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 (ES) 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 AFC 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 (ES1) S2D(E) S2U(E) b g f1 f2 a S2 S1
CF1 and CF2 semantics: an example (2) Second alternative for (ES1) b g f1 f2 a S1 S2=S2U(E) b g f1 f2 a
CF1 and CF2 semantics: an example (3) Third alternative for (ES1) 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