31 research outputs found
More on Descriptive Complexity of Second-Order HORN Logics
This paper concerns Gradel's question asked in 1992: whether all problems
which are in PTIME and closed under substructures are definable in second-order
HORN logic SO-HORN. We introduce revisions of SO-HORN and DATALOG by adding
first-order universal quantifiers over the second-order atoms in the bodies of
HORN clauses and DATALOG rules. We show that both logics are as expressive as
FO(LFP), the least fixed point logic. We also prove that FO(LFP) can not define
all of the problems that are in PTIME and closed under substructures. As a
corollary, we answer Gradel's question negatively
NP-Logic Systems and Model-Equivalence Reductions
In this paper we investigate the existence of model-equivalence reduction
between NP-logic systems which are logic systems with model existence problem
in NP. It is shown that among all NP-systems with model checking problem in NP,
the existentially quantified propositional logic (\exists PF) is maximal with
respect to poly-time model-equivalent reduction. However, \exists PF seems not
a maximal NP-system in general because there exits a NP-system with model
checking problem D^P-complete
A Logic that Captures P on Ordered Structures
We extend the inflationary fixed-point logic, IFP, with a new kind of
second-order quantifiers which have (poly-)logarithmic bounds. We prove that on
ordered structures the new logic captures
the limited nondeterminism class . In order to study its
expressive power, we also design a new version of Ehrenfeucht-Fra\"iss\'e game
for this logic and show that our capturing result will not hold on the general
case, i.e. on all the finite structures.Comment: 15 pages. This article was reported with a title "Logarithmic-Bounded
Second-Order Quantifiers and Limited Nondeterminism" in National Conference
on Modern Logic 2019, on November 9 in Beijin
Jordan Areas and Grids
AbstractJordan curves can be used to represent special subsets of the Euclidean plane, either the (open) interior of the curve or the (compact) union of the interior and the curve itself. We compare the latter with other representations of compact sets using grids of points and we are able to show that knowing the length of a rectifiable curve is sufficient to translate from the grid representation to the Jordan curve
Linear CNF formulas and satisfiability
In this paper, we study {em linear} CNF formulas generalizing linear hypergraphs under combinatorial and complexity theoretical aspects w.r.t. SAT. We establish NP-completeness of SAT for the unrestricted linear formula class, and we show the equivalence of NP-completeness of restricted uniform linear formula classes w.r.t. SAT and the existence of unsatisfiable uniform linear witness formulas. On that basis we prove the NP-completeness of SAT for the uniform linear classes in a proof-theoretic manner by constructing however large-sized formulas. Interested in small witness formulas, we exhibit some combinatorial features of linear hypergraphs closely related to latin squares and finite projective planes helping to construct somehow dense, and significantly smaller unsatisfiable k -uniform linear formulas, at least for the cases k=3,4