Prioritised Default Logic as Argumentation with Partial Order Default

arXiv:1609.05224v1 [cs.AI] 25 Aug 2016

Prioritised Default Logic as Argumentation with Partial Order Default Priorities Anthony P. Young, Sanjay Modgil, Odinaldo Rodrigues Department of Informatics, King’s College London, Strand, London, U.K. {peter.young,sanjay.modgil,odinaldo.rodrigues}@kcl.ac.uk

20th September 2016 Abstract We express Brewka’s prioritised default logic (PDL) as argumentation using ASPIC+ . By representing PDL as argumentation and designing an argument preference relation that takes the argument structure into account, we prove that the conclusions of the justified arguments correspond to the PDL extensions. We will first assume that the default priority is total, and then generalise to the case where it is a partial order. This provides a characterisation of non-monotonic inference in PDL as an exchange of argument and counter-argument, providing a basis for distributed non-monotonic reasoning in the form of dialogue.1

Contents 1 Introduction

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2 Background 2.1 Notation Used in this Paper . . . . . . . . . . . . . . . . . . . . . 2.2 The ASPIC+ Framework . . . . . . . . . . . . . . . . . . . . . . . 2.3 Brewka’s Prioritised Default Logic . . . . . . . . . . . . . . . . .

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3 From ASPIC+ to PDL 3.1 Representing PDL in ASPIC+ . . . . . . . . . . . . . . . . . 3.2 A Suitable Argument Preference Relation . . . . . . . . . . 3.3 The Representation Theorem . . . . . . . . . . . . . . . . . 3.3.1 Non-Blocked Defaults . . . . . . . . . . . . . . . . . 3.3.2 Existence and Uniqueness of Stable Extensions . . . 3.3.3 The Representation Theorem: Statement and Proof 3.4 Satisfaction of Rationality Postulates . . . . . . . . . . . . . 3.4.1 The Stable Extension is Grounded . . . . . . . . . .

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1 The results of Section 3 first appeared in the preprint [23] and have been published in the conference proceedings of AAMAS2016 [24]. This paper gives the full proofs of these results.

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3.4.2 The Trivialisation and Rationality Theorems . . . . . . . 3.4.3 Inconsistent Arguments . . . . . . . . . . . . . . . . . . . Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4 On Lifting the Assumption of a Total Order Default Priority 4.1 The Argument Preference Relation based on Partial Order Default Priorities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1 A Representation of Rules and their Ordering using Strings 4.1.2 Algorithm and Example Calculation . . . . . . . . . . . . 4.1.3 Properties of the Generalised SP Order . . . . . . . . . . 4.1.4 The Generalised Argument Preference Relation for