2,207 research outputs found
The Connectivity of Boolean Satisfiability: No-Constants and Quantified Variants
For Boolean satisfiability problems, the structure of the solution space is
characterized by the solution graph, where the vertices are the solutions, and
two solutions are connected iff they differ in exactly one variable. Motivated
by research on heuristics and the satisfiability threshold, Gopalan et al. in
2006 studied connectivity properties of the solution graph and related
complexity issues for constraint satisfaction problems in Schaefer's framework.
They found dichotomies for the diameter of connected components and for the
complexity of the st-connectivity question, and conjectured a trichotomy for
the connectivity question that we recently were able to prove.
While Gopalan et al. considered CNF(S)-formulas with constants, we here look
at two important variants: CNF(S)-formulas without constants, and partially
quantified formulas. For the diameter and the st-connectivity question, we
prove dichotomies analogous to those of Gopalan et al. in these settings. While
we cannot give a complete classification for the connectivity problem yet, we
identify fragments where it is in P, where it is coNP-complete, and where it is
PSPACE-complete, in analogy to Gopalan et al.'s trichotomy.Comment: superseded by chapter 3 of arXiv:1510.0670
Existentially Restricted Quantified Constraint Satisfaction
The quantified constraint satisfaction problem (QCSP) is a powerful framework
for modelling computational problems. The general intractability of the QCSP
has motivated the pursuit of restricted cases that avoid its maximal
complexity. In this paper, we introduce and study a new model for investigating
QCSP complexity in which the types of constraints given by the existentially
quantified variables, is restricted. Our primary technical contribution is the
development and application of a general technology for proving positive
results on parameterizations of the model, of inclusion in the complexity class
coNP
A Fragment of Dependence Logic Capturing Polynomial Time
In this paper we study the expressive power of Horn-formulae in dependence
logic and show that they can express NP-complete problems. Therefore we define
an even smaller fragment D-Horn* and show that over finite successor structures
it captures the complexity class P of all sets decidable in polynomial time.
Furthermore we study the question which of our results can ge generalized to
the case of open formulae of D-Horn* and so-called downwards monotone
polynomial time properties of teams
Complexity of Nested Circumscription and Nested Abnormality Theories
The need for a circumscriptive formalism that allows for simple yet elegant
modular problem representation has led Lifschitz (AIJ, 1995) to introduce
nested abnormality theories (NATs) as a tool for modular knowledge
representation, tailored for applying circumscription to minimize exceptional
circumstances. Abstracting from this particular objective, we propose L_{CIRC},
which is an extension of generic propositional circumscription by allowing
propositional combinations and nesting of circumscriptive theories. As shown,
NATs are naturally embedded into this language, and are in fact of equal
expressive capability. We then analyze the complexity of L_{CIRC} and NATs, and
in particular the effect of nesting. The latter is found to be a source of
complexity, which climbs the Polynomial Hierarchy as the nesting depth
increases and reaches PSPACE-completeness in the general case. We also identify
meaningful syntactic fragments of NATs which have lower complexity. In
particular, we show that the generalization of Horn circumscription in the NAT
framework remains CONP-complete, and that Horn NATs without fixed letters can
be efficiently transformed into an equivalent Horn CNF, which implies
polynomial solvability of principal reasoning tasks. Finally, we also study
extensions of NATs and briefly address the complexity in the first-order case.
Our results give insight into the ``cost'' of using L_{CIRC} (resp. NATs) as a
host language for expressing other formalisms such as action theories,
narratives, or spatial theories.Comment: A preliminary abstract of this paper appeared in Proc. Seventeenth
International Joint Conference on Artificial Intelligence (IJCAI-01), pages
169--174. Morgan Kaufmann, 200
Complexity of Non-Monotonic Logics
Over the past few decades, non-monotonic reasoning has developed to be one of
the most important topics in computational logic and artificial intelligence.
Different ways to introduce non-monotonic aspects to classical logic have been
considered, e.g., extension with default rules, extension with modal belief
operators, or modification of the semantics. In this survey we consider a
logical formalism from each of the above possibilities, namely Reiter's default
logic, Moore's autoepistemic logic and McCarthy's circumscription.
Additionally, we consider abduction, where one is not interested in inferences
from a given knowledge base but in computing possible explanations for an
observation with respect to a given knowledge base.
Complexity results for different reasoning tasks for propositional variants
of these logics have been studied already in the nineties. In recent years,
however, a renewed interest in complexity issues can be observed. One current
focal approach is to consider parameterized problems and identify reasonable
parameters that allow for FPT algorithms. In another approach, the emphasis
lies on identifying fragments, i.e., restriction of the logical language, that
allow more efficient algorithms for the most important reasoning tasks. In this
survey we focus on this second aspect. We describe complexity results for
fragments of logical languages obtained by either restricting the allowed set
of operators (e.g., forbidding negations one might consider only monotone
formulae) or by considering only formulae in conjunctive normal form but with
generalized clause types.
The algorithmic problems we consider are suitable variants of satisfiability
and implication in each of the logics, but also counting problems, where one is
not only interested in the existence of certain objects (e.g., models of a
formula) but asks for their number.Comment: To appear in Bulletin of the EATC
On Sub-Propositional Fragments of Modal Logic
In this paper, we consider the well-known modal logics ,
, , and , and we study some of their
sub-propositional fragments, namely the classical Horn fragment, the Krom
fragment, the so-called core fragment, defined as the intersection of the Horn
and the Krom fragments, plus their sub-fragments obtained by limiting the use
of boxes and diamonds in clauses. We focus, first, on the relative expressive
power of such languages: we introduce a suitable measure of expressive power,
and we obtain a complex hierarchy that encompasses all fragments of the
considered logics. Then, after observing the low expressive power, in
particular, of the Horn fragments without diamonds, we study the computational
complexity of their satisfiability problem, proving that, in general, it
becomes polynomial
Complexity of Prioritized Default Logics
In default reasoning, usually not all possible ways of resolving conflicts
between default rules are acceptable. Criteria expressing acceptable ways of
resolving the conflicts may be hardwired in the inference mechanism, for
example specificity in inheritance reasoning can be handled this way, or they
may be given abstractly as an ordering on the default rules. In this article we
investigate formalizations of the latter approach in Reiter's default logic.
Our goal is to analyze and compare the computational properties of three such
formalizations in terms of their computational complexity: the prioritized
default logics of Baader and Hollunder, and Brewka, and a prioritized default
logic that is based on lexicographic comparison. The analysis locates the
propositional variants of these logics on the second and third levels of the
polynomial hierarchy, and identifies the boundary between tractable and
intractable inference for restricted classes of prioritized default theories
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