M.Y. Vardi. Automata theory for database theoreticians. In Proc. 8th ACM Symp. on Principles of Database Systems, pages 83--92, 1989.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....22, 27] The present paper is an attempt to provide a gentle introduction to unranked tree automata and to give references to some applications. We mention that Vardi, already in 1989, wrote a paper demonstrating the usefulness of ranked tree automata for the static analysis of datalog programs [37]. # Database Principles Column. Column editor: Leonid Libkin, Department of Computer Science, University of Toronto, Toronto, Ontario M5S 3H5, Canada. E mail: libkin cs.toronto.edu. 2 Trees Every XML document can be represented by a tree. See, for instance, Figure 1. In this view, inner nodes ....

M. Y. Vardi. Automata theory for database theoreticians. In Proceedings of the Eighth ACM Symposium on Principles of Database Systems, pages 83--92. ACM Press, 1989.

....Suciu [2] We do not give many proofs and the purpose of the few ones we discuss is merely to arouse interest and demonstrate underlying ideas. Finally, we mention that automata have been used in database research before: Vardi, for instance, used automata to statically analyze datalog programs [59]. The paper is further organized as follows. In Section 2, we discuss XML. In Section 3, we provide the necessary background definitions concerning trees and logic. In Section 4, we consider unranked tree automata. In brief, unranked trees are trees where every node has a finite but arbitrary ....

M. Y. Vardi. Automata theory for database theoreticians. In Proceedings of the Eighth ACM Symposium on Principles of Database Systems, pages 83--92. ACM Press, 1989.

....1 2 T and 2 2 T [ N , 1 ; 2 ) f g if 1 = 2 ; otherwise, and for every X r 2 P and Y 2 N [ T (X r; Y ) L(r) if X = Y ; otherwise. 11 The proof of the following lemma is a straightforward generalization of the ranked case (see, e.g. the survey paper by Vardi [46]) Lemma 2.8 Deciding whether the tree language accepted by an NBTA is non empty is in PTIME. Proof. Let B = Q; F; be an NBTA. We inductively compute the set of reachable states R de ned as follows: q 2 R i there exists a tree t with q 2 (t) Obviously, L(B) 6= if and only if R F ....

M. Y. Vardi. Automata theory for database theoreticians. In Proceedings of the Eighth ACM Symposium on Principles of Database Systems, pages 83-92. ACM Press, 1989.

....exploited this idea further in our previous paper [NVdB97] where we investigated BAG evaluation mechanisms using deductive rules. the least fixpoint of a system of first order formulas with monadic induction variables is definable in MSO, as had to be proven. If. The proof uses tree automata [GS84, Var89]. A (bottom up deterministic) tree automaton M = Q; ffi; F ) working on derivation trees of context free grammar G, consists of a finite set Q of states, a set F Q of final states, and a transition function ffi mapping the terminal symbols of G, as well as all tuples of the form (p; q1 ; ....

M. Vardi. Automata theory for database theoreticians. Proceedings of the 8th ACM Symposium on Principles of Database Systems, pages 83-92. ACM Press, 1989.

....characterize the expressive power of BAGs in terms of monadic second order logic (MSO) As a corollary we obtain a bottom up property for Boolean BAG queries. First we recall the de nitions of tree automata and MSO. 3.1 Tree automata and MSO 3.1. 1 Tree automata In the theory of tree languages [11, 12, 30], trees are usually viewed as terms over a ranked alphabet. A ranked alphabet is a vocabulary of function symbols with associated arities. The set T of trees is inductively de ned in the following manner: if f is a function symbol in of arity n, and t 1 , t n are trees, then also ....

M. Y. Vardi. Automata theory for database theoreticians. In Proceedings of the Eighth ACM Symposium on Principles of Database Systems, pages 83-92. ACM Press, 1989.

.... is a node that is selected (non emptiness) ii) Given two QAs, is the query computed by one contained in the query computed by the other (containment) and (iii) Given two QAs, do they compute the 2 These automata are very di erent from the (alternating) tree walking automata used in, e.g. [46]. 6 same query (equivalence) One cannot hope to do better than EXPTIME for these decision problems, as non emptiness of two way deterministic tree automata over ranked trees, i.e. even without selecting nodes, is already complete for EXPTIME. We show that the non emptiness, the containment, ....

....We represent the string languages (q; a) by NFAs. The size of B then is the sum of the sizes of Q, and the NFAs de ning the transition function. We need the following lemma in Section 6. Its proof is a straightforward generalization of the ranked case (see, e.g. the survey paper by Vardi [46]) Lemma 5.2 Deciding whether the tree language accepted by an NBTA is non empty is in PTIME. Proof. Let B = Q; F; be an NBTA u . We inductively compute the set of reachable states R de ned as follows: q 2 R i there exists a tree t with q 2 (t) Obviously, L(B) 6= if and only if ....

M. Y. Vardi. Automata theory for database theoreticians. In Proceedings of the Eighth ACM Symposium on Principles of Database Systems, pages 83-92. ACM Press, 1989. 62

....tallness. An inspection of typical results for standard (fixed branching) tree automata shows that these carry over to our slightly more general notion of tree automata, with the above notion of size. We refer in particular to M. Vardi s discussion of tree automata and their applications in [16], and to the handbook article [15] by W. Thomas for background. Theorem 13 Bounded tallness of L(A; F ) is decidable in time polynomial in the size of A. For a fixed finite set Gamma L [ let tp Gamma (A) Phi 2 Gamma fi fi A j= Psi , and Tp Gamma (A; a) S a 0 2E A ....

M.Y. Vardi, Automata theory for database theoreticians, in Theoretical Studies in Computer Science, J.D. Ullman, ed., Academic Press, 1992, pp. 153--180.

....Murata and Wood [9, 30] The size of MF is exponential in the size of F and the non emptiness test of NBTAs can be done in PTIME. The latter is just a modification of the algorithm for testing non emptiness for non deterministic bottom up tree automata over ranked trees (see, e.g. Vardi [45]) Hence, testing non emptiness of extended AGs can be done in EXPTIME. The automaton MF essentially guesses the values of the attributes (there are exponentially many since A and D are not fixed, hence MF has exponentially many states) and verifies whether they satisfy all semantic rules. The ....

M. Y. Vardi. Automata theory for database theoreticians. In Proceedings of the Eighth ACM Symposium on Principles of Database Systems, pages 83--92. ACM Press, 1989.

....We denote these automata with QA u (u stands for unranked) Although these automata 2 A tree language is a set of trees. We say, a QA accepts a tree if the underlying tree automaton accepts it. 3 These automata are very different from the (alternating) tree walking automata used in, e.g. [29]. can accept all recognizable tree languages, they cannot even express all unary queries definable in firstorder logic. The reason for this weakness is that information cannot be passed from one sibling to another. To resolve this, we introduce stay transitions where a two way string automaton ....

M. Y. Vardi. Automata theory for database theoreticians. In Proceedings of the Eighth ACM Symposium on Principles of Database Systems, pages 83--92. ACM Press, 1989.

....a powerful conceptual framework in which devising new labeling schemes becomes much easier. 3.6.1 Related Work The intuition behind the query tree algorithm comes from translating the problem into a decision problem for tree automata. 10 In fact, we have argued that a finite 10 See [ Vardi, 1989 ] for a discussion of the importance of tree automata in database theory. 3.6. SUMMARY 75 labeling scheme essentially guarantees that the set of derivations can be recognized by a reachability test on a finite tree automaton. With respect to tree automata, the contribution of our work is ....

.... Static analysis in Explanation Based Learning [ Etzioni, 1993; Etzioni, 1990 ] 75, 97 Pushing constraint selections [ Srivastava and Ramakrishnan, 1992 ] 75 Partial evaluation of logic programs [ Smith and Hickey, 1990; Lloyd and Shepherdson, 1991; Bruynooghe et al. 1991 ] 75, 100 Tree automata [ Vardi, 1989 ] 74 Automated reasoning and query evaluation: Knowledge compilation [ Selman and Kautz, 1991 ] 100 Deriving optimal search strategies [ Smith, 1986; Greiner, 1991 ] 97 Message passing based query evaluation [ Van Gelder, 1986 ] 96 Magic set transformation [ Ullman, 1989; Mumick et al. 1990 ] ....

Vardi, Moshe Y. 1989. Automata theory for database theoreticians. In Proceedings of the Eighth Symposium on Principles of Database Systems (PODS). 83--92.

....Observation 8) 2 An inspection of corresponding results for standard (fixed branching) tree automata shows that these carry over to our slightly more general notion of tree automata in the following form. We refer in particular to M. Vardi s discussion of tree automata and their applications in [V], and to the handbook article [T97] by W. Thomas for background. Theorem 16 The following are decidable in time polynomial in the size of A: emptiness, bounded height, and bounded tallness of L(A; F ) For a fixed finite set Gamma L [ A 2 T [ and a 2 A let tp Gamma (A; a) Phi ....

M. Vardi, Automata theory for database theoreticians, extended version of a contribution to Proc. 8th ACM Symp. Principles of database Systems, 1989, pp. 83--92.

....of a goal node are rule nodes for all the rules that can be unified with that goal node. The children of a rule node are the goal nodes for the subgoals in the body of that rule. Intuitively, the query tree can be viewed as a tree automaton that accepts only the possible derivations of the query [Var89] Once a query tree is constructed for a given program, a rewritten program can be obtained by forming a rule for every rule node in the tree. The key challenge in building a query tree is to encode precisely a possibly infinite set of symbolic derivations using a finite structure. The key idea ....

Moshe Y. Vardi. Automata theory for database theoreticians. In Proceedings of the Eighth Symposium on Principles of Database Systems (PODS), pages 83--92, March 1989.

....deterministic pushdown automata (2dpda s) 9] We present this proof in an appendix. In the case of recursion free schemas, the consistency problem is coNP complete for a single coded method with two arguments. Some special cases can be shown to be in PTIME, using tree automata techniques [11, 31]. Covariance does not help in the recursive case. However, in the recursion free covariant case we show that there is a PTIME test for a fixed arity coded method. This is interesting in practice, because it motivates our heuristic for the general case. In an object oriented context, it is not ....

....but finding a tight lower bound is an open problem. Remark: If the k terms of the methods to be checked are trees (not dag s) where each input i, for 1 i k appears at most once, consistency checking is in PTIME. This bound is obtained by generalizing the automaton technique to tree automata [11, 31]. 5 Covariance In this section, we consider covariance and show that it simplifies the consistency problem. We first consider simple schemas. For simple schemas, covariance is the following constraint on the signatures of base methods. A simple schema is covariant if for each m and for each ....

M. Y. Vardi. Automata Theory for Database Theoreticians. Proc. ACM PODS, 83--91, 1989.

....how to push constraints in those cases. Query trees can be viewed as a refinement of tree automata techniques for the special purpose of representing symbolic derivation trees of datalog programs. The importance of tree automata techniques for decision problems of datalog programs was shown in [Var89]. Conceivably the results we present could be obtained by applying tree automata techniques directly, but doing so would be considerably more intricate. 2 Satisfiability, Query Reachability and Equivalence In this paper we consider datalog programs that may have safe stratified negation and ....

Moshe Y. Vardi. Automata theory for database theoreticians. In Proceedings of the Eighth Symposium on Principles of Database Systems (PODS), pages 83--92, Philadelphia, PA, March 1989. ACM SIGACTSIGMOD -SIGART.

....respect to the standard semantics just in case some defined predicate occurring in the set D is empty in 6. This means that inconsistency of recursively indefinite databases with respect to the standard semantics can be tested in polynomial time, since this is the complexity of predicate emptiness [125]. For the unique names semantics, things are a little more subtle. Example 3.2.2: Suppose that A; B are basic predicates and that a; b are constant symbols. Let 6 be the program with rules R(x) 0S(x; b) S(x; x) 0B(x) Suppose that D is the database 6; B; D] with basic facts B = fA(b)g and ....

M. Vardi. Automata theory for database theoreticians. In Proceedings of the Eighth ACM SIGACT-SIGMOD-SIGART Symposium on Principles of Database Systems, pages 83--92, 1989.

....to the standard semantics just in case some defined predicate occurring in the set D is empty in Sigma. This means that inconsistency of recursively indefinite databases with respect to the standard semantics can be tested in polynomial time, since this is the complexity of predicate emptiness [30]. For the unique names semantics, things are a little more subtle. Example 2.2: Suppose that A; B are basic predicates and that a; b are constant symbols. Let Sigma be the program with rules R(x) GammaS(x; b) S(x; x) GammaB(x) Suppose that D is the database Sigma; B; D] with basic ....

M. Vardi. Automata theory for database theoreticians. In Proceedings of the Eighth annual SIGACT-SIGMOD-SIGART Symposium on the Principles of Database Systems, pages 83--92, 1989.

....two way deterministic pushdown automata (2dpda s) 9] We present this proof in an appendix. In the case of recursion free schemas, the consistency problem is coNP complete for a single coded method with two arguments. Some special cases can be shown to be in PTIME, using tree automata techniques [11, 32]. Covariance does not help in the recursive case. However, in the recursion free covariant case we show that there is a PTIME test for a fixed arity coded method. This is interesting in practice, because it motivates our heuristic for the general case. In an object oriented context, it is not ....

....but finding a tight lower bound is an open problem. Remark: If the k terms of the methods to be checked are trees (not dag s) where each input i, for 1 i k appears at most once, consistency checking is in PTIME. This bound is obtained by generalizing the automaton technique to tree automata [11, 32]. 5 Covariance In this section, we consider covariance and show that it simplifies the consistency problem. We first consider simple schemas. For simple schemas, covariance is the following constraint on the signatures of base methods. A simple schema is covariant if for each m and for each pair ....

M. Y. Vardi. Automata Theory for Database Theoreticians. Proc. ACM PODS, 83--91, 1989.

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M.Y. Vardi. Automata theory for database theoreticians. In Proc. 8th ACM Symp. on Principles of Database Systems, pages 83--92, 1989.

....yields the decidability of the containment problem, it provides only nonelementary time bounds [Cou90] The main body of the paper is dedicated to a detailed study of the computational complexity of containment and equivalence. For upper bounds, we use the automaton theoretic approach advocated in [Va92]. The key idea is that a recursive program can be viewed as an infinite union of conjunctive queries. These conjunctive queries can be represented by proof trees, and the set of proof trees corresponding to a given recursive program can be represented by a tree automaton. This representation ....

....review some of the relevant results from automata theory on emptiness and containment of automata. We will use these results for proving the upper bound on the complexity of deciding containment of a Datalog predicate in a union of conjunctive queries. The material in this section is quoted from [Va92]. 4.1 Automata on Words An automaton A is a tuple ( Sigma; S; S 0 ; ffi; F ) where Sigma is a finite alphabet, S is a finite set of states, S 0 S is the set of initial states, F S is the set of accepting states, and ffi : S Theta Sigma 2 S is a transition function. Note that the ....

Vardi, M.Y.: Automata theory for database theoreticians. In Theoretical Studies in Computer Science (J.D. Ullman, ed.), Academic Press, 1992, pp. 153-180.

....words or trees is not widely known. Furthermore, my experience has been that many researchers are not very comfortable working with this theory, In this paper I will describe two applications of the theory of automata on infinite words and trees: the first application, taken from [CV92] see also [Va92]) is to optimization of logic database programs and the second application, taken from [VW86b] is to verification of finite state programs. The description here is sketchy on some of the details; for more details the reader is referred to the original papers. My goal in this paper is twofold. ....

Vardi, M.Y.: Automata theory for database theoreticians, in Theoretical Studies in Computer Science (J.D. Ullman, ed.), Academic Press, 1992, pp. 153--180.

No context found.

M. Y. Vardi. Automata theory for database theoreticians. In Proceedings of the Eighth ACM Symposium on Principles of Database Systems, pages 83--92. ACM Press, 1989.

No context found.

M. Y. Vardi. Automata theory for database theoreticians. In Proceedings of the Eighth ACM Symposium on Principles of Database Systems, pages 83-92. ACM Press, 1989.

No context found.

Moshe Y. Vardi. Automata theory for database theoreticians. In Proceedings of the Eighth Symposium on Principles of Database Systems (PODS), pages 83--92, March 1989.

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