DatalogZ

DatalogZ (stylized as Datalog_ℤ) is an extension of Datalog with integer arithmetic and comparisons. The decision problem of whether or not a given ground atom (fact) is entailed by a DatalogZ program is RE-complete (hence, undecidable), which can be shown by a reduction to diophantine equations.^[1]

Syntax

The syntax of DatalogZ extends that of Datalog with numeric terms, which are integer constants, integer variables, or terms built up from these with addition, subtraction, and multiplication. Furthermore, DatalogZ allows comparison atoms, which are atoms of the form t < s or t <= s for numeric terms t, s.^[2]

Semantics

The semantics of DatalogZ are based on the model-theoretic (Herbrand) semantics of Datalog.^[2]

Limit DatalogZ

The undecidability of entailment of DatalogZ motivates the definition of limit DatalogZ. Limit DatalogZ restricts predicates to a single numeric position, which is marked maximal or minimal. The semantics are based on the model-theoretic (Herbrand) semantics of Datalog. The semantics require that Herbrand interpretations be limit-closed to qualify as models, in the following sense: Given a ground atom $a=r(c_{1},\ldots ,c_{n})$ of a limit predicate $r$ where the last position is a max (resp. min) position, if $a$ is in a Herbrand interpretation $I$ , then the ground atoms $r(c_{1},\ldots ,k)$ for $k>c_{n}$ (resp. $k<c_{n}$ ) must also be in $I$ for $I$ to be limit-closed.^[3]

Example

Given a constant w, a binary relation edge that represents the edges of a graph, and a binary relation sp with the last position of sp minimal, the following limit DatalogZ program computes the relation sp, which represents the length of the shortest path from w to any other node in the graph:

sp(w, 0) :- .
sp(y, m + 1) :- sp(x, m), edge(x, y).

References

Notes

^ Dantsin, Evgeny; Eiter, Thomas; Gottlob, Georg; Voronkov, Andrei (2001-09-01). "Complexity and expressive power of logic programming". ACM Computing Surveys. 33 (3): 374–425. doi:10.1145/502807.502810. ISSN 0360-0300.. "For example, datalog (which is EXPTIME-complete) with linear arithmetic constraints [...] is undecidable." (Theorem 10.1)
^ ^a ^b Kaminski et al. 2017, pp. 2.
^ Kaminski et al. 2017, pp. 3.

Sources

Grau, Bernardo Cuenca; Horrocks, Ian; Kaminski, Mark; Kostylev, Egor V.; Motik, Boris (2020-02-25). "Limit Datalog: A Declarative Query Language for Data Analysis". ACM SIGMOD Record. 48 (4): 6–17. doi:10.1145/3385658.3385660. ISSN 0163-5808. S2CID 211520719.
Kaminski, Mark; Grau, Bernardo Cuenca; Kostylev, Egor V.; Motik, Boris; Horrocks, Ian (2017-11-12). "Foundations of Declarative Data Analysis Using Limit Datalog Programs". arXiv:1705.06927 [cs.AI].
Kaminski, Mark; Kostylev, Egor V.; Grau, Bernardo Cuenca; Motik, Boris; Horrocks, Ian (2021-12-22). "The Complexity and Expressive Power of Limit Datalog". Journal of the ACM. 69 (1): 6:1–6:83. doi:10.1145/3495009. ISSN 0004-5411. S2CID 246702614.

[1] Dantsin, Evgeny; Eiter, Thomas; Gottlob, Georg; Voronkov, Andrei (2001-09-01). "Complexity and expressive power of logic programming". ACM Computing Surveys. 33 (3): 374–425. doi:10.1145/502807.502810. ISSN 0360-0300.. "For example, datalog (which is EXPTIME-complete) with linear arithmetic constraints [...] is undecidable." (Theorem 10.1)

[FOOTNOTEKaminskiGrauKostylevMotik20172-2] Kaminski et al. 2017, pp. 2.

[FOOTNOTEKaminskiGrauKostylevMotik20173-3] Kaminski et al. 2017, pp. 3.

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