 78--92.

....computer science, due to the fact that they are among relatively few results in rst order model theory that apply to both nite and in nite structures. Gaifman s theorem itself works for both nite and in nite structures, while for Hanf s theorem an extension to nite structures was formulated by Fagin, Stockmeyer, and Vardi [1995]. More recently, the statements underlying Hanf s and Gaifman s theorems have been abstracted from the statements of the theorems, and used in their own right. In essence, Hanf s theorem states that two structures cannot be distinguished by sentences of quanti er rank k whenever they realize the ....

....That is, A d B if there exists a bijection f : A B such that N A d (a) N B d (f(a) for every a 2 A. We also write (A; a) d (B; b) if there is a bijection f : A B such that N A d ( ac) N B d ( bf(c) for every c 2 A. De nition 2. 3 (Hanf Locality) See [Hanf 1965; Fagin et al. 1995; Hella et al. 1999a] An m ary query Q, m 0, is called Hanf local if there exist a number d 0 such that for any two structures A; B and any a 2 A m ; b 2 B m , A; a) d (B; b) implies a 2 Q(A) i b 2 Q(B) The minimum d for which this holds is called Hanf locality rank of Q, ....

[Article contains additional citation context not shown here]

Fagin, R., Stockmeyer, L., and Vardi, M.Y. 1995. On monadic NP vs monadic co-NP, Information and Computation 120, 1, 78-92.

....of SQL queries What kind of general statement can one provide that would give us strong evidence that SQL cannot express recursive queries For that purpose, we shall use the locality of queries. Locality was the basis of a number of tools for proving expressivity bounds of first order logic [15, 13, 11], and it was recently studied on its own and applied to more expressive logics [17, 22] b : oe : r r : a oe Upsilon Sigma Pi Xi Upsilon Sigma Pi Xi Fig. 1. A local formula cannot distinguish (a; b) from (b; a) ....

R. Fagin, L. Stockmeyer and M. Vardi, On monadic NP vs monadic co-NP, Information and Computation, 120 (1995), 78--92.

....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 nice behavior: its formulae satisfy analogs of Hanf s [8, 12] and Gaifman s [10] theorems, meaning that it can only express local properties. In particular, properties such as connectivity of graphs cannot be expressed. We then consider a theoretical language RL aggr , similar to those defined in [3, 24] and explain how it models all the features of ....

....10 6 ) 4 Aggregate logic: Expressive power In this section we deal with expressiveness of the aggregate logic. Our main goal is to show that it satisfies a very strong locality property. Locality properties were introduced in model theory by Hanf [12] and Gaifman [10] and recently, following [8], they were a subject of renewed attention (see, e.g. 5, 21, 22, 24, 28] and references therein) Intuitively, those properties say that the behavior of logical formulae depends on the structure of small neighborhoods. They imply strong expressivity bounds for queries definable by logical ....

[Article contains additional citation context not shown here]

R. Fagin, L. Stockmeyer and M. Vardi, On monadic NP vs monadic co-NP, Information and Computation, 120 (1994), 78--92.

....order to move closer to proving expressivity bounds in the presence of an order relation, one has to at least be able to lift the results from constant degrees to those that depend on the size of the input. A result in this direction was proved in [16] using a definition on moderate degree from [11]. A class C of graphs (more generally, relational structures) is of moderate degree, if degmax C (n) the maximal in or out degree of an n element graph from C, is at most log o(1) n. That is, for some function ffi (n) such that lim n 1 ffi (n) 0, we have degmax C (n) log ffi(n) n. Fact 3 ....

....of an n element graph from C, is at most log o(1) n. That is, for some function ffi (n) such that lim n 1 ffi (n) 0, we have degmax C (n) log ffi(n) n. Fact 3 ( 16] Deterministic transitive closure cannot be defined by FO(C) in the presence of auxiliary relations of moderate degree. 2 In [11], auxiliary relations of moderate degree were shown to be of no help for expressing connectivity of graphs in monadic Sigma 1 1 . This was extended to degrees n o(1) 22] and to a linear order [21] So one may wonder if a similar program can be carried out for FO(C) There is a significant ....

[Article contains additional citation context not shown here]

R. Fagin, L. Stockmeyer, M. Vardi, On monadic NP vs monadic co-NP, Information and Computation, 120 (1994), 78--92.

.... games for different logics, cf. 14] For example, playing the Ehrenfeucht Fraisse game is one of the steps in the Ajtai Fagin game for monadic Sigma 1 1 [3] Since playing the game often involves an intricate combinatorial argument, it was suggested by Fagin, Stockmeyer and Vardi in [15] to build a library of winning strategies for those games. Or, more generally, one would like to have a collection of versatile and easily applicable tools for proving expressivity bounds of first order logic. A number of results proving expressivity bounds explain the nature of the limitations ....

....at small portions of the input. Several proposals have been made to formalize the notion of locality. Gaifman [16] proved that every firstorder formula is equivalent to a local one, in the sense that only a small part of the input is relevant for evaluating a query. Fagin, Stockmeyer and Vardi [15], modifying a result by Hanf [19] for the finite case, proved that if a certain criterion relating the numbers of small neighborhoods in two structures holds, then these structures agree on sentences whose quantifier rank is determined by the size of those neighborhoods. The author and Wong [25] ....

[Article contains additional citation context not shown here]

R. Fagin, L. Stockmeyer, M. Vardi, On monadic NP vs monadic co-NP, Information and Computation, 120 (1994), 78--92.

....they are among relatively few results in first order model theory that apply to both finite and infinite structures. Gaifman s theorem itself works for both finite and infinite structures, while for Hanf s theorem an extension to finite structures was formulated by Fagin, Stockmeyer, and Vardi [FSV95]. More recently, the statements underlying Hanf s and Gaifman s theorems have been abstracted from the statements of the theorems, and used in their own right. In essence, Hanf s theorem states that two structures cannot be distinguished by sentences of quantifier rank k whenever they realize the ....

....this means that Q cannot distinguish two structures A and B whenever A d B. Theorem 2. 4 (Hanf, Fagin Stockmeyer Vardi [FSV95; Ha65] Every FO sentence defines a Hanf local Boolean query Q with hlr(Q) 3 qr( Phi) 2 An extension to open formulae, although easily derivable from the proof of [FSV95], was probably first explicitly stated in [HLN99] every FO formula ( x) defines a Hanf local query. Better bounds on hlr(Q) of the order O(2 qr( are also known for Hanf locality [Im99; Li00] We shall use the following result the connects the binary relations and . Lemma 2.5 (see ....

[Article contains additional citation context not shown here]

R. Fagin, L. Stockmeyer and M. Vardi, On monadic NP vs monadic co-NP, Information and Computation, 120 (1995), 78--92.

No context found.

R. Fagin, L. Stockmeyer, M. Vardi, On monadic NP vs monadic co-NP, Inf.& Comput., 120 (1994), 78--92.

No context found.

R. Fagin, L. Stockmeyer and M. Vardi. On monadic NP vs monadic co-NP. Information and Computation 120(1), pages 78--92, 1995.

No context found.

R. Fagin, L. Stockmeyer and M. Vardi. On monadic NP vs monadic co-NP. Information and Computation 120(1), pages 78--92, 1995.

No context found.

R. Fagin, L. Stockmeyer, M. Vardi, On monadic NP vs monadic co-NP, Information and Computation, 120 (1994), 78--92.

No context found.

R. Fagin, L. Stockmeyer, M. Vardi, On monadic NP vs monadic co-NP, Information and Computation, 120 (1994), 78--92.

No context found.

R. Fagin, L. Stockmeyer, M. Vardi, On monadic NP vs monadic co-NP, in "Proceedings of 8th IEEE Conference on Structure in Complexity Theory," May 1993.

No context found.

R. Fagin, L. Stockmeyer, M. Vardi, On monadic NP vs monadic co-NP, Information and Computation, 120 (1994), 78--92.

No context found.

R. Fagin, L. Stockmeyer, M. Vardi, On monadic NP vs monadic co-NP, Information and Computation, 120 (1994), 78--92.

No context found.

Ronald Fagin, Larry J. Stockmeyer, and Moshe Y. Vardi. On monadic NP vs monadic co-NP. Information and Computation, 120(1), 1995.

No context found.

R. Fagin, L. Stockmeyer and M. Vardi. On monadic NP vs monadic co-NP. Information and Computation 120(1): 78-92 (1994).

No context found.

R. Fagin, L. Stockmeyer and M. Vardi. On monadic NP vs monadic co-NP. Information and Computation 120(1): 78--92 (1994).

No context found.

R. Fagin, L. Stockmeyer, M. Vardi, On monadic NP vs monadic co-NP, Information and Computation, 120 (1994), 78--92.

First 50 documents

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC