221,831 research outputs found
Backdoors to Normality for Disjunctive Logic Programs
Over the last two decades, propositional satisfiability (SAT) has become one
of the most successful and widely applied techniques for the solution of
NP-complete problems. The aim of this paper is to investigate theoretically how
Sat can be utilized for the efficient solution of problems that are harder than
NP or co-NP. In particular, we consider the fundamental reasoning problems in
propositional disjunctive answer set programming (ASP), Brave Reasoning and
Skeptical Reasoning, which ask whether a given atom is contained in at least
one or in all answer sets, respectively. Both problems are located at the
second level of the Polynomial Hierarchy and thus assumed to be harder than NP
or co-NP. One cannot transform these two reasoning problems into SAT in
polynomial time, unless the Polynomial Hierarchy collapses. We show that
certain structural aspects of disjunctive logic programs can be utilized to
break through this complexity barrier, using new techniques from Parameterized
Complexity. In particular, we exhibit transformations from Brave and Skeptical
Reasoning to SAT that run in time O(2^k n^2) where k is a structural parameter
of the instance and n the input size. In other words, the reduction is
fixed-parameter tractable for parameter k. As the parameter k we take the size
of a smallest backdoor with respect to the class of normal (i.e.,
disjunction-free) programs. Such a backdoor is a set of atoms that when deleted
makes the program normal. In consequence, the combinatorial explosion, which is
expected when transforming a problem from the second level of the Polynomial
Hierarchy to the first level, can now be confined to the parameter k, while the
running time of the reduction is polynomial in the input size n, where the
order of the polynomial is independent of k.Comment: A short version will appear in the Proceedings of the Proceedings of
the 27th AAAI Conference on Artificial Intelligence (AAAI'13). A preliminary
version of the paper was presented on the workshop Answer Set Programming and
Other Computing Paradigms (ASPOCP 2012), 5th International Workshop,
September 4, 2012, Budapest, Hungar
A new look at nonnegativity on closed sets and polynomial optimization
We first show that a continuous function f is nonnegative on a closed set
if and only if (countably many) moment matrices of some signed
measure with support equal to K, are all positive semidefinite
(if is compact is an arbitrary finite Borel measure with support
equal to K. In particular, we obtain a convergent explicit hierarchy of
semidefinite (outer) approximations with {\it no} lifting, of the cone of
nonnegative polynomials of degree at most . Wen used in polynomial
optimization on certain simple closed sets \K (like e.g., the whole space
, the positive orthant, a box, a simplex, or the vertices of the
hypercube), it provides a nonincreasing sequence of upper bounds which
converges to the global minimum by solving a hierarchy of semidefinite programs
with only one variable. This convergent sequence of upper bounds complements
the convergent sequence of lower bounds obtained by solving a hierarchy of
semidefinite relaxations
Ramsey's Theorem for Pairs and Colors as a Sub-Classical Principle of Arithmetic
The purpose is to study the strength of Ramsey's Theorem for pairs restricted
to recursive assignments of -many colors, with respect to Intuitionistic
Heyting Arithmetic. We prove that for every natural number , Ramsey's
Theorem for pairs and recursive assignments of colors is equivalent to the
Limited Lesser Principle of Omniscience for formulas over Heyting
Arithmetic. Alternatively, the same theorem over intuitionistic arithmetic is
equivalent to: for every recursively enumerable infinite -ary tree there is
some and some branch with infinitely many children of index .Comment: 17 page
A bounded degree SOS hierarchy for polynomial optimization
We consider a new hierarchy of semidefinite relaxations for the general
polynomial optimization problem on a
compact basic semi-algebraic set . This hierarchy combines some
advantages of the standard LP-relaxations associated with Krivine's positivity
certificate and some advantages of the standard SOS-hierarchy. In particular it
has the following attractive features: (a) In contrast to the standard
SOS-hierarchy, for each relaxation in the hierarchy, the size of the matrix
associated with the semidefinite constraint is the same and fixed in advance by
the user. (b) In contrast to the LP-hierarchy, finite convergence occurs at the
first step of the hierarchy for an important class of convex problems. Finally
(c) some important techniques related to the use of point evaluations for
declaring a polynomial to be zero and to the use of rank-one matrices make an
efficient implementation possible. Preliminary results on a sample of non
convex problems are encouraging
A hierarchy of eigencomputations for polynomial optimization on the sphere
We introduce a convergent hierarchy of lower bounds on the minimum value of a
real homogeneous polynomial over the sphere. The main practical advantage of
our hierarchy over the sum-of-squares (SOS) hierarchy is that the lower bound
at each level of our hierarchy is obtained by a minimum eigenvalue computation,
as opposed to the full semidefinite program (SDP) required at each level of
SOS. In practice, this allows us to go to much higher levels than are
computationally feasible for the SOS hierarchy. For both hierarchies, the
underlying space at the -th level is the set of homogeneous polynomials of
degree . We prove that our hierarchy converges as in the level
, matching the best-known convergence of the SOS hierarchy when the number
of variables is less than the half-degree (the best-known convergence
of SOS when is ). More generally, we introduce a
convergent hierarchy of minimum eigenvalue computations for minimizing the
inner product between a real tensor and an element of the spherical
Segre-Veronese variety, with similar convergence guarantees. As examples, we
obtain hierarchies for computing the (real) tensor spectral norm, and for
minimizing biquadratic forms over the sphere. Hierarchies of eigencomputations
for more general constrained polynomial optimization problems are discussed.Comment: 31 pages. Comments welcome
On Modal {\mu}-Calculus over Finite Graphs with Bounded Strongly Connected Components
For every positive integer k we consider the class SCCk of all finite graphs
whose strongly connected components have size at most k. We show that for every
k, the Modal mu-Calculus fixpoint hierarchy on SCCk collapses to the level
Delta2, but not to Comp(Sigma1,Pi1) (compositions of formulas of level Sigma1
and Pi1). This contrasts with the class of all graphs, where
Delta2=Comp(Sigma1,Pi1)
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