Tractable Fragments of Datalog with Metric Temporal Operators

IJCAI, pp. 1919-1925, 2020.

Cited by: 0|Bibtex|Views42|DOI:https://doi.org/10.24963/ijcai.2020/266
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We have focused on fragments which mention only the past temporal operators and, analogous results immediately follow for the future temporal operators and

Abstract:

We study the data complexity of reasoning for several fragments of MTL - an extension of Datalog with metric temporal operators over the rational numbers. Reasoning in the full MTL language is PSPACE-complete, which handicaps its application in practice. To achieve tractability we first study the core fragment, which disallows conjunctio...More

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Introduction
  • DatalogMTL [Brandt et al, 2017; Brandt et al, 2018; Wałega et al, 2019b] is a temporal extension of the fundamental rule language Datalog [Abiteboul et al, 1995] in which atoms in rules may contain metric temporal logic (MTL) operators interpreted over a rational timeline [Ouaknine and Worrell, 2008; Hunter et al, 2013].
  • The authors study the data complexity of recursive fragments of DatalogMTL under continuous semantics.
  • In Section 3 the authors show that this does not apply to the setting, as DatalogMTLcore remains PSPACE-hard in data complexity, even if the since operator S is the only temporal operator allowed in rules.
  • Operator S fully expresses the past diamond operator, and the rule (1) can be written using only S as a temporal operator; in contrast, S does allow them to express
Highlights
  • DatalogMTL [Brandt et al, 2017; Brandt et al, 2018; Wałega et al, 2019b] is a temporal extension of the fundamental rule language Datalog [Abiteboul et al, 1995] in which atoms in rules may contain metric temporal logic (MTL) operators interpreted over a rational timeline [Ouaknine and Worrell, 2008; Hunter et al, 2013]
  • Metric temporal logic is equipped with two alternative semantics: pointwise and continuous; in this paper we consider the latter, which is the one typically adopted in works on DatalogMTL
  • In Section 3 we show that this does not apply to our setting, as DatalogMTLcore remains PSPACE-hard in data complexity, even if the since operator S is the only temporal operator allowed in rules
  • We have focused on fragments which mention only the past temporal operators and, analogous results immediately follow for the future temporal operators and
  • We see many possibilities for future work. It is unclear if the bound in Theorem 7 is tight, as well as whether the upper bound applies to DatalogMTLlin
  • We want to consider DatalogMTL over integer timelines, which may lead to lower complexity and which have been already considered for metric temporal logic [Gutiérrez-Basulto et al, 2016]
Conclusion
  • Discussion and Future

    Work

    The authors have focused on fragments which mention only the past temporal operators and , analogous results immediately follow for the future temporal operators and.
  • Consistency checking is TC0-complete, P-hard, and NL-complete for DatalogMTLcore, DatalogMTLcore, and DatalogMTLlin, respectively, and the lower bounds hold already for the propositional fragments.
  • Low complexity fragments have been identified for propositional DatalogMTL under the pointwise semantics [Kikot et al, 2018; Ryzhikov et al, 2019].
  • The authors see many possibilities for future work
  • It is unclear if the bound in Theorem 7 is tight, as well as whether the upper bound applies to DatalogMTLlin.
  • The authors' complexity bounds do not directly provide practical reasoning algorithms, which the authors would like to construct
Summary
  • Introduction:

    DatalogMTL [Brandt et al, 2017; Brandt et al, 2018; Wałega et al, 2019b] is a temporal extension of the fundamental rule language Datalog [Abiteboul et al, 1995] in which atoms in rules may contain metric temporal logic (MTL) operators interpreted over a rational timeline [Ouaknine and Worrell, 2008; Hunter et al, 2013].
  • The authors study the data complexity of recursive fragments of DatalogMTL under continuous semantics.
  • In Section 3 the authors show that this does not apply to the setting, as DatalogMTLcore remains PSPACE-hard in data complexity, even if the since operator S is the only temporal operator allowed in rules.
  • Operator S fully expresses the past diamond operator, and the rule (1) can be written using only S as a temporal operator; in contrast, S does allow them to express
  • Conclusion:

    Discussion and Future

    Work

    The authors have focused on fragments which mention only the past temporal operators and , analogous results immediately follow for the future temporal operators and.
  • Consistency checking is TC0-complete, P-hard, and NL-complete for DatalogMTLcore, DatalogMTLcore, and DatalogMTLlin, respectively, and the lower bounds hold already for the propositional fragments.
  • Low complexity fragments have been identified for propositional DatalogMTL under the pointwise semantics [Kikot et al, 2018; Ryzhikov et al, 2019].
  • The authors see many possibilities for future work
  • It is unclear if the bound in Theorem 7 is tight, as well as whether the upper bound applies to DatalogMTLlin.
  • The authors' complexity bounds do not directly provide practical reasoning algorithms, which the authors would like to construct
Tables
  • Table1: Semantics of DatalogMTL ground literals
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Funding
  • This work was supported by the AIDA project (Alan Turing Institute), the SIRIUS Centre for Scalable Data Access (Research Council of Norway), Samsung Research UK, Siemens AG, and the EPSRC projects AnaLOG (EP/P025943/1), OASIS (EP/S032347/1) and UK FIRES (EP/S019111/1)
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