216 research outputs found
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
Reasoning about Minimal Belief and Negation as Failure
We investigate the problem of reasoning in the propositional fragment of
MBNF, the logic of minimal belief and negation as failure introduced by
Lifschitz, which can be considered as a unifying framework for several
nonmonotonic formalisms, including default logic, autoepistemic logic,
circumscription, epistemic queries, and logic programming. We characterize the
complexity and provide algorithms for reasoning in propositional MBNF. In
particular, we show that entailment in propositional MBNF lies at the third
level of the polynomial hierarchy, hence it is harder than reasoning in all the
above mentioned propositional formalisms for nonmonotonic reasoning. We also
prove the exact correspondence between negation as failure in MBNF and negative
introspection in Moore's autoepistemic logic
The Complexity of Reasoning for Fragments of Autoepistemic Logic
Autoepistemic logic extends propositional logic by the modal operator L. A
formula that is preceded by an L is said to be "believed". The logic was
introduced by Moore 1985 for modeling an ideally rational agent's behavior and
reasoning about his own beliefs. In this paper we analyze all Boolean fragments
of autoepistemic logic with respect to the computational complexity of the
three most common decision problems expansion existence, brave reasoning and
cautious reasoning. As a second contribution we classify the computational
complexity of counting the number of stable expansions of a given knowledge
base. To the best of our knowledge this is the first paper analyzing the
counting problem for autoepistemic logic
Implementing Default and Autoepistemic Logics via the Logic of GK
The logic of knowledge and justified assumptions, also known as logic of
grounded knowledge (GK), was proposed by Lin and Shoham as a general logic for
nonmonotonic reasoning. To date, it has been used to embed in it default logic
(propositional case), autoepistemic logic, Turner's logic of universal
causation, and general logic programming under stable model semantics. Besides
showing the generality of GK as a logic for nonmonotonic reasoning, these
embeddings shed light on the relationships among these other logics. In this
paper, for the first time, we show how the logic of GK can be embedded into
disjunctive logic programming in a polynomial but non-modular translation with
new variables. The result can then be used to compute the extension/expansion
semantics of default logic, autoepistemic logic and Turner's logic of universal
causation by disjunctive ASP solvers such as claspD(-2), DLV, GNT and cmodels.Comment: Proceedings of the 15th International Workshop on Non-Monotonic
Reasoning (NMR 2014
Methods for Solving Necessary Equivalences
Nonmonotonic Logics such as Autoepistemic Logic, Reflective Logic, and Default Logic, are usually
defined in terms of set-theoretic fixed-point equations defined over deductively closed sets of sentences of First
Order Logic. Such systems may also be represented as necessary equivalences in a Modal Logic stronger than
S5 with the added advantage that such representations may be generalized to allow quantified variables crossing
modal scopes resulting in a Quantified Autoepistemic Logic, a Quantified Autoepistemic Kernel, a Quantified
Reflective Logic, and a Quantified Default Logic. Quantifiers in all these generalizations obey all the normal laws
of logic including both the Barcan formula and its converse. Herein, we address the problem of solving some
necessary equivalences containing universal quantifiers over modal scopes. Solutions obtained by these
methods are then compared to related results obtained in the literature by Circumscription in Second Order Logic
since the disjunction of all the solutions of a necessary equivalence containing just normal defaults in these
Quantified Logics, is equivalent to that system
Embedding Non-Ground Logic Programs into Autoepistemic Logic for Knowledge Base Combination
In the context of the Semantic Web, several approaches to the combination of
ontologies, given in terms of theories of classical first-order logic and rule
bases, have been proposed. They either cast rules into classical logic or limit
the interaction between rules and ontologies. Autoepistemic logic (AEL) is an
attractive formalism which allows to overcome these limitations, by serving as
a uniform host language to embed ontologies and nonmonotonic logic programs
into it. For the latter, so far only the propositional setting has been
considered. In this paper, we present three embeddings of normal and three
embeddings of disjunctive non-ground logic programs under the stable model
semantics into first-order AEL. While the embeddings all correspond with
respect to objective ground atoms, differences arise when considering
non-atomic formulas and combinations with first-order theories. We compare the
embeddings with respect to stable expansions and autoepistemic consequences,
considering the embeddings by themselves, as well as combinations with
classical theories. Our results reveal differences and correspondences of the
embeddings and provide useful guidance in the choice of a particular embedding
for knowledge combination.Comment: 52 pages, submitte
Semantic Matchmaking of Web Resources with Local Closed-World Reasoning
Ontology languages like OWL allow for semantically rich annotation of resources (e.g., products advertised at on-line electronic marketplaces). The description logic (DL) formalism underlying OWL provides reasoning techniques that perform match-making on such annotations. This paper identifies peculiarities in the use of DL inferences for matchmaking that derive from OWL\u27s open-world semantics, analyzes local closed-world reasoning for its applicability to matchmaking, and investigates the suitability of two nonmonotonic extensions to DL, autoepistemic DLs and DLs with circumscription, for local closed-world reasoning in the matchmaking context. An elaborate example of an electronic marketplace for PC product catalogs from the e-commerce domain demonstrates how these formalisms can be used to realize such scenarios
Optimizing the computation of overriding
We introduce optimization techniques for reasoning in DLN---a recently
introduced family of nonmonotonic description logics whose characterizing
features appear well-suited to model the applicative examples naturally arising
in biomedical domains and semantic web access control policies. Such
optimizations are validated experimentally on large KBs with more than 30K
axioms. Speedups exceed 1 order of magnitude. For the first time, response
times compatible with real-time reasoning are obtained with nonmonotonic KBs of
this size
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