Boundedness is undecidable for datalog programs with a single recursive rule

doi:10.1016/0020-0190(89)90019-7

Information Processing Letters

Volume 32, Issue 6, 3 October 1989, Pages 281-287

https://doi.org/10.1016/0020-0190(89)90019-7 Get rights and content

Abstract

We prove that boundedness is undecidable even for Datalog programs with a single recursive rule. This strengthens the undecidability results of Gaifman et al. (1987) for multiple recursive rule Datalog programs. Our proof technique involves approximate simulation of a fixpoint of a formula with disjunctions, by a fixpoint of a formula without disjunctions.

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Active database systems enhance the functionality of traditional databases through the use of active rules or ‘triggers’. One of the principal analysis questions for such systems is that of termination—is it possible for the rules to recursively activate one another indefinitely, given an initial triggering event. In this paper, we study the decidability of the termination problem, our aim being to delimit the boundary between the decidable and the undecidable. We present results for two broad types of variations, variations in rule syntax and variations in meta level features. Within each of these, we identify members close to the boundary of (un)decidability and also look at the effect of combining members of each type. The maximal decidable class we present is capable of expressing some useful kinds of application requirements, such as checking and repairing inclusion constraints. The work is also interesting from a theoretical point of view, since the context is similar to the while query language and the dynamics gives an interesting contrast to Datalog with negation.
On the complexity of single-rule datalog queries
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Citation Excerpt :
These additional atoms make sure that over databases expanded by the or and eq relations, the set of ground instances of XP is equivalent to the union of the ground instances of the single rules of P, and hence XP encodes simultaneously all rules of P and is in essence equivalent to P. All our complexity and expressiveness results follow rather straightforwardly from this construction and from the corresponding results for general datalog programs (see Table 1). Note that a different product construction was given by Abiteboul [2] in order to show that the boundedness problem for datalog sirups is undecidable. That construction does preserve program boundedness, but not equivalence.
Datalog programs containing a unique rule and possibly some facts are known as single rule programs, or sirups. We study the complexity of evaluating sirups over variable and fixed databases, respectively, as well as the descriptive complexity of sirups, i.e., their expressive power. In all cases it turns out that even very restricted classes of sirups have the same complexity and essentially the same expressive power as general datalog programs. In particular, the evaluation of single clause programs is EXPTIME complete (combined complexity) and, if restricted to linear recursive rules, PSPACE complete. Moreover, sirups with one recursive rule and one fact capture PTIME on ordered structures, if a certain data representation is assumed and certain predefined relations are provided. We also prove that the datalog clause implication problem, i.e., deciding whether a datalog clause implies another one, is EXPTIME complete. Our main technical tool is a product construction which maps a datalog programs to an essentially equivalent sirup.
Undecidable boundedness problems for datalog programs
1995, The Journal of Logic Programming
A given Datalog program is bounded if its depth of recursion is independent of the input database. Deciding boundedness is a basic task for the analysis of database logic programs. The undecidability of Datalog boundedness was first demonstrated by Gaifman et al. [7]. We introduce new techniques for proving the undecidability of (various kinds of) boundedness, which allow us to considerably strengthen the results of Gaifman et al. [7]. In particular, (1) we use a new generic reduction technique to show that program boundedness is undecidable for arity 2 predicates, even with linear rules; (2) we use the mortality problem of Turing machines to show that uniform boundedness is undecidable for arity 3 predicates and for arity 1 predicates when ≠ is also allowed; (3) by encoding all possible transitions of a two-counter machine in a single rule, we show that program (resp., predicate) boundedness is undecidable for two linear rules (resp., one rule and a projection) and one initialization rule, where all predicates have small arities (6 or 7).
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Boundedness is undecidable for datalog programs with a single recursive rule

Abstract

Decidable optimization problems for database logic programs

Proc. 20th ACM STOC

Parallel evaluation of recursive rule queries

Proc. 5th ACM Symp. on Principles of Database Systems

Undecidable optimization problems for database logic programs

Proc. 2nd Symp. on Logic in Computer Science

A time bound on the materialization of some recursively defined views

Proc. 11th Internat. Conf. on Very Large Data Bases

Logic programming and parallel complexity