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Computer Science, Logic in Computer Science, cs.LO
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 $\exists \vec{z} \forall
\vec{x}_1 \exists \vec{y}_1 \ldots \forall \vec{x}_n \exists \vec{y}_n . \psi$
in which $\psi$ is quantifier free, satisfiability can be decided in
nondeterministic $n$-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 $SF_{\leq k}$ 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.
