A. Aho, Y. Sagiv, and J. D. Ullman. Equivalence of relational expressions. SIAM Journal of computing, (8)2:218--246, 1979.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....distinct rows does not mean that these actually occur in the evaluation of a query. 8 Related Work The problem of query containment on which our system is based dates to the work of Chandra and Merlin [8] Subsequent work has considered containment for various extensions of conjunctive queries [21, 3], datalog [24, 22, 9] queries with order predicates [17, 28, 19, 30, 16] with complex objects [20] with regular expressions [13] and under bag semantics [10] In [7] Bodlaender shows that given a constant bound on the treewidth, there is a linear time algorithm to recognize whether a graph ....

Alfred Aho, Yehoshua Sagiv, and Jerey D. Ullman. Equivalence of relational expressions. SIAM Journal of Computing, (8)2:218-246, 1979.

....obtain the finite #nplication problem. Solving the implication problem is the main computational task associated with a class of dependencies. As a rule, algorithmic approaches to database schema design and query optimization are based on efficient solutions of the implication problem (see, e.g. [12, 6, 3, 18, 62, 51]) Evidenfiy, if we are concerned with applications then the finlie implication problem is the one which is most relevant. However, it tends to be much more difficult to deal with. Moreover, for the classes of dependencies for which implication is decidable, it generally happens that finite ....

.... as possible (i.e. contains a minimum number of instances of expensive operators, such as join) Since equivalence of two queries is a data dependency, the problem of testing equivalence of queries in the presence of dependencies can be approached with fle sumdard tools for implication problems [3, 18, 62]. The equivalence of relational database queries in the presence of FD s and IND s has been examined in [43, 48] essentially by extending classical techniques (namely the chase) The authors of [43] show that under reasonable restrictions on the IND s, query equivalence can be reduced to ....

Aho, A.V., Sagiv, Y. and Ullman, J.D. Equivalences of Relational Expressions. SlAM Journal on Computing 8, 2 (1979), 218-246.

....it. The de nition is straightforward. 98 De nition 7.1 (Schema containment) A schema s 1 contains a schema s 2 (denoted by s 1 s 2 ) if for all databases o all matches of s 2 are also matches of s 1 . This de nition is related to the more traditional notion of query containment as de ned in [ASU79] Intuitively, the potential for optimization using schema containment is the following. From the de nition we immediately observe that if a schema s 1 contains another schema s 2 then 1. matches of s 2 can only be found among the matches of s 1 . If we want to nd the matches of s 2 and already ....

....q 2 , written q 1 q 2 , if for all databases the answers of q 2 are a subset of the answers of q 1 . The queries q 1 and q 2 are equivalent, written q 1 q 2 , if q 1 q 2 and q 2 q 1 . Query containment has a variety of applications. Originally it was used for query optimization [CM77, ASU79, SY81] More recently, it has become an important notion in the context of rewriting of queries using views [LMSS95, CKPS95] In particular, this notion plays a signi cant role for materializing views in data warehouses. Levy and Sagiv use the notion of query containment to investigate kinds of ....

A. Aho, Y. Sagiv, and J. D. Ullman. Equivalence of relational expressions. SIAM Journal on Computing, 8(2):218-246, 1979.

....of one query is a subset of the answer set of another. Containment between extensional, conjunctive queries was first studied in [5] and the problem was shown to be NP complete. 11 Several sub classes of extensional, conjunctive queries have been identified to have polynomial time algorithms [2, 3, 14]. Containment tests for extensional, conjunctive queries that permit negation have been presented in [18] and for those that involve arithmetic comparisons in [16] Containment between intensional queries with respect to a Datalog program is computationally harder: the containment question is ....

A. Aho, Y. Sagiv, and J. Ullman. Equivalence of relational expressions. SIAM Journal of COmputing, 8(2):218--246, 1979.

....our results. 2 Definitions and Notation In this section, we give most of the notation required for the rest of the paper. 2.1 Basic Definitions We shall follow standard notation [Ma, U] and only give some definitions here. A tableau is a set of tuples or rows defined on a set of attributes U [ASU]. For each A j 2 U , the domain of a tableau on A j consists of countable many nondistinguished variables (ndv s) a distinguished variable (dv) a j , and constants taken from dom(A j ) the domain of A j . We say that a tuple [X ] is total if [A i ] is a constant, for all A i 2 X , X U . A ....

....A i 2 X , X U . A symbol is either a constant or a variable, and we say that a symbol is unique if it is distinct from any other symbol appearing anywhere else. A tableau T is minimal if none of its proper subsets is equivalent to T ; for details on equivalence and minimization of tableaux see [ASU]. Let R=fR 1 ; R k g be a database scheme. Then a (database) state for R is a function r that 5 maps each relation scheme R i 2 R to a relation r i defined on R i . We write r = r 1 ; r k . Let r = r 1 ; r k be a state for R and let U = R. We define a tableau T r ....

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Aho, A.V., Sagiv, Y., Ullman, J.D., "Equivalence of Relational Expressions," SIAM J. on Computing 8:2 (1979), pp. 218-246.

....second query can be answered using a view defined by the first query. It is shown to be undecidable for relational queries [1] and Datalog queries [18] The problem is NP complete for conjunctive queries [4] unions of conjunctive queries [17] and conjunctive queries under functional dependencies [2, 3]. Query containment is Pi p 2 complete for conjunctive queries with built in predicates [21] and NP complete for the subclasses of such queries called the left and right semiinterval queries [11] The complexity of query containment is also studied for conjunctive queries under functional and ....

A. V. Aho, Y. Sagiv, and J. D. Ullman. Equivalence of relational expressions. SIAM Journal of Computing, 8(2):218--246, 1979.

....mapping from T i Gamma1 to T i Gamma1 1 . Then, consider the i th step in the chase process of T : it equates two symbols, say s 1 and s 2 , in T i Gamma1 , because 2 Different, but equivalent definitions of containment and equivalence of tableaux can be found in the literature [2, 3]. 12 of an FD Y A 2 F and of two tuples t 1 ; t 2 2 T i Gamma1 , with t 1 [Y ] t 2 [Y ] and t 1 [A] s 1 6= s 2 = t 2 [A] since is a containment mapping, we have that t 0 1 = t 1 ) and t 0 2 = t 2 ) are tuples in T i Gamma1 1 , with t 0 1 [X] t 0 2 [X] Now, we have two ....

A.V. Aho, Y. Sagiv, and J.D. Ullman. Equivalence of relational expressions. SIAM Journal on Computing, 8(2):218--246, 1979.

....F , with variables z = x1 , xn # y1 , ym and clauses c = c1 , cp . Clause c i contains the three variables (either positive or negated) z i,1 , z i,2 , and z i,3 . We begin by building Q # 1 and Q # 2 , in a similar fashion to the reduction proof used in [3] to show NP hardness of conjunctive query containment. We use the EDB predicates r1 , rp of arity 3, each predicate representing a di#erent clause in F . Query Q # 1 simply records which variables are in each clause. It is defined as follows: q # 1 ( r1(z1,1 , z1,2 , z1,3 ) ....

.... following two queries: q # 1 ( r1(x1 , x2 , y1) r2(x1 , x2 , y2) q # 2 ( r1(1, 1, 1) r1(1, 1, 0) r1(1, 0, 1) r1(1, 0, 0) r1(0, 1, 1) r1(0, 1, 0) r1(0, 0, 1) r2(0, 0, 0) r2(0, 0, 1) r2(0, 1, 0) r2(0, 1, 1) r2(1, 0, 0) r2(1, 0, 1) r2(1, 1, 0) Similar to the argument in [3], it can be shown that this is a valid reduction from the CNF satisfiability problem to the conjunctive query containment problem: F is satisfiable if and only if Q # 2 # Q # 1 . In particular, any satisfying truth assignment for F is also a valid containment mapping from Q # 1 to Q # 2 , and ....

A. Aho, Y. Sagiv, and J. D. Ullman. Equivalence of relational expressions. SIAM Journal of Computing, (8)2:218-- 246, 1979.

.... recently, query containment has been used to determine when queries are independent of updates to the database [LS93] rewriting queries using views [CKPS95, LMSS95] and maintenance of integrity constraints [GSUW94] Previous work on containment has considered queries in the relational algebra [CM77, SY81, ASU79] and datalog [Shm93, Sag88, CV92] Several works have considered the extension of containment algorithms for queries involving order [Klu88, vdM92, LS93, ZO93, GSUW94] Queries over bags were considered in [CV93] and modifications of containment algorithms to consider semantics of class ....

Alfred Aho, Yehoshua Sagiv, and Jeffrey D. Ullman. Equivalence of relational expressions. SIAM Journal of Computing, (8)2:218--246, 1979.

....state on this database schema gives rise to a non empty answer. 2 The query in Example 1.1 is an instance of conjunctive query proposed in this paper. This class of conjunctive queries captures the essence of conjunctive queries in the relational model. The class of relational conjunctive queries [11, 3] represents a natural and important 3 subclass of queries that most often asked by a user. As we will see, determining satisfiability of the proposed conjunctive queries is difficult while the same problem is trivial for relational conjunctive queries. The cause of complication in this setting is ....

Aho, A.V., Sagiv, Y. and Ullman, J.D., "Equivalence of Relational Expressions," SIAM J. of Computing 8(2), 1979, pp. 218-246.

....in the local as global approach, the problem of answering queries using local relations becomes a problem of reformulation of queries in terms of views. This problem is closely related to the problem of query containment that has been extensively studied in the database literature (e.g. [4, 1, 13, 8, 14, 17, 5, 15, 12, 3]) This problem has been shown as being generally harder than the problem of answering queries over a database. In particular, it is undecidable for Datalog. In this paper, we describe the framework of the PICSEL information integration system, which makes use of the expressive power of the carin ....

Alfred Aho, Yehoshua Sagiv, and Jeffrey D. Ullman. Equivalence of relational expressions. SIAM Journal of Computing, (8)2:218--246, 1979.

....For such queries, we have obtained an algorithm which avoids the complications of inferring arithmetical constraints [SRSS94, NSS98] thus, it becomes possible to use algorithms for optimizing queries 2 Consider the semantics of the last conjunct of the where clause of Q. 9 without constraints [DBS90, CR97, ASU79a, ASU79b, JKlug84, CM77] to optimize nested SQL query blocks with max, min. We believe our approach will be fruitfully applicable in other cases. A natural proposal is to apply it to aggregation operators which are known to be delicate to analyze, such as count [Kim82, GW87, Mur92] Finally, it should be possible to ....

A. Aho, Y. Sagiv, J. Ullman. Equivalence of relational expressions. SIAM J. on Computing 8(2), 1979.

....ARO grant DAAH04 95 1 0192, and USAF contract F33615 93 1 1339. well known to be NP complete [7] In view of its practical significance, considerable attention has been devoted to finding classes of conjunctive queries that admit polynomial time algorithms for equivalence and minimization [1, 2, 12, 4]. Acyclic queries in particular have been extensively studied in the context of query optimization in distributed database systems, and are well known to have desirable algorithmic properties [23] In this paper, we first present polynomial time algorithms to test containment of an arbitrary ....

....there is an equivalent rewriting of Q using V, and if so, produces a rewriting of Q using V that has no more than n subgoals. The algorithm runs in time O(k 2 NQ jVj k log jVj) and uses space O(NQ jVj k ) where NQ is the size of Q and jVj is the size of V. 7 Related Work Aho et al. [1, 2] gave polynomial time minimization and equivalence algorithms for conjunctive queries corresponding to simple tableaux. Their results were extended by Johnson and Klug [12] to the class of fanout free queries. Biskup et al. 4] extending the ideas in [1, 2] and [12] considered a class of typed ....

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A.V. Aho, Y. Sagiv, and J.D. Ullman. Equivalence of relational expressions. SIAM Journal on Computing, 8(2):218--246, May 1979.

....and complexity results for the three instances of the VVT problem mentioned above. The main tool we use to obtain our results is the connection that we establish between the VVT problem and the problem of query containment, that has been extensively studied in the database literature (e.g. [7, 1, 28, 16, 29, 36, 8, 32, 21, 22]) We show that viewing the VVT problem from the perspective of query containment provides a uniform view of the VVT problem which covers the different cases mentioned above. Specifically, our contributions are the following: 1. We show that for function free Horn rule KBs, the I O consistency and ....

....corresponds to the case of Definition 2 where the sentence C is Phi ) P out . 4 Verification and Query Containment Our approach to solving the verification problem is based on showing a close connection to the problem of query containment, that has been considered in the database literature [7, 1, 28, 16, 29, 36, 8, 32, 21, 22]. In this section we formalize the connection between the VVT problem and the query containment problem in the presence of I O consistency and I O dependency constraints. As a result, in Section 5 we obtain novel algorithms for solving these problems as well as the fundamental complexity results ....

Alfred Aho, Yehoshua Sagiv, and Jeffrey D. Ullman. Equivalence of relational expressions. SIAM Journal of Computing, (8)2:218--246, 1979.

.... problem is known to be undecidable [Shm93] However, algorithms for restricted cases are given in [CV92, CV94, Sag88, LS93] Finally, if the database relations are known to satisfy integrity constraints (e.g. functional dependencies, tuple generating dependencies) the algorithms in [CM77, ASU79b, ASU79a, JK83] can be used for deciding equivalence. We obtain the following decidability results for the answer completeness problem. Theorem 3.2 : Let Q be a union of conjunctive queries over the relations R 1 ; Rn and comparison predicates, and let Gamma be a set of local ....

Alfred Aho, Yehoshua Sagiv, and Jeffrey D. Ullman. Equivalence of relational expressions. SIAM Journal of Computing, (8)2:218--246, 1979.

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A. Aho, Y. Sagiv, and J. D. Ullman. Equivalence of relational expressions. SIAM Journal of computing, (8)2:218--246, 1979.

No context found.

A. Aho, Y. Sagiv., and J. Ullman. Equivalences of relational expressions. SlAM Journal on Computing, 8(2):218--246, 1979.

No context found.

A. Aho, Y. Sagiv., and J. Ullman. Equivalences of relational expressions. SlAM Journal on Computing, 8(2):218--246, 1979.

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A. Aho, Y. Sagiv, and J.D. Ullman. Equivalence of relational expressions. SIAM Journal on Computing, 8:218--246, 1979.

No context found.

A. Aho, Y. Sagiv, and J. D. Ullman. Equivalence of relational expressions. SIAM Journal of computing, (8)2:218--246, 1979.

No context found.

A. Aho, Y. Sagiv., and J. Ullman. Equivalences of relational expressions. SlAM Journal on Computing, 8(2):218--246, 1979.

No context found.

A. Aho, Y. Sagiv, and J. D. Ullman. Equivalence of relational expressions. SIAM Journal of computing, (8)2:218--246, 1979.

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A. Aho, Y. Sagiv, and J. D. Ullman. Equivalence of relational expressions. SIAM Journal of Computing, (8)2:218-246, 1979.

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AS79 Aho, A.V., Sagiv, Y. and Ullman, J.D., Equivalence of Relational Expressions. SIAM J. of Computing, 8, 2 (1979), 218-246.

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Alfred Aho, Yehoshua Sagiv, and Jeffrey D. Ullman. Equivalence of relational expressions. SIAM Journal of Computing, (8)2:218--246, 1979.

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