2,287 research outputs found
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
Space Efficiency of Propositional Knowledge Representation Formalisms
We investigate the space efficiency of a Propositional Knowledge
Representation (PKR) formalism. Intuitively, the space efficiency of a
formalism F in representing a certain piece of knowledge A, is the size of the
shortest formula of F that represents A. In this paper we assume that knowledge
is either a set of propositional interpretations (models) or a set of
propositional formulae (theorems). We provide a formal way of talking about the
relative ability of PKR formalisms to compactly represent a set of models or a
set of theorems. We introduce two new compactness measures, the corresponding
classes, and show that the relative space efficiency of a PKR formalism in
representing models/theorems is directly related to such classes. In
particular, we consider formalisms for nonmonotonic reasoning, such as
circumscription and default logic, as well as belief revision operators and the
stable model semantics for logic programs with negation. One interesting result
is that formalisms with the same time complexity do not necessarily belong to
the same space efficiency class
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
Reformulating Non-Monotonic Theories for Inference and Updating
We aim to help build programs that do large-scale, expressive non-monotonic reasoning (NMR): especially, 'learning agents' that store, and revise, a body of conclusions while continually acquiring new, possibly defeasible, premise beliefs. Currently available procedures for forward inference and belief revision are exhaustive, and thus impractical: they compute the entire non-monotonic theory, then re-compute from scratch upon updating with new axioms. These methods are thus badly intractable. In most theories of interest, even backward reasoning is combinatoric (at least NP-hard). Here, we give theoretical results for prioritized circumscription that show how to reformulate default theories so as to make forward inference be selective, as well as concurrent; and to restrict belief revision to a part of the theory. We elaborate a detailed divide-and-conquer strategy. We develop concepts of structure in NM theories, by showing how to reformulate them in a particular fashion: to be conjunctively decomposed into a collection of smaller 'part' theories. We identify two well-behaved special cases that are easily recognized in terms of syntactic properties: disjoint appearances of predicates, and disjoint appearances of individuals (terms). As part of this, we also definitionally reformulate the global axioms, one by one, in addition to applying decomposition. We identify a broad class of prioritized default theories, generalizing default inheritance, for which our results especially bear fruit. For this asocially monadic class, decomposition permits reasoning to be localized to individuals (ground terms), and reduced to propositional. Our reformulation methods are implementable in polynomial time, and apply to several other NM formalisms beyond circumscription
Reasoning about Action: An Argumentation - Theoretic Approach
We present a uniform non-monotonic solution to the problems of reasoning
about action on the basis of an argumentation-theoretic approach. Our theory is
provably correct relative to a sensible minimisation policy introduced on top
of a temporal propositional logic. Sophisticated problem domains can be
formalised in our framework. As much attention of researchers in the field has
been paid to the traditional and basic problems in reasoning about actions such
as the frame, the qualification and the ramification problems, approaches to
these problems within our formalisation lie at heart of the expositions
presented in this paper
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