S. Abiteboul and E. Simon. Fundamental properties of deterministic and nondeterministic extensions of Datalog. Technical report, INRIA, July 1990. to appear in Theoretical Computer Science.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....involving OK are removed, because OK is declared locally within the procedure. This example is reminiscent of the LOGRES system 90] which permits the sequential application of different rule modules . The example also embodies some of the spirit of the theoretical language DATALOG [AV88, AS90] a variant of DATALOG in which rule heads can be positive or negative. Unlike DATALOG , which supports either inflationary semantics and a semantics based on nondeterministic application of rules, this example uses a semantics reminiscent of stratified logic programming. In the next example, ....

S. Abiteboul and E. Simon. Fundamental properties of deterministic and nondeterministic extensions of Datalog. Technical report, INRIA, July 1990. to appear in Theoretical Computer Science.

....q n as interpretation of q # ) Hence, the evaluation of # in the WF semantics corresponds to a nested #xpoint. While for the NI semantics, the results mentioned above (besides the game theoretic normal form) are consequences of known facts and are more or less explicit in the literature (see [2, 4, 7]) the corresponding results for the WF semantics are new; in particular, they solve an open problem stated in [1] The parts on the NI semantics (Section 3) and the WF semantics (Section 4) may be read independently. For the WF semantics we use a normal form for least #xpoint logic LFP due to ....

....queries (#; q t ) 2 As already remarked in the introduction, the reader only interested in the well founded semantics may skip this section. 262 J. Flum et al. Theoretical Computer Science 239 (2000) 257 276 The following theorem is a straightforward generalization of a result of [2] (cf. also [7] Theorem 1 (NI Datalog6NI Datalog 2 ) For every NI Datalog program # there is an equivalent total program. Moreover; there is an equivalent total program having as truth set the false set of #. Proof (sketch) Let # be a Datalog program and (for simplicity) q its unique ....

S. Abiteboul, E. Simon, Fundamental properties of deterministic and nondeterministic extensions of datalog, Theoret. Comput. Sci. 78(1) (1991) 137--158.

....q n as interpretation of q 0 ) Hence, the evaluation of Pi in the WF semantics corresponds to a nested fixpoint. While for the NI semantics the results mentioned above (besides the gametheoretic normal form) are consequences of known facts and are more or less explicit in the literature (see [AS91], Gro92] EF95] the corresponding results for the WF semantics are new; in particular, they solve an open problem stated in [AHV95] The parts on the NI semantics (Section 3) and the WFsemantics (Section 4) may be read independently. For the WF semantics we use a normal form for least fixpoint ....

....semantics may skip this section. 6 where q c f denotes the complement of q f with respect to D m . NI Datalog 2 is the restriction of NI Datalog to total programs and to the corresponding queries ( Pi; q t ) The following theorem is a straightforward generalization of a result of [AS91] (cf. also [Gro92] Theorem 1 (NI Datalog NI Datalog 2 ) For every NI Datalog program Pi there is an equivalent total program. Moreover, there is an equivalent total program having as truth set the false set of Pi. Proof (Sketch) Let Pi be a Datalog : program and (for simplicity) q its ....

S. Abiteboul and E. Simon. Fundamental Properties of Deterministic and Nondeterministic Extensions of Datalog. Theoretical Computer Science, 78(1):137--158, 1991.

.... for I Datalog termination is guaranteed, this is no longer the case of Datalog : it is undecidable whether a Datalog : program reaches a fixpoint for all databases; moreover, confluence is no longer guaranteed if instead of the presented semantics, a nondeterministic semantics is used [AS91] On the other hand, nondeterminism can be a powerful programming paradigm which increases the (theoretical and practical) expressiveness of a language [AV91,GGSZ97] 1 For efficiency reasons however, the distinction between events and conditions may be crucial in practice. 2 Another ....

S. Abiteboul and E. Simon. Fundamental Properties of Deterministic and Nondeterministic Extensions of Datalog. Theoretical Computer Science, 78(1):137--158, 1991.

....more formally. The important notion of Sigma stratification is defined which implies the existence of a unique perfect model. Finally, we show that termination of arbitrary Statelog programs is undecidable (a similar result for the one state update language Datalog : has been given in [2]) 3.1 Syntax Definition 3.1 (Statelog) A Statelog rule r is of the form r : S k 0 ]H [S k 1 ]B 1 ; S k n ]B n where n 1, the head H is a Datalog atom, B i are Datalog literals (i.e. either atoms A or negated atoms :A) S is the distinguished state variable and the offsets k ....

.... : S] o(X) S] m(X,Y) u(Y) S 1] u(X) S] m(X,Y) o(Y) S 1] m(X,Y) S] m(X,Y) S] u(X) S] n(X) Given an empty database D as input, both P (i) and P (ii) do not terminate: P (i) violates positive recursion which leads to the non terminating sequence of M: 0] q, 1] q,[2] : q, 3] q, Similarly, P (ii) violates 1 Cycles and the sequence oscillates between fq,rg and f: q, rg. Note that P (ii) contains gaps between states, in the sense that [S 2]q , S]q is valid. This possibility of gaps is ruled out by (ii) The requirement of isolation, together with ....

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S. Abiteboul and E. Simon. Fundamental properties of deterministic and nondeterministic extensions of datalog. Theoretical Computer Science, 78, 1991.

....and data manipulation languages and, more recently, as high level protocols to interact with heterogeneous data repositories, is the most remarkable success of declarative languages in this field. The natural evolution of this research has led to the use of logic based languages, like Datalog [1, 28], LDL [34] or RDL [17] that add deductive power to the simple databases we were used to work with. These deductive databases can efficiently handle vast amounts of derived knowledge, moving most semantic issues from (diffuse) applicative logic to (centralized) database design. The second main ....

S. Abiteboul and E. Simon. Fundamental properties of deterministic and nondeterministic extensions of Datalog. Theoretical Computer Science, (78):137--158, 1991.

....balancing facility. This last point is the focus of this paper. 3 The PARULEL Language Here, we review the main features of PARULEL to lay the groundwork for our subsequent discussion. 3. 1 Syntax PARULEL syntax is derived from OPS5, while its operational semantics resemble those of Datalog : [1, 5]. As in Datalog : PARULEL is capable of set oriented updates involving all rule actions that do not conflict according to certain criteria. The relationship between PARULEL and Datalog : is discussed at length in [5] PARULEL programs consist of object level production rules, metarules ....

S. Abiteboul and E. Simon. Fundamental properties of deterministic and nondeterministic extensions of datalog . Journal of Theoretical Computer Science, 1991.

....difference between A RDL and RDL is that A RDL rules are considered with respect to a database state transition along with the current database state and they may refer to this state transition in their antecedent. 3 This is very similar to suggested extensions of Datalog as described in, e.g. [AS91]. 6 Ariel In Ariel there are two forms of rules: rules without activators, called pattern based rules, and rules with activators, called event based rules. For convenience, we refer to these two forms of rules as Ariel p and Ariel e respectively. Along our deductive to active axis, Ariel p ....

S. Abiteboul and E. Simon. Fundamental properties of deterministic and nondeterministic extensions of datalog. Theoretical Computer Science, 78:137--158, 1991.

....more formally. The important notion of Sigma stratification is defined which implies the existence of a unique perfect model. Finally, we show that termination of arbitrary Statelog programs is undecidable (a similar result for the one state update language Datalog : has been given in [2]) 3.1 Syntax Definition 3.1 (Statelog) A Statelog rule r is of the form r : S k 0 ]H [S k 1 ]B 1 ; S k n ]B n where n 1, the head H is a Datalog atom, B i are Datalog literals (i.e. either atoms A or negated atoms :A) S is the distinguished state variable and the offsets ....

....) Then P afp is [S 1] win(X) S 1] move(X,Y) S] win(Y) As usually, we assume that the input relation move is kept constant by a frame rule. Given the input move = f(a; b) b; a) b; c) c; d)g the following non terminating computation results: 0] win = 1] win = fa,b,cg [2] win = fcg [3] win = fa,b,cg [4] win = fcg : In accordance with [20] even numbered states correspond to a nondecreasing sequence of underestimates while odd numbered states represent a nonincreasing sequence of overestimates of true atoms. It is easy to see that, independent of the ....

S. Abiteboul and E. Simon. Fundamental properties of deterministic and nondeterministic extensions of datalog. Theoretical Computer Science, 78, 1991.

....an equational rule based (EQL) program. Corresponding analysis tools were developed to estimate the time responses for programs written in other production languages, i.e. in MRL [21] and in OPS5 [6] Similar concepts within the deductive databases research are presented by Abiteboul and Simon [1]. They discuss the totalness and loop freeness of a deductive system, which, in the terminology of rule based systems, describes the stability of the systems in terms of the initial points reaching their fixed points in finite time. If the analysis finds that the given real time rule based ....

S. Abiteboul and E. Simon, "Fundamental properties of deterministic and nondeterministic extensions of Datalog," Theoretical Computer Science, vol. 78, pp. 137--158, 1991.

....fact we only need an undo phase, and for an asserted fact we only need a redo phase. However, for more powerful rule languages, we need both an undo and a redo phase to incrementally update the database. These two phases are not always straightforward. Rule languages such as OPS5 and Datalog : [3] usually operate in match select act cycles, and at each cycle (or iteration) some conflict resolution strategy is applied. Consider for example redoing the consequences of f . This involves matching all the rules (or some subset of them in a language such as stratified datalog) and considering ....

S. Abiteboul and E. Simon; Fundamental properties of deterministic and nondeterministic extensions of Datalog ; Journal of Theoretical Computer Science, 1990.

....In [KU91] the issue of rule set termination is discussed, along with the issue of conflicting updates determining when one rule may undo changes made by a previous rule. Although models and a problem solving architecture for rule analysis are proposed, no algorithms are given. In [AS91] issues of termination and unique fixed points are considered in the context of various extensions to Datalog. In addition to the very different semantics of Datalog (logic) and production rules, AS91] does not address the issue of determining whether a given rule set exhibits certain ....

....and a problem solving architecture for rule analysis are proposed, no algorithms are given. In [AS91] issues of termination and unique fixed points are considered in the context of various extensions to Datalog. In addition to the very different semantics of Datalog (logic) and production rules, AS91] does not address the issue of determining whether a given rule set exhibits certain properties (as we do) but rather states results about whether all rule sets in a given language are guaranteed to exhibit the properties. In [CW90] we presented initial methods for analyzing termination in the ....

S. Abiteboul and E. Simon. Fundamental properties of deterministic and nondeterministic extensions of datalog. Theoretical Computer Science, 78:137--158, 1991.

....expressed in t while as follows: shared A(1) Even(0) A : fx j S(x)g; Even : f( g; while :Last do right; Even : f( g Gamma Even ; if Even then A : A fx j S(x)g od. 3 If the execution loops indefinitely, the result is defined to be empty by default. Such loops can always be detected [3]. 4 In the first state, left has no effect; in the last state, right has no effect. In the above program, A and Even are both shared relations. Note how they are declared as variables in the beginning of the program, indicating their status of shared relation and their arity; we will always ....

S. Abiteboul and E. Simon. Fundamental properties of deterministic and nondeterministic extensions of Datalog. Theoretical Computer Science, 78:137--158, 1991. 7 Actually, the collection of k-types with an order on them.

....applying two FO queries. The previous analysis fails for non terminating while programs, since we cannot detect termination even for instances satisfying the extension axioms. The problem of termination disappears in the case of restricted languages like while, where termination can be tested [AS91]. Remark 5.1 The results in this section are closely related to recent work of Kolaitis and Vardi on 0 1 laws [KV87, KV90a, KV92] They state a non effective version of Theorem 5.2 for formulas in infinitary firstorder logic with k variables 2 , denoted L k 1 . More precisely, they show that ....

S. Abiteboul and E. Simon. Fundamental properties of deterministic and nondeterministic extensions of datalog. Theoretical Computer Science, 78:137--158, 1991. 8

....direction 4 . Furthermore, we partition the auxiliary relations into state relations, which are stored in the different states, and shared relations, which are stored in 3 If the execution loops indefinitely, the result is defined to be empty by default. Such loops can always be detected [AS91] 4 In the first state, left has no effect; in the last state, right has no effect. the memory of the machine itself. So, the values of (and assignments to) state relations depend on the current state the machine is looking at, while this is not the case for shared relations. Finally, we ....

S. Abiteboul and E. Simon. Fundamental properties of deterministic and nondeterministic extensions of Datalog. Theoretical Computer Science, 78:137--158, 1991.

....state. The machine always starts execution from the first state. The temporal query language while extended with left and right moves just described is denoted by t while. 4 If the execution loops indefinitely, the result is undefined. Infinite loops can always be detected at run time in while [2]. 5 In the first state, left has no effect; in the last state, right has no effect. Example 4.2 The query from Example 4.1 can be expressed in t while as follows: shared A(1) Even(0) A : fx j S(x)g; Even : f( g; while :Last do right; Even : f( g Gamma Even; if Even 6= then A : A ....

S. Abiteboul and E. Simon. Fundamental properties of deterministic and nondeterministic extensions of Datalog. Theoretical Computer Science, 78:137--158, 1991.

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