9,450 research outputs found
The Algorithmic Complexity of Modular Decomposition
Modular decomposition is a thoroughly investigated topic inmany areas such as switching theory, reliability theory, game theory andgraph theory. We propose an O(mn)-algorithm for the recognition of amodular set of a monotone Boolean function f with m prime implicantsand n variables. Using this result we show that the computation ofthe modular closure of a set can be done in time O(mn2). On the otherhand, we prove that the recognition problem for general Boolean functions is NP-complete. Moreover, we introduce the so called generalizedShannon decomposition of a Boolean functions as an efficient tool forproving theorems on Boolean function decompositions.computational complexity;Boolean functions;decomposition algorithm;modular decomposition;substitution decomposition
Efficient enumeration of solutions produced by closure operations
In this paper we address the problem of generating all elements obtained by
the saturation of an initial set by some operations. More precisely, we prove
that we can generate the closure of a boolean relation (a set of boolean
vectors) by polymorphisms with a polynomial delay. Therefore we can compute
with polynomial delay the closure of a family of sets by any set of "set
operations": union, intersection, symmetric difference, subsets, supersets
). To do so, we study the problem: for a set
of operations , decide whether an element belongs to the closure
by of a family of elements. In the boolean case, we prove that
is in P for any set of boolean operations
. When the input vectors are over a domain larger than two
elements, we prove that the generic enumeration method fails, since
is NP-hard for some . We also study the
problem of generating minimal or maximal elements of closures and prove that
some of them are related to well known enumeration problems such as the
enumeration of the circuits of a matroid or the enumeration of maximal
independent sets of a hypergraph. This article improves on previous works of
the same authors.Comment: 30 pages, 1 figure. Long version of the article arXiv:1509.05623 of
the same name which appeared in STACS 2016. Final version for DMTCS journa
The Algorithmic Complexity of Modular Decomposition
Modular decomposition is a thoroughly investigated topic in
many areas such as switching theory, reliability theory, game theory and
graph theory. We propose an O(mn)-algorithm for the recognition of a
modular set of a monotone Boolean function f with m prime implicants
and n variables. Using this result we show that the computation of
the modular closure of a set can be done in time O(mn2). On the other
hand, we prove that the recognition problem for general Boolean func
tions is NP-complete. Moreover, we introduce the so called generalized
Shannon decomposition of a Boolean functions as an efficient tool for
proving theorems on Boolean function decompositions
On implicational bases of closure systems with unique critical sets
We show that every optimum basis of a finite closure system, in D.Maier's
sense, is also right-side optimum, which is a parameter of a minimum CNF
representation of a Horn Boolean function. New parameters for the size of the
binary part are also established. We introduce a K-basis of a general closure
system, which is a refinement of the canonical basis of Duquenne and Guigues,
and discuss a polynomial algorithm to obtain it. We study closure systems with
the unique criticals and some of its subclasses, where the K-basis is unique. A
further refinement in the form of the E-basis is possible for closure systems
without D-cycles. There is a polynomial algorithm to recognize the D-relation
from a K-basis. Thus, closure systems without D-cycles can be effectively
recognized. While E-basis achieves an optimum in one of its parts, the
optimization of the others is an NP-complete problem.Comment: Presented on International Symposium of Artificial Intelligence and
Mathematics (ISAIM-2012), Ft. Lauderdale, FL, USA Results are included into
plenary talk on conference Universal Algebra and Lattice Theory, June 2012,
Szeged, Hungary 29 pages and 2 figure
Simple universal models capture all classical spin physics
Spin models are used in many studies of complex systems---be it condensed
matter physics, neural networks, or economics---as they exhibit rich
macroscopic behaviour despite their microscopic simplicity.
Here we prove that all the physics of every classical spin model is
reproduced in the low-energy sector of certain `universal models'.
This means that (i) the low energy spectrum of the universal model reproduces
the entire spectrum of the original model to any desired precision, (ii) the
corresponding spin configurations of the original model are also reproduced in
the universal model, (iii) the partition function is approximated to any
desired precision, and (iv) the overhead in terms of number of spins and
interactions is at most polynomial.
This holds for classical models with discrete or continuous degrees of
freedom.
We prove necessary and sufficient conditions for a spin model to be
universal, and show that one of the simplest and most widely studied spin
models, the 2D Ising model with fields, is universal.Comment: v1: 4 pages with 2 figures (main text) + 4 pages with 3 figures
(supplementary info). v2: 12 pages with 3 figures (main text) + 35 pages with
6 figures (supplementary info) (all single column). v2 contains new results
and major revisions (results for spin models with continuous degrees of
freedom, explicit constructions, examples...). Close to published version.
v3: minor typo correcte
Boolean Operations, Joins, and the Extended Low Hierarchy
We prove that the join of two sets may actually fall into a lower level of
the extended low hierarchy than either of the sets. In particular, there exist
sets that are not in the second level of the extended low hierarchy, EL_2, yet
their join is in EL_2. That is, in terms of extended lowness, the join operator
can lower complexity. Since in a strong intuitive sense the join does not lower
complexity, our result suggests that the extended low hierarchy is unnatural as
a complexity measure. We also study the closure properties of EL_ and prove
that EL_2 is not closed under certain Boolean operations. To this end, we
establish the first known (and optimal) EL_2 lower bounds for certain notions
generalizing Selman's P-selectivity, which may be regarded as an interesting
result in its own right.Comment: 12 page
Circuit complexity, proof complexity, and polynomial identity testing
We introduce a new algebraic proof system, which has tight connections to
(algebraic) circuit complexity. In particular, we show that any
super-polynomial lower bound on any Boolean tautology in our proof system
implies that the permanent does not have polynomial-size algebraic circuits
(VNP is not equal to VP). As a corollary to the proof, we also show that
super-polynomial lower bounds on the number of lines in Polynomial Calculus
proofs (as opposed to the usual measure of number of monomials) imply the
Permanent versus Determinant Conjecture. Note that, prior to our work, there
was no proof system for which lower bounds on an arbitrary tautology implied
any computational lower bound.
Our proof system helps clarify the relationships between previous algebraic
proof systems, and begins to shed light on why proof complexity lower bounds
for various proof systems have been so much harder than lower bounds on the
corresponding circuit classes. In doing so, we highlight the importance of
polynomial identity testing (PIT) for understanding proof complexity.
More specifically, we introduce certain propositional axioms satisfied by any
Boolean circuit computing PIT. We use these PIT axioms to shed light on
AC^0[p]-Frege lower bounds, which have been open for nearly 30 years, with no
satisfactory explanation as to their apparent difficulty. We show that either:
a) Proving super-polynomial lower bounds on AC^0[p]-Frege implies VNP does not
have polynomial-size circuits of depth d - a notoriously open question for d at
least 4 - thus explaining the difficulty of lower bounds on AC^0[p]-Frege, or
b) AC^0[p]-Frege cannot efficiently prove the depth d PIT axioms, and hence we
have a lower bound on AC^0[p]-Frege.
Using the algebraic structure of our proof system, we propose a novel way to
extend techniques from algebraic circuit complexity to prove lower bounds in
proof complexity
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