250 research outputs found
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
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
Type-elimination-based reasoning for the description logic SHIQbs using decision diagrams and disjunctive datalog
We propose a novel, type-elimination-based method for reasoning in the
description logic SHIQbs including DL-safe rules. To this end, we first
establish a knowledge compilation method converting the terminological part of
an ALCIb knowledge base into an ordered binary decision diagram (OBDD) which
represents a canonical model. This OBDD can in turn be transformed into
disjunctive Datalog and merged with the assertional part of the knowledge base
in order to perform combined reasoning. In order to leverage our technique for
full SHIQbs, we provide a stepwise reduction from SHIQbs to ALCIb that
preserves satisfiability and entailment of positive and negative ground facts.
The proposed technique is shown to be worst case optimal w.r.t. combined and
data complexity and easily admits extensions with ground conjunctive queries.Comment: 38 pages, 3 figures, camera ready version of paper accepted for
publication in Logical Methods in Computer Scienc
The Language of Search
This paper is concerned with a class of algorithms that perform exhaustive
search on propositional knowledge bases. We show that each of these algorithms
defines and generates a propositional language. Specifically, we show that the
trace of a search can be interpreted as a combinational circuit, and a search
algorithm then defines a propositional language consisting of circuits that are
generated across all possible executions of the algorithm. In particular, we
show that several versions of exhaustive DPLL search correspond to such
well-known languages as FBDD, OBDD, and a precisely-defined subset of d-DNNF.
By thus mapping search algorithms to propositional languages, we provide a
uniform and practical framework in which successful search techniques can be
harnessed for compilation of knowledge into various languages of interest, and
a new methodology whereby the power and limitations of search algorithms can be
understood by looking up the tractability and succinctness of the corresponding
propositional languages
Logic-Based Explainability in Machine Learning
The last decade witnessed an ever-increasing stream of successes in Machine
Learning (ML). These successes offer clear evidence that ML is bound to become
pervasive in a wide range of practical uses, including many that directly
affect humans. Unfortunately, the operation of the most successful ML models is
incomprehensible for human decision makers. As a result, the use of ML models,
especially in high-risk and safety-critical settings is not without concern. In
recent years, there have been efforts on devising approaches for explaining ML
models. Most of these efforts have focused on so-called model-agnostic
approaches. However, all model-agnostic and related approaches offer no
guarantees of rigor, hence being referred to as non-formal. For example, such
non-formal explanations can be consistent with different predictions, which
renders them useless in practice. This paper overviews the ongoing research
efforts on computing rigorous model-based explanations of ML models; these
being referred to as formal explanations. These efforts encompass a variety of
topics, that include the actual definitions of explanations, the
characterization of the complexity of computing explanations, the currently
best logical encodings for reasoning about different ML models, and also how to
make explanations interpretable for human decision makers, among others
Compiling CSPs: A Complexity Map of (Non-Deterministic) Multivalued Decision Diagrams
International audienceConstraint Satisfaction Problems (CSPs) offer a powerful framework for representing a great variety of problems. The difficulty is that most of the requests associated with CSPs are NP-hard. When these requests have to be addressed online, Multivalued Decision Diagrams (MDDs) have been proposed as a way to compile CSPs. In the present paper, we draw a compilation map of MDDs, in the spirit of the NNF compilation map, analyzing MDDs according to their succinctness and to their tractable transformations and queries. Deterministic ordered MDDs are a generalization of ordered binary decision diagrams to non-Boolean domains: unsurprisingly, they have similar capabilities. More interestingly, our study puts forward the interest of non-deterministic ordered MDDs: when restricted to Boolean domains, they capture OBDDs and DNFs as proper subsets and have performances close to those of DNNFs. The comparison to classical, deterministic MDDs shows that relaxing the determinism requirement leads to an increase in succinctness and allows more transformations to be satisfied in polynomial time (typically, the disjunctive ones). Experiments on random problems confirm the gain in succinctness
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