5,289 research outputs found
Abduction in Well-Founded Semantics and Generalized Stable Models
Abductive logic programming offers a formalism to declaratively express and
solve problems in areas such as diagnosis, planning, belief revision and
hypothetical reasoning. Tabled logic programming offers a computational
mechanism that provides a level of declarativity superior to that of Prolog,
and which has supported successful applications in fields such as parsing,
program analysis, and model checking. In this paper we show how to use tabled
logic programming to evaluate queries to abductive frameworks with integrity
constraints when these frameworks contain both default and explicit negation.
The result is the ability to compute abduction over well-founded semantics with
explicit negation and answer sets. Our approach consists of a transformation
and an evaluation method. The transformation adjoins to each objective literal
in a program, an objective literal along with rules that ensure
that will be true if and only if is false. We call the resulting
program a {\em dual} program. The evaluation method, \wfsmeth, then operates on
the dual program. \wfsmeth{} is sound and complete for evaluating queries to
abductive frameworks whose entailment method is based on either the
well-founded semantics with explicit negation, or on answer sets. Further,
\wfsmeth{} is asymptotically as efficient as any known method for either class
of problems. In addition, when abduction is not desired, \wfsmeth{} operating
on a dual program provides a novel tabling method for evaluating queries to
ground extended programs whose complexity and termination properties are
similar to those of the best tabling methods for the well-founded semantics. A
publicly available meta-interpreter has been developed for \wfsmeth{} using the
XSB system.Comment: 48 pages; To appear in Theory and Practice in Logic Programmin
Deterministic and Probabilistic Binary Search in Graphs
We consider the following natural generalization of Binary Search: in a given
undirected, positively weighted graph, one vertex is a target. The algorithm's
task is to identify the target by adaptively querying vertices. In response to
querying a node , the algorithm learns either that is the target, or is
given an edge out of that lies on a shortest path from to the target.
We study this problem in a general noisy model in which each query
independently receives a correct answer with probability (a
known constant), and an (adversarial) incorrect one with probability .
Our main positive result is that when (i.e., all answers are
correct), queries are always sufficient. For general , we give an
(almost information-theoretically optimal) algorithm that uses, in expectation,
no more than queries, and identifies the target correctly with probability at
leas . Here, denotes the
entropy. The first bound is achieved by the algorithm that iteratively queries
a 1-median of the nodes not ruled out yet; the second bound by careful repeated
invocations of a multiplicative weights algorithm.
Even for , we show several hardness results for the problem of
determining whether a target can be found using queries. Our upper bound of
implies a quasipolynomial-time algorithm for undirected connected
graphs; we show that this is best-possible under the Strong Exponential Time
Hypothesis (SETH). Furthermore, for directed graphs, or for undirected graphs
with non-uniform node querying costs, the problem is PSPACE-complete. For a
semi-adaptive version, in which one may query nodes each in rounds, we
show membership in in the polynomial hierarchy, and hardness
for
Finding Cycles and Trees in Sublinear Time
We present sublinear-time (randomized) algorithms for finding simple cycles
of length at least and tree-minors in bounded-degree graphs. The
complexity of these algorithms is related to the distance of the graph from
being -minor-free (resp., free from having the corresponding tree-minor).
In particular, if the graph is far (i.e., -far) {from} being
cycle-free, i.e. if one has to delete a constant fraction of edges to make it
cycle-free, then the algorithm finds a cycle of polylogarithmic length in time
\tildeO(\sqrt{N}), where denotes the number of vertices. This time
complexity is optimal up to polylogarithmic factors.
The foregoing results are the outcome of our study of the complexity of {\em
one-sided error} property testing algorithms in the bounded-degree graphs
model. For example, we show that cycle-freeness of -vertex graphs can be
tested with one-sided error within time complexity
\tildeO(\poly(1/\e)\cdot\sqrt{N}). This matches the known
query lower bound, and contrasts with the fact that any minor-free property
admits a {\em two-sided error} tester of query complexity that only depends on
the proximity parameter \e. For any constant , we extend this result
to testing whether the input graph has a simple cycle of length at least .
On the other hand, for any fixed tree , we show that -minor-freeness has
a one-sided error tester of query complexity that only depends on the proximity
parameter \e.
Our algorithm for finding cycles in bounded-degree graphs extends to general
graphs, where distances are measured with respect to the actual number of
edges. Such an extension is not possible with respect to finding tree-minors in
complexity.Comment: Keywords: Sublinear-Time Algorithms, Property Testing, Bounded-Degree
Graphs, One-Sided vs Two-Sided Error Probability Updated versio
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