Expressing Default Logic Variants in Default Logic

Realism for agents with unconditional beliefs, desires and intentions (BDI agents) has been analyzed in modal logic. This paper provides a logical analysis of realism for agents with conditional beliefs and desires in a rule based approach analogous to Reiter's default logic. We distinguish two types of realism, which we call ‘a priori' and ‘a posteriori' realism. We analyze whether these two new properties are compatible with other properties discussed in the literature, such as existence of extensions. We show that Reiter's default logic is too strong, in the sense that a weaker notion of maximality of extensions is needed to satisfy realism. Finally we show that several existing approaches do not satisfy the new realism properties, and we introduce a new construction that does satisfy them.

Knowledge formalization and reasoning automation are central within Artificial Intelligence. Classical logic has been traditionally used for such purposes. However, it is better suited to deal with complete knowledge in ideal circumstances. In real situations, in which the knowledge is partial, classical logic is not sufficient since it is monotonic. No monotonic logics have been proposed to better cope with practical reasoning. A successful formalization of no monotonic reasoning is Reiter's default logic which extends classical logic with default rules. Unfortunately, default logic is undividable. One reason for that is the use of classical logic as its monotonic basis. In this work, we change the default logic monotonic basis and propose a new default logic based on the description logic ALC. This new default logic is decidable and useful to formalize practical reasoning on hierarchical ontology's with exceptions, specially the ones that deals with legal knowledge and reasoning. On the default counterpart, we add some restrictions to the application of defaults in order to obtain nice properties such as coherence and elimination of anomalous extensions. We present the main algorithms used to build the extensions for this logic with its complexity analysis.

Cumulative default logic: Finite characterization, algorithms, and complexity

This paper presents the most influential formalizations of nonmonotonic reasoning: McDermott and Doyle's Nonmonotonic Logic I, Reiter's Default Logic, and McCarthy's Circumscription. Unfortunately all these formalizations are not semidecidable. In consequence implementations of nonmonotonic systems either must give up the idea of theoremhood and replace it by a weaker concept, or the expressive power of the logical language has to be restricted. Examples for both approaches are given.

Conservative extensions of logical theories play an important role in software engineering. They provide a formal basis for program refinement and guarantee the integrity and transparency of modules and objects. This paper studies conservative extension ideas in the context of default logic. In particular, we define several alternative concepts, study their properties, and derive interconnections among the concepts. The main result provides an interesting distinctive feature of Reiter's default logic over some well-known variants.

In Reiter's default logic, the operator in the fixed-point definition of extension is not appropriate to compute extensions by its iterated applications. This paper presents a class of alternative operators, called compatible ones, such that, at least for normal default theories and so-called well-founded, ordered default theories, we can get extensions by iterated applications of them. In addition, we completely answer Etherington's conjectures about both his procedure for generating extensions and a modified version of it. In particular, we give an example of a finite, ordered default theory, for which the original procedure fails to converge, and show that the computation of the modified one is essentially the iteration of a compatible operator and converges for finite, ordered theories.

In this paper, we show that Reiter's default logic in the propositional case can be translated into answer set programming by identifying the internal relationships among formulas in a default theory. Based on this idea, we implement a new default logic solver - dl2asp. We report some experimental results, in particular the application of dl2asp for solving the fair division problem in social choice theory.

The authors introduce a dynamic-input problem solver and contrast it with the traditional static approach. A domain-independent hypothesis revision strategy for solving dynamic-input interpretation problems is presented, and its reasoning techniques are compared with R. Reiter's (1980) default logic. A novel problem solver architecture based on a truth maintenance system is described. Advantages of this approach include a domain-independent strategy for generation and revising hypotheses in a dynamic environment, and a clear separation between the referential level and the inferencing level in a default reasoning system.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Consistency-based approaches in nonmonotonic reasoning may be expected to yield multiple sets of default conclusions for a given default theory. Reasoning about such extensions is carried out at the meta-level. In this paper, we show how such reasoning may be carried out at the object level for a large class of default theories. Essentially we show how one can translate a (semi-monotonic) default theory Δ, obtaining a second Δ', such that Δ has a single extension that encodes every extension of _. Moreover, our translated theory is only a constant factor larger than the original (with the exception of unique names axioms). We prove that our translation behaves correctly. In the approach we can now encode the notion of extension from within the framework of standard default logic. Hence one can encode notions such as skeptical and credulous conclusions, and can reason about such conclusions within a single extension. This result has some theoretical interest, in that it shows how multiple extensions of semi-monotonic default theories are encodable with manageable overhead in a single extension.

Two general approaches to reasoning with imperfect information are discussed: nonmonotonic reasoning and a calculus of uncertainty. Default reasoning is posed as an approach that is potentially capable of integrating many facets of these two approaches. Practical requirements for default reasoning are then established. This is done by identifying a number of cases that involve incomplete and uncertain information and showing how they can be addressed by default reasoning. Parametric and symbolic reasoning are differentiated, and it is shown that both types are necessary. This distinction is important, as most approaches tend to neglect either the parametric or the symbolic aspect of default reasoning, thereby restricting its use to one of the two approaches discussed above. Five capabilities that are necessary to effect default reasoning are identified. The major characteristics for systems that handle incomplete and uncertain information as well as other types of imperfect information are established.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Compiling specificity into approaches to nonmonotonic reasoning

The default logic (Rei 80) and the autoepistemic logic (Mo 85) are non-monotonic logics. In default logic a default theory consists of a set of propositions sentences W and a set of defaults D. A sentence is defined to be derivable from a given default theory if it belongs to an extension of the default theory. A central question is: Given a default theory Δ = (D, W) and a sentence β, is there an extension of Δ which contains β?

We consider the connections between belief revision, conditional logic and nonmonotonic reasoning, using as a foundation the approach to theory change developed by Alchourron, Gardenfors and Makinson (the AGM approach). This is first generalized to allow the iteration of theory change operations to capture the dynamics of epistemic states according to a principle of minimal change of entrenchment. The iterative operations of expansion, contraction and revision are characterized both by a set of postulates and by Grove’s construction based on total pre-orders on the set of complete theories of the belief logic. We present a sound and complete conditional logic whose semantics is based on our iterative revision operation, but which avoids Gardenfors’s triviality result because of a severely restricted language of beliefs and hence the weakened scope of our extended postulates. In the second part of the paper, we develop a computational approach to theory dynamics using Rott’s E-bases as a representation for epistemic states. Under this approach, a ranked E-base is interpreted as standing for the most conservative entrenchment compatible with the base, reflecting a kind of foundationalism in the acceptance of evidence for a belief. Algorithms for the computation of our iterative versions of expansion, contraction and revision are presented. Finally, we consider the relationship between nonmonotonic reasoning and both conditional logic and belief revision. Adapting the approach of Delgrande, we show that the unique extension of a default theory expressed in our conditional logic of belief revision corresponds to the most conservative belief state which respects the theory: however, this correspondence is limited to propositional default theories. Considering first order default theories, we present a belief revision algorithm which incorporates the assumption of independence of default instances and propose the use of a base logic for default reasoning which incorporates uniqueness of names. We conclude with an examination of the behavior of an implemented system on some of Lifschitz’s benchmark problems in nonmonotonic reasoning.

Reiter’s pioneering work on default reasoning aimed to formalize the notion of quickly “jumping to conclusions”. Unfortunately, his resulting formalism, default logic, proved more computationally complex than classical logics. The same has been true for the other major formalisms for nonmonotonic reasoning, circumscription [20], nonmonotonic logic [22], [21], and autoepistemic logic [25]. This has dramatically limited the applicability of formal methods vis a vis real problems involving defaults. The complexity of consistency checking is one of the two problems that must be addressed to reduce the complexity of default reasoning. We propose an approach to default reasoning based on approximate consistency checking, using a novel synthesis of limited contexts and fast incomplete checks, and argue that this combination overcomes the limitations of its component parts. Our approach trades correctness for speed, but we argue that the nature of default reasoning makes this trade relatively inexpensive and intuitively plausible. This approach not only accords well with Reiter’s original motivations, but converges to Default Logic in the computational limit. We describe a prototype implementation of a default reasoner based on these ideas, and a preliminary empirical evaluation.KeywordsConsistency CheckDefense Advance Research Project AgencyDefault LogicNonmonotonic ReasoningDefault ReasoningThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Operational characterization of extensions in some logics for default reasoning