20,418 research outputs found
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
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
More on Descriptive Complexity of Second-Order HORN Logics
This paper concerns Gradel's question asked in 1992: whether all problems
which are in PTIME and closed under substructures are definable in second-order
HORN logic SO-HORN. We introduce revisions of SO-HORN and DATALOG by adding
first-order universal quantifiers over the second-order atoms in the bodies of
HORN clauses and DATALOG rules. We show that both logics are as expressive as
FO(LFP), the least fixed point logic. We also prove that FO(LFP) can not define
all of the problems that are in PTIME and closed under substructures. As a
corollary, we answer Gradel's question negatively
Minimization for Generalized Boolean Formulas
The minimization problem for propositional formulas is an important
optimization problem in the second level of the polynomial hierarchy. In
general, the problem is Sigma-2-complete under Turing reductions, but
restricted versions are tractable. We study the complexity of minimization for
formulas in two established frameworks for restricted propositional logic: The
Post framework allowing arbitrarily nested formulas over a set of Boolean
connectors, and the constraint setting, allowing generalizations of CNF
formulas. In the Post case, we obtain a dichotomy result: Minimization is
solvable in polynomial time or coNP-hard. This result also applies to Boolean
circuits. For CNF formulas, we obtain new minimization algorithms for a large
class of formulas, and give strong evidence that we have covered all
polynomial-time cases
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
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