P. Revesz. A closed form evaluation for datalog queries with integer (gap)-order constraints. Theoretical Computer Science, 116:117--149, 1993.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....constraints: the constraints over = fcg c2C ) We will interpret the polynomial and linear constraints over the reals, the rationals or the integers, the order constraints over the rationals or the integers, and the inequality constraints over arbitrary in nite sets. It is well known [KKR95, Rev93] that all those structures admit e ective quanti er elimination. 1. A constraint k tuple, in variables x 1 ; x k , over is a nite conjunction 1 N , where each i , 1 i N , is an constraint with variables that are among x 1 ; x k . 2. A constraint relation ....

....r2.year cannot be allowed. 27 7.3 Gap Order Datalog Gap order constraints in Z are atomic formulas of the form x c y, x d, and d x for c; d 2 Z; c 0. Constraint tuples formed from gap order constraints over variables x 1 ; x k can be represented by a directed graph on k 2 [Rev93] or equivalently by a matrix representation of such a graph [GK96] In both cases, the representation is quadratic in k (worst case) A Gap Order Datalog query (program) is a nite set of Horn clauses with variables constrained by gap order constraints. Thus both the bottom up evaluation [Rev93] ....

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P. Z. Revesz. A Closed-Form Evaluation for Datalog Queries with Integer (Gap)- Order Constraints. Theoretical Computer Science, 116:117-149, 1993.

....that, under the bottom up semantics, for any Turing machine one can e ectively construct a Datalog : z program that computes the same function and is safe whenever the machine is total (Theorem 2.2) Although by appearance, the result looks similar to the one by P. Revesz (Proposition 2. 3 in [7]) that any Turing computable function is expressible by a query of Datalog : z , a closer look reveals that the the two results are altogether di erent. To emphasize only one distinction, the programs that express (total) Turing computable functions in [7] need not terminate under the bottom up ....

....one by P. Revesz (Proposition 2.3 in [7] that any Turing computable function is expressible by a query of Datalog : z , a closer look reveals that the the two results are altogether di erent. To emphasize only one distinction, the programs that express (total) Turing computable functions in [7] need not terminate under the bottom up semantics, hence, they may not be safe. 1 De nitions Throughout this paper, we deal with the domain Z of integer numbers together with the relations = of equality, of the integer linear order, and g of the integer gap order, for all natural numbers g. ....

P.Z. Revesz. A closed form evaluation for Datalog queries with integer (gap)-order constraints. Theoretical Computer Science, 116(1):117-149, 1993.

....the automaton is divided into progressively larger cycles. The major difference from [6] is, however, that we can calculate least fixed points (i.e. exact invariants) for some of the cycles. The method of [3, 5] is used to calculate least fixed points for the innermost loops, and the method of [10] is used to calculate least fixed points for the outermost loops. This leads us to choose a different representation scheme to that used by Halbwachs. A significant advantage of our method is that we do not need to specify the initial inputs to the automaton, but can leave them as uninstantiated ....

....to be duplicated, with one copy for each of the new states. In practice, many of these extra transitions can never be made and so can be eliminated. Our method combines two methods for computing invariants, one for computing the invariants of internal cycles [3] and another for simple circuits [10]. Figure 2 shows how the various stages of the method fit together. 3 Method The basis of the method (see figure 3) is the replacement of simple circuits in the automaton with meta transitions on a single state, and then using bottom up evaluation to calculate the state invariant. s2 s1 b c d a ....

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P. Z. Revesz. A closed-form evaluation for Datalog queries with integer (gap)-order constraints. Theoretical Computer Science, 1993. vol. 116, pages 117--149. This article was processed using the L A T E X macro package with LLNCS style

....of these constraints can be represented by a directed graph with one node for each variable and constant, and an edge from x to y with label k for each of the conjuncts x # k y. See [9] for an introduction, in a linear programming context, to di#erence constraints over real variables. See [32] for constraint processing algorithms for di#erence constraints with integer variables and non negative gap values. These algorithms are implemented in a system described in [3] Linear Arithmetic Constraints A conjunction of linear equality constraints can be represented by a matrix in canonical ....

....queries, and, more generally, 4) how and under what conditions can we perform e#cient logic based queries Two basic paradigms for constraint databases have emerged: constraint databases in the form of logic programs ( datalog constraints ) and constraint databases as systems of objects. See [2, 32, 14] for some reports on early prototypes and applications. 44 relational databases[21] Before the advent of constraint logic programming, there was a well known connection between classical relational databases and logic programming, viz. that the datalog sublanguage of logic programs su#ces to ....

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P. Revesz. A closed-form evaluation for datalog queries with integer (gap)- order constraints. Theoretical Computer Science, 116:117--149, 1993.

....where S is the spatial database predicate and B is the rest of the rule body. In other words, it is syntactically guaranteed that derived relations in the program can only contain points from the given bounded data set. Our observations stand in contrast to results achieved by Revesz and others [8, 13, 14], which show that in other, non spatial, instances of the constraint database framework (more specifically, databases and queries involving order constraints only) useful termination guarantees can be found. We conclude that the termination of particular spatial recursive queries will have to be ....

P.Z. Revesz. A closed-form evaluation for Datalog queries with integer (gap)-order constraints. Theoretical Computer Science, 116:117--149, 1993.

....(PCm GCm ) corresponding to the intuitive semantics of Xi. Xi denotes all the instants t such that there exists i 2 [1; m] with t satisfying both PC i and GC i . Some basic operations have been defined for simple periodicity constraints in [Toman et al. 1994] and for gap order constraints in [Revesz 1993], using a graph for the representation of PC s and GC s. For example, rules are given to combine two constraints on the same variables, either detecting an inconsistency or deriving a new constraint equivalent to the conjunction of the given ones. We now define conjunction ( and complement ( ....

....2 GC 0 1 ) PC 1 PC 0 2 ; GC 1 GC 0 2 ) PC 2 PC 0 2 ; GC 2 GC 0 2 ) g; Xi = f ( PC 1 :PC 2 ; ftrueg) ftrueg; GC 1 :GC 2 ) PC 1 ; GC 2 ) PC 2 ; GC 1 ) g. Note that conjunction among PC s and among GC 0 s is defined among the operations in [Toman et al. 1994; Revesz 1993] and that negation can be easily eliminated. For example, if PC = t j k c t 0 j k 0 c 0 ) PC is the disjunction of the An Access Control Model Supporting Periodicity Constraints and Temporal Reasoning Delta 9 constraints (PC s) t j k r with r = 0; c Gamma 1; c 1; k ....

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Revesz, P. 1993. A closed form evaluation for Datalog queries with integer (gap)-order constraints. Theoretical Computer Science 116, 1, 117--149.

....under development at CNET. Verification with GAP. In [9] we achieved a first mechanical proof of U by encoding the successor relation of the system as a logic program with arithmetical constraints, and computing a fixed point of the program through the bottom up evaluation procedure of Revesz [15]. The encoding required an approximation of the successor relation, so that only an upper approximation of P ost was generated. Nevertheless this approximation was sufficient to prove U , because it did not contain any state violating U . With respect to that approach, we have used here HyTech ....

P.Z. Revesz. "A Closed-Form Evaluation for Datalog Queries with Integer (Gap)- Order Constraints", Theoretical Computer Science, 1993, vol. 116, pp. 117-149.

....ability and entailment tests) The use of deduction to compute temporal properties allows us to enhance the model checking procedure by enriching the set of inference rules used to generate logical consequences of a CLP program. Similar techniques are used, e.g. in Constraint Databases [KKR95, Rev93,RSS92] to improve the eciency of bottom up query evaluation. 1 If A and A B hold then B holds. 2 Giorgio Delzanno and Andreas Podelski: Constraint based Deductive Model Checking We explore the potential of this approach practically by using one of the existing CLP systems with di erent ....

....tems and temporal properties. In [FV94] Fribourg and Veloso Pexoto de ne the notion of automata with constraints and study their properties (e.g. language inclusion) through a representation as CLP programs. In [FR96] Fribourg and Richardson use CLP programs over gap order integer constraints [Rev93] in order to compute the set of reachable states for a decidable class of in nite state systems. Constraints of the form x = y 1 (as needed in our examples) are not gap order constraints. In [FO97] Fribourg and Olsen study reachability for system with integer counters via a translation to ....

P. Z. Revesz. A Closed-form Evaluation for Datalog Queries with Integer (Gap)-order Constraints. Theoretical Computer Science, 116(1):117-149, 1993.

....is divided into progressively larger cycles. The major difference from [Hal93] is, however, that we can calculate least fixed points (i.e. exact invariants) for some of the cycles. The method of [FO95, FVP94] is used to calculate least fixed points for the innermost loops, and the method of [Rev93] is used to calculate least fixed points for the outermost loops. This leads us to choose a different representation scheme to that used by Halbwachs. A significant advantage of our method is that we do not need to specify the initial inputs to the automaton, but can leave them as uninstantiated ....

....s 2 , connected by the transition s 1 t s 2 . Simple circuits are paths, s s 1 : s n s, which start and finish in the same state, s. Our method combines two methods for computing invariants, one for computing the invariants of internal cycles [FO95] and another for simple circuits [Rev93]. 3 Method More precisely, there are five steps to the method: 1. Identify the internal cycles in the automaton. We exploit the method of [FO95] to calculate least fixed points for the internal cycles. This allows us to calculate more accurate invariants than if, for example, we just treated ....

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P. Z. Revesz. A closed-form evaluation for Datalog queries with integer (gap)-order constraints. Theoretical Computer Science, 1993. vol. 116, pages 117--149. 9

....formulas in order to accelerate the processing of queries, and, more generally, 4) how and under what conditions can we perform efficient logic based queries At present, a major challenge is to advance from the theory to the actual construction of useful constraint databases. See [Bro96, Rev93] for some reports on early prototypes and applications, and also [GBG 97] for proceedings of a conference on constraint databases and applications. A classical application of constraint databases is to the representation of spatial information. Consider a geographical database that stores ....

....A conjunction of order constraints among integer variables can naturally be put into the canonical form of a graph whose nodes are variables and whose edges indicate order dependencies; we can have edges that represent strict equalities and edges that represent nonstrict equalities. In [Rev93] a representation for gap order constraints of the form X g Y, which is taken to mean that the gap between X and Y is at least g, i.e. X Y g. Version 3 of CLP inference engine: Introduction of the Constraint Store constraintStore CStore; global constraint store, ....

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P. Revesz. A closed-form evaluation for datalog queries with integer (gap)- order constraints. Theoretical Computer Science, 116:117--149, 1993. 92

....Databases [KKR95, 1 If A and A B hold, then B holds. 2 Giorgio Delzanno and Andreas Podelski: Constraint based Deductive Model Checking Concurrent Systems ; CLP programs Temporal Properties ; Models of CLP Programs Model Checking ; Deduction Fig. 1. A connection between two elds. Rev93,RSS92] to improve the eciency of bottom up query evaluation. We explore the potential of this approach practically by using one of the existing CLP systems with di erent constraint domains as an implementation platform. We have implemented an algorithm to generate models for CLP programs using ....

....(essentially, mixed forms of backward and forward analysis) by applying transformations such as the magic sets templates algorithm [RSS92] to the CLP programs PS F . Such transformations are natural in the context of CLP programs which may also be viewed as constraint data bases (see [RSS92,Rev93] The application of a kind of magic set transformation on the CLP program P = PS F , where the clauses have a restricted form (one or no predicate in the body) yields the following CLP program e P (with new predicates e p and g init) e P = fp(x) body; e p(x 0 ) j p(x) body 2 Pg [ ....

[Article contains additional citation context not shown here]

P. Z. Revesz. A Closed-form Evaluation for Datalog Queries with Integer (Gap)-order Constraints. Theoretical Computer Science, 116(1):117-149, 1993.

....(essentially, mixed forms of backward and forward analysis) by applying transformations such as the magic sets templates algorithm [RSS92] to the CLP programs PS PhiF . Such transformations are natural in the context of CLP programs which may also be viewed as constraint data bases (see [RSS92, Rev93] The application of a kind of magic set transformation on the CLP program P = PS Phi F , where the clauses have a restricted form (one or no predicate in the body) yields the following CLP program e P (with new predicates e p and g init) e P = fp(x) body; e p(x 0 ) j p(x) body 2 Pg ....

....6 Related Work There have been other attempts to connect logic programming and verification, none of which has our generality with respect to the applicable concurrent systems and temporal properties. In [FR96] Fribourg and Richardson use CLP programs over gap order integer constraints [Rev93] in order to compute the set of reachable states for a decidable class of infinite state systems. Constraints of the form x = y 1 (as needed in our examples) are not gap order constraints. In [FO97] Fribourg and Olsen study reachability for system with integer counters. These approaches are ....

P. Z. Revesz. A Closed-form Evaluation for Datalog Queries with Integer (Gap)-order Constraints. Theoretical Computer Science, 116(1):117--149, 1993.

....or meta programming in CLP(R) Revesz has shown that bottom up evaluation (with subsumption) terminates for a special class of logic programs with gap order constraints. Gap order constraints are inequalities of the form X 1 X 2 c where X 1 and X 2 are variables and c a non negative integer [44]. Originally in [44] the domain of variables was assumed to be Z, but the procedure can be slightly modify in order to involve real valued variables instead. The reachability problem can be easily encoded under the form of a logic program with constraints. For example, consider the timed ....

....in CLP(R) Revesz has shown that bottom up evaluation (with subsumption) terminates for a special class of logic programs with gap order constraints. Gap order constraints are inequalities of the form X 1 X 2 c where X 1 and X 2 are variables and c a non negative integer [44] Originally in [44], the domain of variables was assumed to be Z, but the procedure can be slightly modify in order to involve real valued variables instead. The reachability problem can be easily encoded under the form of a logic program with constraints. For example, consider the timed automaton with three clocks ....

P.Z. Revesz. "A Closed-Form Evaluation for Datalog Queries with Integer (Gap)- Order Constraints", Theoretical Computer Science, 1993, vol. 116, pp. 117-149.

.... on any but the most trivial instances [BW98] 1 Applying history dependent widening and narrowing techniques as already foreseen in the abstract interpretation scheme [CC77] and basing our intuition on techniques from Constraint Logic Programming (see [DP98] and Constraint Data Bases [KKR95, Rev93] we show that a set of abstractions of the modelchecking fixpoint operator yields an accurate model checking algorithm (i.e. a full test if terminating) These abstractions are able to drastically decrease the number of iterations or even to enforce termination of an otherwise non terminating ....

P. Z. Revesz. A closed-form evaluation for Datalog queries with integer (gap)-order constraints. Theoretical Computer Science, 116(1), pages 117--149, 1993.

....which interprets arithmetic constraints over integers and equality disequality constraints over terms in the usual way [6, 8] There is no complete constraint solver for arbitrary integer arithmetic constraints. However, there are powerful decidable subsets of integer arithmetic constraints [27, 31, 19, 39, 20, 44, 41, 18]. This paper focuses on decidable subsets of integer arithmetic constraints and assumes that D is satisfaction complete. The assumption, which can be relaxed, helps us to separate the issue of the completeness of the constraint solver from that of the completeness of the operational semantics ....

P. Z. Revesz. A closed-form evaluation for Datalog queries with integer (gap)-order constraints. Journal of Theoretical Computer Science, 116(1):117--149, 1993.

....of [ST95c] is that, under the bottom up semantics, for any Turing machine one can effectively construct a Datalog : z program that computes the same function and is safe whenever the machine is total. Although by appearance, the result looks similar to the one by P. Revesz (Proposition 2. 3 in [Rev93] that any Turingcomputable function is expressible by a query of Datalog : z , a closer look reveals that the the two results are altogether different. To emphasize only one distinction, the programs that express (total) Turing computable functions in [Rev93] need not terminate under the ....

....by P. Revesz (Proposition 2.3 in [Rev93] that any Turingcomputable function is expressible by a query of Datalog : z , a closer look reveals that the the two results are altogether different. To emphasize only one distinction, the programs that express (total) Turing computable functions in [Rev93] need not terminate under the bottom up semantics, hence, they may not be safe. 1.8. Notation. Notation is usual. denotes either the end of the proof or that the proof of the statement is omitted. 2. A recursive domain with decidable FO theory which has no recursive syntax for safe queries The ....

P.Z. Revesz, A closed form evaluation for Datalog queries with integer (gap)-order constraints, Theoretical Computer Science 116 (1993), no. 1, 117--149.

....language the arithmetic over N becomes definable and thus the query language undecidable. The situation changes drastically if structures with a discrete order as universe are considered. It is known that positive Datalog queries on discrete order databases can be evaluated in closed form (see [14]) but the data complexity is still unknown. For first order queries a better result can be shown. Theorem 20. First order queries on discrete order databases can be evaluated in Logspace. See [13] for a proof of the theorem. In Section 3 we have shown that the data complexity of first order ....

P. Revesz. A closed form evaluation for datalog queries with integer (gap)-order constraints. Theoretical Computer Science, 116:117--149, 1993.

....with stratified negation, or Datalog : z , where negations are allowed, but only w.r.t. the intensional predicates whose computation already terminated (cf. 1] This machinery only works well, however, if the Datalog program terminates. If it does not, the construction collapses. Revesz [6] established that termination of Datalog : z programs is undecidable. One remedy is to consider only those Datalog : z programs whose termination is guaranteed for all inputs. Such programs often are called safe. Notice that this definition is semantical in nature. Revesz [7] introduced a ....

P.Z. Revesz. A closed form evaluation for Datalog queries with integer (gap)-order constraints. Theoretical Computer Science, 116(1):117--149, 1993.

....conjunctions S i and S j of half addition constraints such that the set of solutions of S i is included in the set of solutions of S j . This shows that the fixpoint evaluation could be modified to add only points that do not dominate any earlier point in the sequence. By the geometric Lemma in [24], in any fixed dimension any sequence of distinct points with non negative integer coordinates must be finite, if no point dominates any earlier point in the sequence. This shows that using a modified constraint fixpoint evaluation: constraint form when the domain is the integer numbers. For ....

....constraints are half addition constraints, but some half addition constraints are not gap order constraints. For example, x y 5 is a half addition constraint but it is not expressible by gap order constraints. A least fixpoint evaluation for Datalog with gap order constraints is described in [24]. The recognition problem is also studied in [8] The DISCO system [5] implements Datalog queries with integer gap order constraints. Adding negation in a safe way to Datalog with gap order queries is studied in [25] A temporal constraint is like a gap order constraint but the gap value can be ....

P. Z. Revesz. A Closed Form Evaluation for Datalog Queries with Integer (Gap)-Order Constraints. Theoretical Computer Science, vol. 116, no. 1, 117-149, 1993.

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P. Revesz. A closed form evaluation for datalog queries with integer (gap)-order constraints. Theoretical Computer Science, 116:117--149, 1993.

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P. Z. Revesz. A Closed-Form Evaluation for Datalog Queries with Integer (Gap)- Order Constraints. Theoretical Computer Science, 116:117-149, 1993. 38

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Peter Z. Revesz. A Closed-Form Evaluation for Datalog Queries with Integer (Gap)-Order Constraints. TCS, 116(1):117--149, 1993.

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P.Z. Revesz. "A Closed-Form Evaluation for Datalog Queries with Integer (Gap)- Order Constraints", Theoretical Computer Science, 1993, vol. 116, pp. 117-149.

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P. Z. Revesz. A Closed-Form Evaluation for Datalog Queries with Integer (Gap)- Order Constraints. Theoretical Computer Science, 116:117-149, 1993.

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P.Z. Revesz. "A Closed-Form Evaluation for Datalog Queries with Integer (Gap)- Order Constraints", Theoretical Computer Science, 1993, vol. 116, pp. 117-149.

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