1,856 research outputs found
Probabilities on Sentences in an Expressive Logic
Automated reasoning about uncertain knowledge has many applications. One
difficulty when developing such systems is the lack of a completely
satisfactory integration of logic and probability. We address this problem
directly. Expressive languages like higher-order logic are ideally suited for
representing and reasoning about structured knowledge. Uncertain knowledge can
be modeled by using graded probabilities rather than binary truth-values. The
main technical problem studied in this paper is the following: Given a set of
sentences, each having some probability of being true, what probability should
be ascribed to other (query) sentences? A natural wish-list, among others, is
that the probability distribution (i) is consistent with the knowledge base,
(ii) allows for a consistent inference procedure and in particular (iii)
reduces to deductive logic in the limit of probabilities being 0 and 1, (iv)
allows (Bayesian) inductive reasoning and (v) learning in the limit and in
particular (vi) allows confirmation of universally quantified
hypotheses/sentences. We translate this wish-list into technical requirements
for a prior probability and show that probabilities satisfying all our criteria
exist. We also give explicit constructions and several general
characterizations of probabilities that satisfy some or all of the criteria and
various (counter) examples. We also derive necessary and sufficient conditions
for extending beliefs about finitely many sentences to suitable probabilities
over all sentences, and in particular least dogmatic or least biased ones. We
conclude with a brief outlook on how the developed theory might be used and
approximated in autonomous reasoning agents. Our theory is a step towards a
globally consistent and empirically satisfactory unification of probability and
logic.Comment: 52 LaTeX pages, 64 definiton/theorems/etc, presented at conference
Progic 2011 in New Yor
A structural approach to kernels for ILPs: Treewidth and Total Unimodularity
Kernelization is a theoretical formalization of efficient preprocessing for
NP-hard problems. Empirically, preprocessing is highly successful in practice,
for example in state-of-the-art ILP-solvers like CPLEX. Motivated by this,
previous work studied the existence of kernelizations for ILP related problems,
e.g., for testing feasibility of Ax <= b. In contrast to the observed success
of CPLEX, however, the results were largely negative. Intuitively, practical
instances have far more useful structure than the worst-case instances used to
prove these lower bounds.
In the present paper, we study the effect that subsystems with (Gaifman graph
of) bounded treewidth or totally unimodularity have on the kernelizability of
the ILP feasibility problem. We show that, on the positive side, if these
subsystems have a small number of variables on which they interact with the
remaining instance, then we can efficiently replace them by smaller subsystems
of size polynomial in the domain without changing feasibility. Thus, if large
parts of an instance consist of such subsystems, then this yields a substantial
size reduction. We complement this by proving that relaxations to the
considered structures, e.g., larger boundaries of the subsystems, allow
worst-case lower bounds against kernelization. Thus, these relaxed structures
can be used to build instance families that cannot be efficiently reduced, by
any approach.Comment: Extended abstract in the Proceedings of the 23rd European Symposium
on Algorithms (ESA 2015
Successor-Invariant First-Order Logic on Graphs with Excluded Topological Subgraphs
We show that the model-checking problem for successor-invariant first-order
logic is fixed-parameter tractable on graphs with excluded topological
subgraphs when parameterised by both the size of the input formula and the size
of the exluded topological subgraph. Furthermore, we show that model-checking
for order-invariant first-order logic is tractable on coloured posets of
bounded width, parameterised by both the size of the input formula and the
width of the poset.
Our result for successor-invariant FO extends previous results for this logic
on planar graphs (Engelmann et al., LICS 2012) and graphs with excluded minors
(Eickmeyer et al., LICS 2013), further narrowing the gap between what is known
for FO and what is known for successor-invariant FO. The proof uses Grohe and
Marx's structure theorem for graphs with excluded topological subgraphs. For
order-invariant FO we show that Gajarsk\'y et al.'s recent result for FO
carries over to order-invariant FO
Distance Constraint Satisfaction Problems
We study the complexity of constraint satisfaction problems for templates
that are first-order definable in , the integers with
the successor relation. Assuming a widely believed conjecture from finite
domain constraint satisfaction (we require the tractability conjecture by
Bulatov, Jeavons and Krokhin in the special case of transitive finite
templates), we provide a full classification for the case that Gamma is locally
finite (i.e., the Gaifman graph of has finite degree). We show that
one of the following is true: The structure Gamma is homomorphically equivalent
to a structure with a d-modular maximum or minimum polymorphism and
can be solved in polynomial time, or is
homomorphically equivalent to a finite transitive structure, or
is NP-complete.Comment: 35 pages, 2 figure
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