A fine-grained hierarchy of hard problems in the separated fragment

Marco Voigt 1, 2
1 VERIDIS - Modeling and Verification of Distributed Algorithms and Systems
MPII - Max-Planck-Institut für Informatik, Inria Nancy - Grand Est, LORIA - FM - Department of Formal Methods
Abstract : Recently, the separated fragment (SF) has been introduced and proved to be decidable. Its defining principle is that universally and existentially quantified variables may not occur together in atoms. The known upper bound on the time required to decide SF's satisfiability problem is formulated in terms of quantifier alternations: Given an SF sentence ∃ z ∀ x1∃ y1. .. ∀ xn∃ yn. ψ in which ψ is quantifier free, satisfiability can be decided in non-deterministic n-fold exponential time. In the present paper, we conduct a more fine-grained analysis of the complexity of SF-satisfiability. We derive an upper and a lower bound in terms of the degree ∂ of interaction of existential variables (short: degree) — a novel measure of how many separate existential quantifier blocks in a sentence are connected via joint occurrences of variables in atoms. Our main result is the k-NEXPTIME-completeness of the satisfiability problem for the set of all SF sentences that have degree k or smaller. Consequently, we show that SF-satisfiability is non-elementary in general, since SF is defined without restrictions on the degree. Beyond trivial lower bounds, nothing has been known about the hardness of SF-satisfiability so far.
Document type :
Conference papers
Complete list of metadatas

Cited literature [16 references]  Display  Hide  Download

https://hal.inria.fr/hal-01592172
Contributor : Stephan Merz <>
Submitted on : Wednesday, September 27, 2017 - 6:49:59 PM
Last modification on : Tuesday, February 19, 2019 - 3:40:04 PM
Long-term archiving on : Thursday, December 28, 2017 - 12:30:31 PM

File

VoigtLICS.pdf
Files produced by the author(s)

Identifiers

Collections

Citation

Marco Voigt. A fine-grained hierarchy of hard problems in the separated fragment. LICS 2017 - 32nd Annual ACM/IEEE Symposium on Logic in Computer Science, Jun 2017, Reykjavik, Iceland. pp.1 - 12, ⟨10.1109/LICS.2017.8005094⟩. ⟨hal-01592172⟩

Share

Metrics

Record views

342

Files downloads

100