The Complexity of Reasoning for Fragments of Default Logic

Default logic was introduced by Reiter in 1980. In 1992, Gottlob classified the complexity of the extension existence problem for propositional default logic as \SigmaPtwo-complete, and the complexity of the credulous and skeptical reasoning problem as SigmaP2-complete, resp. PiP2-complete. Additionally, he investigated restrictions on the default rules, i.e., semi-normal default rules. Selman made in 1992 a similar approach with disjunction-free and unary default rules. In this paper we systematically restrict the set of allowed propositional connectives. We give a complete complexity classification for all sets of Boolean functions in the meaning of Post's lattice for all three common decision problems for propositional default logic. We show that the complexity is a hexachotomy (SigmaP2-, DeltaP2-, NP-, P-, NL-complete, trivial) for the extension existence problem, while for the credulous and skeptical reasoning problem we obtain similar classifications without trivial cases.Comment: Corrected versio

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

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

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

Description of GADEL

This article describes the first implementation of the GADEL system : a Genetic Algorithm for Default Logic. The goal of GADEL is to compute extensions in Reiter's default logic. It accepts every kind of finite propositional default theories and is based on evolutionary principles of Genetic Algorithms. Its first experimental results on certain instances of the problem show that this new approach of the problem can be successful.Comment: System Descriptions and Demonstrations at Nonmonotonic Reasoning Workshop, 2000 6 pages, 2 figures, 5 table

Named Models in Coalgebraic Hybrid Logic

Hybrid logic extends modal logic with support for reasoning about individual states, designated by so-called nominals. We study hybrid logic in the broad context of coalgebraic semantics, where Kripke frames are replaced with coalgebras for a given functor, thus covering a wide range of reasoning principles including, e.g., probabilistic, graded, default, or coalitional operators. Specifically, we establish generic criteria for a given coalgebraic hybrid logic to admit named canonical models, with ensuing completeness proofs for pure extensions on the one hand, and for an extended hybrid language with local binding on the other. We instantiate our framework with a number of examples. Notably, we prove completeness of graded hybrid logic with local binding

Modelling default and likelihood reasoning as probabilistic

A probabilistic analysis of plausible reasoning about defaults and about likelihood is presented. 'Likely' and 'by default' are in fact treated as duals in the same sense as 'possibility' and 'necessity'. To model these four forms probabilistically, a logic QDP and its quantitative counterpart DP are derived that allow qualitative and corresponding quantitative reasoning. Consistency and consequence results for subsets of the logics are given that require at most a quadratic number of satisfiability tests in the underlying propositional logic. The quantitative logic shows how to track the propagation error inherent in these reasoning forms. The methodology and sound framework of the system highlights their approximate nature, the dualities, and the need for complementary reasoning about relevance