Synthesis of Proof Procedures for Default Reasoning

Artificial Intelligence, 1998

The aim of this work is to show how Prolog technology can he used for efficient implementation of query answering in default logics. The idea is to translate a default theory along with a query into a Prolog program and a Prolog query such that the original query is derivable from the default theory iff the Prolog query is derivable from the Prolog program. In order to comply with the goal-oriented proof search of this approach, we focus on default theories supporting local proof procedures, as exemplified by so-called semi-monotonic default theories. Although this does not capture general default theories under Reiter's interpretation, it does so under alternative ones'.

Argumentation in Artificial Intelligence, 2009

In this paper we explore the thesis that the role of argumentation in practical reasoning in general and legal reasoning in particular is to justify the use of defeasible rules to derive a conclusion in preference to the use of other defeasible rules to derive a conflicting conclusion. The defeasibility of rules is expressed by means of non-provability claims as additional conditions of the rules. We outline an abstract approach to defeasible reasoning and argumentation which includes many existing formalisms, including default logic, extended logic programming, non-monotonic modal logic and auto-epistemic logic, as special cases. We show, in particular, that the "admissibility" semantics for all these formalisms has a natural argumentationtheoretic interpretation and proof procedure, which seem to correspond well with informal argumentation. In the admissibility semantics there is only one way for one argument to attack another, namely by undermining one of its non-provability claims. In this paper, we show how other kinds of attack between arguments, specifically how rebuttal and priority attacks, can be reduced to the undermining of non-provability claims.

Computational Intelligence, 1994

Default logic has been introduced for handling reasoning with incomplete knowledge. It has been widely studied, and various definitions have been proposed for it. Most of the variants have been defined by means of fixed points of some operator. We propose here another approach, which is based on a study of the way in which general rules with exceptions, used in a default reasoning process, can contradict one another. We then isolate sets of noncontradicting rules, as large as possible in order to exploit as much information as possible, and construct, for each of these sets of rules, the set of conclusions that can be deduced from it. We show that our framework encompasses most of the existing variants of default logic, allowing those variants to be compared from a knowledge representation point of view. Our approach also enables us to provide an operational definition of extensions in some interesting cases. Proof-theoretical and semantical aspects are investigated.

Lecture Notes in Computer Science, 1993

We present what we call Constructive Default Logic (CDL) -a default logic in which the fixedpoint definition of extensions is replaced by a constructive definition which yield so-called constructive extensions. Selection functions are used to represent explicitly the control of the reasoning process in this default logic. It is well-known that Reiter's original default logic lacks, in general, a default proof theory. We will show that CDL does have a default proof theory, and we will also show that this is related to the fact that CDL has the existence property for constructive extensions and that it also has the semi-monotonicity property. Furthermore, we will also show that, with respect to some counter-examples that were suggested by Lukaszewicz, constructive extensions yield more intuitive conclusions than Reiter's extensions. Hence, constructive default logic does not only have heuristic advantages over Reiter's default theory from a computational point of view, but it is also more adequate with respect to knowledge representation.

Artificial Intelligence, 1994

Reiter's default logic has proven to be an enduring and versatile approach to nonmonotonic reasoning. Subsequent work in default logic has concentrated in two major areas. First, modifications have been developed to extend and augment the approach. Second, there has been ongoing interest in semantic foundations for default logic. In this paper, a number of variants of default logic are developed to address differing intuitions arising from the original and subsequent formulations. First, we modify the manner in which consistency is used in the definition of a default extension. The idea is that a global rather than local notion of consistency is employed in the formation of a default extension. Second, we argue that in some situations the requirement of proving the antecedent of a default is too strong. A second variant of default logic is developed where this requirement is dropped; subsequently these approaches are combined, leading to a final variant. These variants then lead to default systems which conform to alternative intuitions regarding default reasoning. For all of these approaches, a fixed-point and a pseudo-iterative definition are given; as well a semantic characterisation of these approaches is provided. In the combined approach we argue also that one can now reason about a set of defaults and can determine, for example, if a particular default in a set is redundant. We show the relation of this work to that of Lukaszewicz and Brewka, and to the Theorist system.

2008

Abstract. Default logics represent an important class of the nonmonotonic formalisms. Using simple by powerful inference rules, called defaults, these logic systems model reasoning patterns of the form ”in the absence of information to the contrary of... ”, and thus formalize the default reasoning, a special type of nonmonotonic reasoning. In this paper we propose an automated system, called DARR, with two components: a propositional theorem prover and a theorem prover for constrained and rational propositional default logics. A modified version of semantic tableaux method is used to implement the propositional prover. Also, this theorem proving method is adapted for computing extensions because one of its purpose is to produce models, and extensions are models of the world described by default theories. 1.

Annals of Mathematics and Artificial Intelligence, 1990

In the paper we introduce a variant of autoepistemic logic that is especially suitable for expressing default reasonings. It is based on the notion of iterative expansion. We show a new way of translating default theories into the language of modal logic under which default extensions correspond exactly to iterative expansions. Iterative expansions have some attractive properties. They are more restrictive than autoepistemic expansions, and, for some classes of theories, than moderately grounded expansions. At the same time iterative expansions avoid several undesirable properties of strongly grounded expansions, for example, they are grounded in the whole set of the agent's initial assumptions and do not depend on their syntactic representation. Iterative expansions are defined syntactically. We define a semantics which leads to yet another notion of expansion-weak iterative expansion-and we show that there is an important class of theories, that we call T-programs, for which iterative and weak iterative expansions coincide. Thus, for T-programs, iterative expansions can be equivalently defined by semantic means. The question of existence of a semantic definition of iterative expansions for general theories remains open.

arXiv (Cornell University), 2020

Recently there has been an increasing interest in frameworks extending Dung's abstract Argumentation Framework (AF). Popular extensions include bipolar AFs and AFs with recursive attacks and necessary supports. Although the relationships between AF semantics and Partial Stable Models (PSMs) of logic programs has been deeply investigated, this is not the case for more general frameworks extending AF. In this paper we explore the relationships between AF-based frameworks and PSMs. We show that every AF-based framework ∆ can be translated into a logic program P∆ so that the extensions prescribed by different semantics of ∆ coincide with subsets of the PSMs of P∆. We provide a logic programming approach that characterizes, in an elegant and uniform way, the semantics of several AF-based frameworks. This result allows also to define the semantics for new AF-based frameworks, such as AFs with recursive attacks and recursive deductive supports. Under consideration for publication in Theory and Practice of Logic Programming.

The Journal of Logic Programming, 1995

Logic programs are considered as abductive programs with negative literals as abductive hypotheses. A simple framework for semantics of logic programming is introduced based on the notion of acceptable hypotheses. We show that our framework captures, generalizes, and unifies different semantic concepts (e.g., well-founded models, stable models, stationary semantics, etc.) in logic programming. We demonstrate that our framework accommodates in a natural way both the minimalism and maximalism intuitions to semantics of logic programming. Further, we show that Eshghi and Kowalski's procedure is a proof procedure for the abductive semantics. We also give sufficient conditions for the coincidence between different semantics.

Computational Intelligence, 2004

We present a general approach for representing and reasoning with sets of defaults in default logic, focussing on reasoning about preferences among sets of defaults. First, we consider how to control the application of a set of defaults so that either all apply (if possible) or none do (if not). From this, an approach to dealing with preferences among sets of default rules is developed. We begin with an ordered default theory, consisting of a standard default theory, but with possible preferences on sets of rules. This theory is transformed into a second, standard default theory wherein the preferences are respected. The approach differs from other work, in that we obtain standard default theories and do not rely on prioritised versions of default logic. In practical terms this means we can immediately use existing default logic theorem provers for an implementation. As well, we directly generate just those extensions containing the most preferred applied rules; in contrast, most previous approaches generate all extensions, then select the most preferred. In a major application of the approach, we show how semi-monotonic default theories can be encoded so that reasoning can be carried out at the object level. With this, we can reason about default extensions 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.