12,606 research outputs found
A Fine-Grained Hierarchy of Hard Problems in the Separated Fragment
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
in which is quantifier free, satisfiability can be decided in
nondeterministic -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 -NEXPTIME-completeness of the
satisfiability problem for the set of all SF sentences that have
degree 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.Comment: Full version of the LICS 2017 extended abstract having the same
title, 38 page
Tableau-based decision procedure for the multi-agent epistemic logic with operators of common and distributed knowledge
We develop an incremental-tableau-based decision procedure for the
multi-agent epistemic logic MAEL(CD) (aka S5_n (CD)), whose language contains
operators of individual knowledge for a finite set Ag of agents, as well as
operators of distributed and common knowledge among all agents in Ag. Our
tableau procedure works in (deterministic) exponential time, thus establishing
an upper bound for MAEL(CD)-satisfiability that matches the (implicit)
lower-bound known from earlier results, which implies ExpTime-completeness of
MAEL(CD)-satisfiability. Therefore, our procedure provides a complexity-optimal
algorithm for checking MAEL(CD)-satisfiability, which, however, in most cases
is much more efficient. We prove soundness and completeness of the procedure,
and illustrate it with an example.Comment: To appear in the Proceedings of the 6th IEEE Conference on Software
Engineering and Formal Methods (SEFM 2008
Upper and Lower Bounds for Weak Backdoor Set Detection
We obtain upper and lower bounds for running times of exponential time
algorithms for the detection of weak backdoor sets of 3CNF formulas,
considering various base classes. These results include (omitting polynomial
factors), (i) a 4.54^k algorithm to detect whether there is a weak backdoor set
of at most k variables into the class of Horn formulas; (ii) a 2.27^k algorithm
to detect whether there is a weak backdoor set of at most k variables into the
class of Krom formulas. These bounds improve an earlier known bound of 6^k. We
also prove a 2^k lower bound for these problems, subject to the Strong
Exponential Time Hypothesis.Comment: A short version will appear in the proceedings of the 16th
International Conference on Theory and Applications of Satisfiability Testin
Minimizing energy below the glass thresholds
Focusing on the optimization version of the random K-satisfiability problem,
the MAX-K-SAT problem, we study the performance of the finite energy version of
the Survey Propagation (SP) algorithm. We show that a simple (linear time)
backtrack decimation strategy is sufficient to reach configurations well below
the lower bound for the dynamic threshold energy and very close to the analytic
prediction for the optimal ground states. A comparative numerical study on one
of the most efficient local search procedures is also given.Comment: 12 pages, submitted to Phys. Rev. E, accepted for publicatio
Another look at graph coloring via propositional satisfiability
AbstractThis paper studies the solution of graph coloring problems by encoding into propositional satisfiability problems. The study covers three kinds of satisfiability solvers, based on postorder reasoning (e.g., grasp, chaff), preorder reasoning (e.g., 2cl, 2clsEq), and back-chaining (modoc). The study evaluates three encodings, one of them believed to be new. Some new symmetry-breaking methods, specific to coloring, are used to reduce the redundancy of solutions. A by-product of this research is an implemented lower-bound technique that has shown improved lower bounds for the chromatic numbers of the long-standing unsolved random graphs known as DSJC125.5 and DSJC125.9. Independent-set analysis shows that the chromatic numbers of DSJC125.5 and DSJC125.9 are at least 18 and 40, respectively, but satisfiability encoding was able to demonstrate only that the chromatic numbers are at least 13 and 38, respectively, within available time and space
A Lower Bound of Conditional Branches for Boolean Satisfiability on Post Machines
We establish a lower bound of conditional branches for deciding the
satisfiability of the conjunction of any two Boolean formulas from a set called
a full representation of Boolean functions of variables - a set containing
a Boolean formula to represent each Boolean function of variables. The
contradiction proof first assumes that there exists a Post machine (Post's
Formulation 1) that correctly decides the satisfiability of the conjunction of
any two Boolean formulas from such a set by following an execution path that
includes fewer than conditional branches. By using multiple runs of this
Post machine, with one run for each Boolean function of variables, the
proof derives a contradiction by showing that this Post machine is unable to
correctly decide the satisfiability of the conjunction of at least one pair of
Boolean formulas from a full representation of -variable Boolean functions
if the machine executes fewer than conditional branches. This lower bound
of conditional branches holds for any full representation of Boolean
functions of variables, even if a full representation consists solely of
minimized Boolean formulas derived by a Boolean minimization method. We discuss
why the lower bound fails to hold for satisfiability of certain restricted
formulas, such as 2CNF satisfiability, XOR-SAT, and HORN-SAT. We also relate
the lower bound to 3CNF satisfiability. The lower bound does not depend on
sequentiality of access to the boxes in the symbol space and will hold even if
a machine is capable of non-sequential access.Comment: This article draws heavily from arXiv:1406.597
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