Phan Minh Dung

On the acceptability of arguments and its fundamental role in nonmonotonic reasoning, logic programming and n -person games

The purpose of this paper is to study the fundamental mechanism, humans use in argumentation, and to explore ways to implement this mechanism on computers. We do so by first developing a theory for argumentation whose central notion is the acceptability of arguments. Then we argue for the “correctness” or “appropriateness” of our theory with two strong arguments. The first one shows that most of the major approaches to nonmonotonic reasoning in AI and logic programming are special forms of our theory of argumentation. The second argument illustrates how our theory can be used to investigate the logical structure of many practical problems. This argument is based on a result showing that our theory captures naturally the solutions of the theory of n-person games and of the well-known stable marriage problem. By showing that argumentation can be viewed as a special form of logic programming with negation as failure, we introduce a general logic-programming-based method for generating meta-interpreters for argumentation systems, a method very much similar to the compiler-compiler idea in conventional programming. Keyword: Argumentation; Nonmonotonic reasoning; Logic programming; n-person games; The stable marriage problem

An abstract, argumentation-theoretic approach to default reasoning

Abstract We present an abstract framework for default reasoning, which includes Theorist, default logic, logic programming, autoepistemic logic, non-monotonic modal logics, and certain instances of circumscription as special cases. The framework can be understood as a generalisation of Theorist. The generalisation allows any theory formulated in a monotonic logic to be extended by a defeasible set of assumptions. An assumption can be defeated (or “attacked”) if its “contrary” can be proved, possibly with the aid of other conflicting assumptions. We show that, given such a framework, the standard semantics of most logics for default reasoning can be understood as sanctioning a set of assumptions, as an extension of a given theory, if and only if the set of assumptions is conflict-free (in the sense that it does not attack itself) and it attacks every assumption not in the set. We propose a more liberal, argumentation-theoretic semantics, based upon the notion of admissible extension in logic programming. We regard a set of assumptions, in general, as admissible if and only if it is conflict-free and defends itself (by attacking) every set of assumptions which attacks it. We identify conditions for the existence of extensions and for the equivalence of different semantics.

Computing ideal sceptical argumentation

We present two dialectic procedures for the sceptical ideal semantics for argumentation. The first procedure is defined in terms of dispute trees, for abstract argumentation frameworks. The second procedure is defined in dialectical terms, for assumption-based argumentation frameworks. The procedures are adapted from (variants of) corresponding procedures for computing the credulous admissible semantics for assumption-based argumentation, proposed in [P.M. Dung, R.A. Kowalski, F. Toni, Dialectic proof procedures for assumption-based, admissible argumentation, Artificial Intelligence 170 (2006) 114-159]. We prove that the first procedure is sound and complete, and the second procedure is sound in general and complete for a special but natural class of assumption-based argumentation frameworks, that we refer to as p-acyclic. We also prove that in the case of p-acyclic assumption-based argumentation frameworks (a variant of) the procedure of [P.M. Dung, R.A. Kowalski, F. Toni, Dialectic proof procedures for assumption-based, admissible argumentation, Artificial Intelligence 170 (2006) 114-159] for the admissible semantics is complete. Finally, we present a variant of the procedure of [P.M. Dung, R.A. Kowalski, F. Toni, Dialectic proof procedures for assumption-based, admissible argumentation, Artificial Intelligence 170 (2006) 114-159] that is sound for the sceptical grounded semantics.

Dialectic proof procedures for assumption-based, admissible argumentation

We present a family of dialectic proof procedures for the admissibility semantics of assumption-based argumentation. These proof procedures are defined for any conventional logic formulated as a collection of inference rules and show how any such logic can be extended to a dialectic argumentation system.The proof procedures find a set of assumptions, to defend a given belief, by starting from an initial set of assumptions that supports an argument for the belief and adding defending assumptions incrementally to counter-attack all attacks.The proof procedures share the same notion of winning strategy for a dispute and differ only in the search strategy they use for finding it. The novelty of our approach lies mainly in its use of backward reasoning to construct arguments and potential arguments, and the fact that the proponent and opponent can attack one another before an argument is completed. The definition of winning strategy can be implemented directly as a non-deterministic program, whose search strategy implements the search for defences.

Assumption-Based Argumentation

Assumption-Based Argumentation (ABA) [4, 3, 27, 9, 12, 20, 22] was developed, starting in the 90s, as a computational framework to reconcile and generalise most existing approaches to default reasoning [24, 25, 4, 3, 27, 26]. ABA was inspired by Dung’s preferred extension semantics for logic programming [10, 7], with its dialectical interpretation of the acceptability of negation-as-failure assumptions based on the notion of “no-evidence-to-the-contrary” [10, 7], by the Kakas, Kowalski and Toni interpretation of the preferred extension semantics in argumentation-theoretic terms [24, 25], and by Dung’s abstract argumentation (AA) [6, 8]. Because ABA is an instance of AA, all semantic notions for determining the “acceptability” of arguments in AA also apply to arguments in ABA. Moreover, like AA, ABA is a general-purpose argumentation framework that can be instantiated to support various applications and specialised frameworks, including: most default reasoning frameworks [4, 3, 27, 26] and problems in legal reasoning [27, 13], game-theory [8], practical reasoning and decision-theory [33, 29, 15, 28, 14]. However, whereas in AA arguments and attacks between arguments are abstract and primitive, in ABA arguments are deductions (using inference rules in an underlying logic) supported by assumptions. An attack by one argument against another is a deduction by the first argument of the contrary of an assumption supporting the second argument. Differently from a number of existing approaches to non-abstract argumentation (e.g. argumentation based on classical logic [2] and DeLP [23]) ABA does not have explicit rebuttals and does not impose the restriction that arguments have consistent and minimal supports. However, to a large extent, rebuttals can be obtained “for

An argumentation-theoretic foundation for logic programming

Abstract 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 Kowalskis procedure is a proof procedure for the abductive semantics. We also give sufficient conditions for the coincidence between different semantics.

On the relations between stable and well-founded semantics of logic programs

We study the relations between stable and well-founded semantics of logic programs. 1. We show that stable semantics can be defined in the same way as well-founded semantics based on the basic notion of unfounded sets. Hence, stable semantics can be considered as “two-valued well-founded semantics”. 2. An axiomatic characterization of stable and well-founded semantics of logic programs is given by a new completion theory, called strong completion. Similar to the Clarks completion, the strong completion can be interpreted in either two-valued or three-valued logic. We show that ◦ Two-valued strong completion specifies the stable semantics. ◦ Three-valued strong completion specifies the well-founded semantics. 3. We study the equivalence between stable semantics and well-founded semantics. At first, we prove the equivalence between the two semantics for strict programs. Then we introduce the bottom-stratified and top-strict condition generalizing both the stratifiability and the strictness, and show that the new condition is sufficient for the equivalence between stable and well-founded semantics. Further, we show that the call-consistency condition is sufficient for the existence of at least one stable model.

Integrating data from possibly inconsistent databases

The author addresses the problem of data inconsistencies while integrating data sets from multiple autonomous relational databases. The author starts by arguing that the semantics of integrating possibly inconsistent data is naturally captured by the maximal consistent subsets of the set of all information contained in the collected data. Based on this idea, a simple and intuitive semantical framework is proposed, called the integrated relational calculus which is an extension of the classical relational calculus, for manipulating and querying possibly inconsistent data. It is then shown that the model generalizes the model of flexible relational algebra of Agarwal, Keller, Wiederhold and Saraswat (1995) in the sense that the latter can be embedded into the former. It is also shown that the flexible relational model is not capable of integrating correctly relations with more than one key. The author further argues that flexible relational model provides a rather weak query language. The author then proves that for databases with only one key the flexible model provides a correct integration of inconsistent data.

Towards a Common Framework for Dialectical Proof Procedures in Abstract Argumentation

We present a common framework for dialectical proof procedures for computing credulous, grounded, ideal and sceptical preferred semantics of abstract argumentation. The framework is based on the notions of dispute derivation and base derivation. Dispute derivation is a dialectical notion first introduced for computing credulous semantics in assumption-based argumentation, and adapted here for computing credulous semantics and grounded semantics. Base derivation is introduced for two purposes: (i) to characterize all preferred extensions containing a given argument, and (ii) to represent backtracking in the search for a dispute derivation. We prove the soundness of the proof procedures for any argumentation frameworks and their completeness for general classes of finitary or finite-branching argumentation frameworks containing the class of finite argumentation frameworks as a subclass.We also discuss related results.

An argument-based approach to reasoning with specificity

We present a new priority-based approach to reasoning with specificity which subsumes inheritance reasoning. The new approach differs from other priority-based approaches in the literature in the way priority between defaults is handled. Here, it is conditional rather than unconditional as in other approaches. We show that any unconditional handling of priorities between defaults as advocated in the literature until now is not sufficient to capture general defeasible inheritance reasoning. We propose a simple and novel argumentation semantics for reasoning with specificity taking the conditionality of the priorities between defaults into account. Since the proposed argumentation semantics is a form of stable semantics of nonmonotonic reasoning, it inherits a common problem of the latter where it is not always defined for every default theory. We propose a class of stratified default theories for which the argumentation semantics is always defined. We also show that acyclic and consistent inheritance networks are stratified. We prove that the argumentation semantics satisfies the basic properties of a nonmonotonic consequence relation such as deduction, reduction, conditioning, and cumulativity for well-defined and stratified default theories. We give a modular and polynomial transformation of default theories with specificity into semantically equivalent Reiter default theories.