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On a Flexible Representation for Defeasible Reasoning Variants Session 27: Argumentation AAMAS 2018, July 10-15, 2018, Stockholm, Sweden On a Flexible Representation for Defeasible Reasoning Variants Abdelraouf Hecham Pierre Bisquert Madalina Croitoru University of Montpellier IATE, INRA University of Montpellier Montpellier, France Montpellier, France Montpellier, France [email protected] [email protected] [email protected] ABSTRACT Table 1: Defeasible features supported by tools. We propose Statement Graphs (SG), a new logical formalism for Tool Blocking Propagating Team Defeat No Team Defeat defeasible reasoning based on argumentation. Using a flexible label- ASPIC+ - X - X ing function, SGs can capture the variants of defeasible reasoning DEFT - X - X DeLP - X - X (ambiguity blocking or propagating, with or without team defeat, DR-DEVICE X - X - and circular reasoning). We evaluate our approach with respect Flora-2 X - X - to human reasoning and propose a working first order defeasible reasoning tool that, compared to the state of the art, has richer expressivity at no added computational cost. Such tool could be of great practical use in decision making projects such as H2020 Existing literature has established the link between argumenta- NoAW. tion and defeasible reasoning via grounded semantics [13, 28] and Dialectical Trees [14]. Such approaches only allow for ambiguity KEYWORDS propagation without team defeat [16, 26]. In this paper we propose a new logical formalism called State- Defeasible Reasoning; Defeasible Logics; Existential Rules ment Graph (SGs) that captures all features showed in Table 1 via ACM Reference Format: a flexible labelling function. The SG can be seen as a generalisa- Abdelraouf Hecham, Pierre Bisquert, and Madalina Croitoru. 2018. On a tion of Abstract Dialectical Frameworks (ADF) [8] that enrich ADF Flexible Representation for Defeasible Reasoning Variants. In Proc. of the acceptance condition. 17th International Conference on Autonomous Agents and Multiagent Systems After introducing SGs in Section 3 we show in Section 4 how the (AAMAS 2018), Stockholm, Sweden, July 10–15, 2018, IFAAMAS, 9 pages. flexible labelling function of SG can capture ambiguity blocking (Section 4.1), ambiguity propagating (Section 4.2), team defeat (Sec- 1 INTRODUCTION tion 4.3) and circular reasoning (Section 4.4). In Section 5 we evalu- ate the practical applicability of SGs. We demonstrate (Section 5.2) Defeasible reasoning [25] is used to evaluate claims or statements certain features of human reasoning empirically demonstrated by in an inconsistent setting. It has been successfully applied in many psychologists. Furthermore, we provide a tool (Section 5.1) that multi-agent domains such as legal reasoning [3], business rules [24], implements SGs and, despite its higher expressivity than the tools contracting [17], planning [15], agent negotiations [12], inconsis- in Table 1, performs better or equally good. tency management [23], etc. Defeasible reasoning can also be used for reasoning with uncertain rules in food science. In the EU H2020 NoAW project we are interested in reasoning logically about how 2 BACKGROUND NOTIONS to manage waste from wine by products. Such rules, elicited by ex- Language. We consider a propositional language of literals and the perts, non experts, consumers etc via online surveys have to be put connectives ¹^; !; ); º. A literal is either an atom (an atomic together and used as a whole for decision making. Unfortunately, formula) or the complement of an atom. The complement of atom there is no universally valid way to reason defeasibly. An inherent f is denoted f . > and ? are considered atoms. Given Φ a finite non- characteristic of defeasible reasoning is its systematic reliance on a empty conjunction of literals and a literal ψ , a rule r is a formula set of intuitions and rules of thumb, which have been long debated of the form Φ V ψ such that V2 f!; ); g. We call Φ the body between logicians [1, 19, 22, 26]. For example, could an information of r denoted B¹rº and ψ the head of r denoted H¹rº. The set R of derived from a contested claim be used to contest another claim rules is composed of: (i.e. ambiguity handling)? Could “chains” of reasoning for the same (1) R! the set of strict rules (of the form Φ ! ψ ) expressing claim be combined to defend against challenging statements (i.e. undeniable implications i.e. if Φ then definitely ψ . team defeat)? Is circular reasoning allowed? etc. (2) R) the set of defeasible rules (of the form Φ ) ψ ) express- 1 The main available defeasible reasoning tools are ASPIC+ [27], ing a weaker implication i.e. if Φ then generally ψ . DEFT [18], DeLP [14], DR-DEVICE [5], and Flora-2 [31]. Table 1 R (3) the set of defeater rules (of the form Φ ψ ) used to shows that no tools can support all features. prevent a complement of a conclusion i.e. if Φ then ψ should not be concluded. It does not imply that ψ is concluded. 1Available and runnable as of November 2017. Given a literal f , the rule of the form > ! f or > ) f is called Proc. of the 17th International Conference on Autonomous Agents and Multiagent Systems a fact rule. A derivation for f is a sequence of strict and defeasible (AAMAS 2018), M. Dastani, G. Sukthankar, E. André, S. Koenig (eds.), July 10–15, 2018, 2 R Stockholm, Sweden. © 2018 International Foundation for Autonomous Agents and rules that starts from a fact rule and end with a rule r s.t. Multiagent Systems (www.ifaamas.org). All rights reserved. H¹rº = f . A strict derivation contains only strict rules. 1123 Session 27: Argumentation AAMAS 2018, July 10-15, 2018, Stockholm, Sweden A defeasible knowledge base is a tuple KB = ¹F ; R; ≻º where of “responsible” is propagated to “дuilty” because “дuilty” can be F is a set of fact rules, R is a set of strict, defeasible and defeater derived (arд4 is allowed to defeat arд1), hence, the answer to the rules, and ≻ is a superiority relation on R. A superiority relation is query q is ‘false’ (i.e. KB 2prop дuilty, where ⊨prop denotes an acyclic relation ≻ (i.e. the transitive closure of ≻ is irreflexive). entailment in ambiguity propagating). If r1 ≻ r2, then r1 is called superior to r2, and r2 inferior to r1. This Ambiguity propagation results in fewer conclusions (as more expresses that r1 may override r2. A query (also called claim) on a ambiguities are allowed) and may be preferable when the cost of an knowledge base KB is a conjunction of literals. incorrect conclusion is high. Ambiguity blocking may be appropriate Defeasible reasoning. To reason defeasibly about a conclusion, in situations where contested claims cannot be used to contest other all chains of rule applications (also called arguments) reaching that claims (e.g. in the legal domain) [19]. conclusion must be evaluated along with any conflict (i.e. attack) Team Defeat. The absence of team defeat means that for an ar- that arises from other chains of rules. This can be achieved using gument to be undefeated it has to single-handedly defeat all its two kinds of approaches: (1) approaches based on the evaluation attackers, as shown in Example 2. of arguments during their construction, such as defeasible logics Example 2. Generally, animals do not fly unless they are birds. [6, 25] and (2) approaches using the idea of extensions, where Also, penguins do not fly except magical ones. ‘Tweety’ is an animal, arguments are built then evaluated at a later stage; these encapsulate a bird, and a magical penguin. Can ‘Tweety’ fly? KB = ¹F ; R; ≻º: argumentation-based techniques [13, 14]. •F f> ) ; > ) > ) > ) g An argument is a minimal sequence of rule applications from = animal bird, penдuin, maдical . a fact rule to a rule with a literal in its head called the conclusion •R = frd1 : animal ) f ly; rd2 : bird ) f ly; 0 of that argument. An argument arд attacks another argument arд rd3 : penдuin ) f ly; rd4 : maдical ^ penдuin ) f lyg. on a literal f if its conclusion is f and f is one of the literals that • (rd2 ≻ rd1), (rd4 ≻ rd3). 0 0 appear in arд . We say that arд defeats arд if the rule in arд gen- The query is q = f ly. In the absence of team defeat, the 0 noT D noT D erating f is not inferior to the rule in arд generating f . answer to q is ‘false’ (i.e. KB 2 f ly, where ⊨ denotes entailment in defeasible reasoning without team defeat) because Ambiguity Handling. A literal f is ambiguous if there is an there is no chain of reasoning for “f ly” that can defend itself from undefeated argument for f and another undefeated argument for all attacks: even if rd2 defends itself from rd1 (because rd2 ≻ rd1), f and the superiority relation does not state which argument is it does not defend against rd3 (since rd2 ⊁ rd3), and the same superior, as shown in Example 1. applies for rd4: it defends against rd3 but fails against rd1 because rd4 ⊁ rd1. If team defeat is allowed then the answer to q is ‘true’ Example 1. The following defeasible knowledge base KB = ¹F ; TD TD (i.e. KB ⊨ f ly, where ⊨ denotes entailment in defeasible R; º describes a situation where a piece of evidence ‘A’ suggests that ? reasoning with team defeat) because all attacks are defeated: rd1 is a defendant is responsible while an evidence ‘B’ indicates that he is defeated by rd2 (rd2 ≻ rd1) and rd3 is defeated by rd4 (rd4 ≻ rd3). not responsible; Both evidences are equally reliable. A defendant is Argumentation-based techniques for defeasible reasoning do not presumed not guilty unless responsibility has been proven: allow for team defeat, whereas defeasible logics do [2, 26].
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