253,673 research outputs found
IASCAR: Incremental Answer Set Counting by Anytime Refinement
Answer set programming (ASP) is a popular declarative programming paradigm
with various applications. Programs can easily have many answer sets that
cannot be enumerated in practice, but counting still allows quantifying
solution spaces. If one counts under assumptions on literals, one obtains a
tool to comprehend parts of the solution space, so-called answer set
navigation. However, navigating through parts of the solution space requires
counting many times, which is expensive in theory. Knowledge compilation
compiles instances into representations on which counting works in polynomial
time. However, these techniques exist only for CNF formulas, and compiling ASP
programs into CNF formulas can introduce an exponential overhead. This paper
introduces a technique to iteratively count answer sets under assumptions on
knowledge compilations of CNFs that encode supported models. Our anytime
technique uses the inclusion-exclusion principle to improve bounds by over- and
undercounting systematically. In a preliminary empirical analysis, we
demonstrate promising results. After compiling the input (offline phase), our
approach quickly (re)counts.Comment: Under consideration in Theory and Practice of Logic Programming
(TPLP
An Abstraction of Whitney's Broken Circuit Theorem
We establish a broad generalization of Whitney's broken circuit theorem on
the chromatic polynomial of a graph to sums of type
where is a finite set and is a mapping from the power set of into
an abelian group. We give applications to the domination polynomial and the
subgraph component polynomial of a graph, the chromatic polynomial of a
hypergraph, the characteristic polynomial and Crapo's beta invariant of a
matroid, and the principle of inclusion-exclusion. Thus, we discover several
known and new results in a concise and unified way. As further applications of
our main result, we derive a new generalization of the maximums-minimums
identity and of a theorem due to Blass and Sagan on the M\"obius function of a
finite lattice, which generalizes Rota's crosscut theorem. For the classical
M\"obius function, both Euler's totient function and its Dirichlet inverse, and
the reciprocal of the Riemann zeta function we obtain new expansions involving
the greatest common divisor resp. least common multiple. We finally establish
an even broader generalization of Whitney's broken circuit theorem in the
context of convex geometries (antimatroids).Comment: 18 page
On the asymptotic magnitude of subsets of Euclidean space
Magnitude is a canonical invariant of finite metric spaces which has its
origins in category theory; it is analogous to cardinality of finite sets.
Here, by approximating certain compact subsets of Euclidean space with finite
subsets, the magnitudes of line segments, circles and Cantor sets are defined
and calculated. It is observed that asymptotically these satisfy the
inclusion-exclusion principle, relating them to intrinsic volumes of polyconvex
sets.Comment: 23 pages. Version 2: updated to reflect more recent work, in
particular, the approximation method is now known to calculate (rather than
merely define) the magnitude; also minor alterations such as references adde
Faster exponential-time algorithms in graphs of bounded average degree
We first show that the Traveling Salesman Problem in an n-vertex graph with
average degree bounded by d can be solved in O*(2^{(1-\eps_d)n}) time and
exponential space for a constant \eps_d depending only on d, where the
O*-notation suppresses factors polynomial in the input size. Thus, we
generalize the recent results of Bjorklund et al. [TALG 2012] on graphs of
bounded degree.
Then, we move to the problem of counting perfect matchings in a graph. We
first present a simple algorithm for counting perfect matchings in an n-vertex
graph in O*(2^{n/2}) time and polynomial space; our algorithm matches the
complexity bounds of the algorithm of Bjorklund [SODA 2012], but relies on
inclusion-exclusion principle instead of algebraic transformations. Building
upon this result, we show that the number of perfect matchings in an n-vertex
graph with average degree bounded by d can be computed in
O*(2^{(1-\eps_{2d})n/2}) time and exponential space, where \eps_{2d} is the
constant obtained by us for the Traveling Salesman Problem in graphs of average
degree at most 2d.
Moreover we obtain a simple algorithm that counts the number of perfect
matchings in an n-vertex bipartite graph of average degree at most d in
O*(2^{(1-1/(3.55d))n/2}) time, improving and simplifying the recent result of
Izumi and Wadayama [FOCS 2012].Comment: 10 page
Symmetric inclusion-exclusion
One form of the inclusion-exclusion principle asserts that if A and B are
functions of finite sets then A(S) is the sum of B(T) over all subsets T of S
if and only if B(S) is the sum of (-1)^|S-T| A(T) over all subsets T of S.
If we replace B(S) with (-1)^|S| B(S), we get a symmetric form of
inclusion-exclusion: A(S) is the sum of (-1)^|T| B(T) over all subsets T of S
if and only if B(S) is the sum of (-1)^|T| A(T) over all subsets T of S.
We study instances of symmetric inclusion-exclusion in which the functions A
and B have combinatorial or probabilistic interpretations. In particular, we
study cases related to the Polya-Eggenberger urn model in which A(S) and B(S)
depend only on the cardinality of S.Comment: 10 page
Information as Distinctions: New Foundations for Information Theory
The logical basis for information theory is the newly developed logic of
partitions that is dual to the usual Boolean logic of subsets. The key concept
is a "distinction" of a partition, an ordered pair of elements in distinct
blocks of the partition. The logical concept of entropy based on partition
logic is the normalized counting measure of the set of distinctions of a
partition on a finite set--just as the usual logical notion of probability
based on the Boolean logic of subsets is the normalized counting measure of the
subsets (events). Thus logical entropy is a measure on the set of ordered
pairs, and all the compound notions of entropy (join entropy, conditional
entropy, and mutual information) arise in the usual way from the measure (e.g.,
the inclusion-exclusion principle)--just like the corresponding notions of
probability. The usual Shannon entropy of a partition is developed by replacing
the normalized count of distinctions (dits) by the average number of binary
partitions (bits) necessary to make all the distinctions of the partition
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