A Recursive Approach to Argumentation: Motivation and Perspectives Pietro Baroni, Massimiliano Giacomin {baroni, giacomin}@ing.unibs.it DEA - Dipartimento di Elettronica per l’Automazione Università degli Studi di Brescia (Italy) NMR 2004, Whistler BC, Canada

Aim of the work Context: Dung’s abstract argumentation framework Grounded Semantics Stable semantics Preferred Semantics Argumentation semantics Problems with odd-length cycles: [Baroni & Giacomin, ECSQARU 03]: Solution based on a recursive definition of extensions [Baroni & Giacomin, ECAI 04]: General Recursive Schema including Dung’s semantics In this paper: Role of the Recursive Schema to support novel proposals: definition of four new semantics overcoming the problems

Dung’s Argumentation Framework AF = <A, > attack relation arguments Semantics Defeat graph Extensions Defeat Status Justified arguments : belong to all extensions

The core of Dung’s theory: Admissibility Acceptability  acceptable w.r.t. (“defended by”) S all defeaters of  are defeated by S  S Admissibility of a set S conflict-free every element acceptable w.r.t. S (defends all of its elements) IF Complete extension contains all acceptable elements w.r.t. itself

The Grounded Semantics Grounded extension: Least complete extension (unique status approach) Most skeptical semantics Undefeated Defeated Provisionally Defeated Defeat status

The Preferred Semantics Preferred extensions Maximal complete extensions = max Set: is conflict-free defends all of its elements (multiple status approach) Defeat status of arguments UNDEFEATED: belongs to all of the extensions DEFEATED: out from all of the extensions PROVISIONALLY DEFEATED: belongs to at least one extension, does not belong to all of the extensions

Floating arguments b b g g d d a a b g d Preferred Semantics a VS. b Grounded Semantics g d a

Preferred Semantics: a problematic example involving odd-length cycles NB: grounded semantics yields all arguments provisionally defeated VS b d g a f1 f2 [ECSQARU’03]: solution based on a recursive definition along SCCs

Generalizing the use of recursion: SCC-based computation Computation of extensions along Strongly Connected Components S3 S4 S1 SCCs form an acyclic graph E = S (ES) S2 S5 Possible choices of extension elements in a SCC S only depend on choices made in SCCs that precede S in the graph

A General Recursive Schema: “Edge conditions” SD(E) : nodes attacked by E SU(E) : nodes defended by E SP(E) : nodes not attacked nor defended by E S Choices in preceding components SD(E) SP(E) SU(E) Possible choices in [SP(E)SU(E)] expressed as a function FG ( SU(E) ) AF[SP(E)SU(E)]

A General Recursive Schema: Recursive Computation FG (SU(E)) computed recursively AF[SP(E)SU(E)] AF[SP(E)SU(E)] More than one SCC in AF[SP(E)SU(E)] : function computed recursively on each of them SD(E) SP(E) SU(E) One SCC only in AF[SP(E)SU(E)] : base of the recursion: “base function” FG* (SU(E)) AF[SP(E)SU(E)] Semantics specified by a base function defined on single-SCC argumentation frameworks

A General Recursive Schema: Definition of SCC-recursive semantics Given AF = <A, >, E  A is an extension iff E  FGAF(A) For all C  A, E  FGAF(C) iff in case |SCC(AF)|=1 or A= E  FG*AF(C) otherwise:  S  SCC(AF) (E  S)  FG (SU(E)C) AF[SP(E)SU(E)] Definition parametric w.r.t. FG*AF(C): includes complete, grounded and preferred semantics (see ECAI’04)

Looking for a new notion of extension: Exploring the space of SCC-Recursiveness Exploring SCC-recursive semantics  exploring alternative base functions Preferred extensions = maximal admissible sets Maintain admissibility  give up maximality Give up admissibility NEW NOTION OF EXTENSION

Beyond Preferred Semantics S2P(E) S2U(E) b g f1 f2 a S2 S1 {2}  E admissible  (E  S1) =   1  E Complete Extensions {} if C  A FG* ({2}) = {} FG*AF(C) = AF[S2] ? otherwise

Beyond Preferred Semantics: AD1 {} if C  A FG*AF(C) = P E AF otherwise correctly deals with the above example, but… b A unique (preferred) extension g d  and  undefeated,  and  defeated a

Beyond Preferred Semantics: AD2 g unique defeater of a and f g is not “fully attacked” (b  E) g f a {} if C  A FG*AF(C) = {E | E is maximal in AS*AF } otherwise AS*AF = {F  A : F admissible and  g: g F F includes all attackers of g}

Beyond Preferred Semantics: adherence to Dung’s framework AD2 correctly deals with all the examples considered so far (e.g. floating arguments) AD1 fails in one example AD1 and AD2 extensions are complete extensions  adhere to Dung’s fundamental notion of admissibility Other alternative: relax the admissibility constraint…

b g a b b b g g g a a a Beyond Admissibility  one node at most Non empty extensions NO CONTRADICTIONS  one node at most SYMMETRY  all nodes treated equally b g a b b b g g g a a a Maximal conflict-free sets

Beyond Admissibilty: CF1 and CF2 semantics FG*AF(C) = Maximal Conflict-Free Sets of AFC S SD(E) SP(E) SU(E) CF2 FG*AF(C) = Maximal Conflict-Free Sets of AF SP(E) U SU(E) Role of “defence” completely ignored S SD(E)

CF1 and CF2 semantics: an example First alternative for (ES1) S2D(E) S2U(E) b g f1 f2 a S2 S1

CF1 and CF2 semantics: an example (2) Second alternative for (ES1) b g f1 f2 a S1 S2=S2U(E) b g f1 f2 a

CF1 and CF2 semantics: an example (3) Third alternative for (ES1) S2P(E) S2U(E) b g f1 f2 a S1 S2 b S2P(E) S2U(E) CF2 only! g f1 f2 a

Comparing SCC-Recursive Semantics Any SCC-Recursive semantics agrees with grounded semantics: each extension includes the grounded extension (provided a simple condition on the base function is verified) AD vs. CF: different treatment of odd-length cycles b b g d f g d f a a b Empty set is the only admissible set CF1 and CF2: enforce floating defeat and acceptance g d f a

Comparing SCC-Recursive Semantics: Self-defeating arguments a self-defeating: should not prevent b from being justified {b} should be the unique extension a b Any admissibility-based semantics fails (and CF1 too) Treatment by CF2: a b Max C.F.Set:  S2=S2P(E) Max C.F.Set: {b}

Conclusions A general recursive schema: SCC-recursiveness as a unifying notion SCC-recursiveness as a “source” of novel proposals Overcoming problems of preferred semantics Most satisfactory approach: CF2, which fully departs from the notion of admissibility Future work: General properties of the SCC-recursive schema Further exploration and comparison between semantics Examples in different application domains [Incremental] algorithms based on the SCC-recursive schema