17,111 research outputs found
Algorithms and lower bounds for de Morgan formulas of low-communication leaf gates
The class consists of Boolean functions
computable by size- de Morgan formulas whose leaves are any Boolean
functions from a class . We give lower bounds and (SAT, Learning,
and PRG) algorithms for , for classes
of functions with low communication complexity. Let
be the maximum -party NOF randomized communication
complexity of . We show:
(1) The Generalized Inner Product function cannot be computed in
on more than fraction of inputs
for As a corollary, we get an average-case lower bound for
against .
(2) There is a PRG of seed length that -fools . For
, we get the better seed length . This gives the first
non-trivial PRG (with seed length ) for intersections of half-spaces
in the regime where .
(3) There is a randomized -time SAT algorithm for , where In particular, this implies a nontrivial
#SAT algorithm for .
(4) The Minimum Circuit Size Problem is not in .
On the algorithmic side, we show that can be
PAC-learned in time
On the Succinctness of Query Rewriting over OWL 2 QL Ontologies with Shallow Chases
We investigate the size of first-order rewritings of conjunctive queries over
OWL 2 QL ontologies of depth 1 and 2 by means of hypergraph programs computing
Boolean functions. Both positive and negative results are obtained. Conjunctive
queries over ontologies of depth 1 have polynomial-size nonrecursive datalog
rewritings; tree-shaped queries have polynomial positive existential
rewritings; however, in the worst case, positive existential rewritings can
only be of superpolynomial size. Positive existential and nonrecursive datalog
rewritings of queries over ontologies of depth 2 suffer an exponential blowup
in the worst case, while first-order rewritings are superpolynomial unless
. We also analyse rewritings of
tree-shaped queries over arbitrary ontologies and observe that the query
entailment problem for such queries is fixed-parameter tractable
Complexity of Propositional Proofs under a Promise
We study -- within the framework of propositional proof complexity -- the
problem of certifying unsatisfiability of CNF formulas under the promise that
any satisfiable formula has many satisfying assignments, where ``many'' stands
for an explicitly specified function \Lam in the number of variables . To
this end, we develop propositional proof systems under different measures of
promises (that is, different \Lam) as extensions of resolution. This is done
by augmenting resolution with axioms that, roughly, can eliminate sets of truth
assignments defined by Boolean circuits. We then investigate the complexity of
such systems, obtaining an exponential separation in the average-case between
resolution under different size promises:
1. Resolution has polynomial-size refutations for all unsatisfiable 3CNF
formulas when the promise is \eps\cd2^n, for any constant 0<\eps<1.
2. There are no sub-exponential size resolution refutations for random 3CNF
formulas, when the promise is (and the number of clauses is
), for any constant .Comment: 32 pages; a preliminary version appeared in the Proceedings of
ICALP'0
Dichotomy Results for Fixed-Point Existence Problems for Boolean Dynamical Systems
A complete classification of the computational complexity of the fixed-point
existence problem for boolean dynamical systems, i.e., finite discrete
dynamical systems over the domain {0, 1}, is presented. For function classes F
and graph classes G, an (F, G)-system is a boolean dynamical system such that
all local transition functions lie in F and the underlying graph lies in G. Let
F be a class of boolean functions which is closed under composition and let G
be a class of graphs which is closed under taking minors. The following
dichotomy theorems are shown: (1) If F contains the self-dual functions and G
contains the planar graphs then the fixed-point existence problem for (F,
G)-systems with local transition function given by truth-tables is NP-complete;
otherwise, it is decidable in polynomial time. (2) If F contains the self-dual
functions and G contains the graphs having vertex covers of size one then the
fixed-point existence problem for (F, G)-systems with local transition function
given by formulas or circuits is NP-complete; otherwise, it is decidable in
polynomial time.Comment: 17 pages; this version corrects an error/typo in the 2008/01/24
versio
Faster Query Answering in Probabilistic Databases using Read-Once Functions
A boolean expression is in read-once form if each of its variables appears
exactly once. When the variables denote independent events in a probability
space, the probability of the event denoted by the whole expression in
read-once form can be computed in polynomial time (whereas the general problem
for arbitrary expressions is #P-complete). Known approaches to checking
read-once property seem to require putting these expressions in disjunctive
normal form. In this paper, we tell a better story for a large subclass of
boolean event expressions: those that are generated by conjunctive queries
without self-joins and on tuple-independent probabilistic databases. We first
show that given a tuple-independent representation and the provenance graph of
an SPJ query plan without self-joins, we can, without using the DNF of a result
event expression, efficiently compute its co-occurrence graph. From this, the
read-once form can already, if it exists, be computed efficiently using
existing techniques. Our second and key contribution is a complete, efficient,
and simple to implement algorithm for computing the read-once forms (whenever
they exist) directly, using a new concept, that of co-table graph, which can be
significantly smaller than the co-occurrence graph.Comment: Accepted in ICDT 201
On the succinctness of query rewriting over shallow ontologies
We investigate the succinctness problem for conjunctive query rewritings over OWL2QL ontologies of depth 1 and 2 by means of hypergraph programs computing Boolean functions. Both positive and negative results are obtained. We show that, over ontologies of depth 1, conjunctive queries have polynomial-size nonrecursive datalog rewritings; tree-shaped queries have polynomial positive existential rewritings; however, in the worst case, positive existential rewritings can be superpolynomial. Over ontologies of depth 2, positive existential and nonrecursive datalog rewritings of conjunctive queries can suffer an exponential blowup, while first-order rewritings can be superpolynomial unless NP ïżœis included in P/poly. We also analyse rewritings of tree-shaped queries over arbitrary ontologies and note that query entailment for such queries is fixed-parameter tractable
Circuit Complexity Meets Ontology-Based Data Access
Ontology-based data access is an approach to organizing access to a database
augmented with a logical theory. In this approach query answering proceeds
through a reformulation of a given query into a new one which can be answered
without any use of theory. Thus the problem reduces to the standard database
setting.
However, the size of the query may increase substantially during the
reformulation. In this survey we review a recently developed framework on
proving lower and upper bounds on the size of this reformulation by employing
methods and results from Boolean circuit complexity.Comment: To appear in proceedings of CSR 2015, LNCS 9139, Springe
Nondeterminism and an abstract formulation of Ne\v{c}iporuk's lower bound method
A formulation of "Ne\v{c}iporuk's lower bound method" slightly more inclusive
than the usual complexity-measure-specific formulation is presented. Using this
general formulation, limitations to lower bounds achievable by the method are
obtained for several computation models, such as branching programs and Boolean
formulas having access to a sublinear number of nondeterministic bits. In
particular, it is shown that any lower bound achievable by the method of
Ne\v{c}iporuk for the size of nondeterministic and parity branching programs is
at most
Lower Bounds for (Non-Monotone) Comparator Circuits
Comparator circuits are a natural circuit model for studying the concept of bounded fan-out computations, which intuitively corresponds to whether or not a computational model can make "copies" of intermediate computational steps. Comparator circuits are believed to be weaker than general Boolean circuits, but they can simulate Branching Programs and Boolean formulas. In this paper we prove the first superlinear lower bounds in the general (non-monotone) version of this model for an explicitly defined function. More precisely, we prove that the n-bit Element Distinctness function requires ?((n/ log n)^(3/2)) size comparator circuits
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