Introduction Incremental Computation Experiments Conclusions and future work Efficient Computation of Extensions for Dynamic Abstract Argumentation Frameworks: An Incremental Approach Gianvincenzo Alfano, Sergio Greco, Francesco Parisi { g.alfano, greco, fparisi } @dimes.unical.it Department of Informatics, Modeling, Electronics and System Engineering University of Calabria Italy 26 th International Joint Conference on Artificial Intelligence August 19-25, 2017 Melbourne, Australia
Introduction Incremental Computation Experiments Conclusions and future work Motivation Argumentation in AI A general way for representing arguments and relationships (rebuttals) between them It allows representing dialogues, making decisions, and handling inconsistency and uncertainty Abstract Argumentation Framework (AF) [Dung 1995]: arguments are abstract entities (no attention is paid to their internal structure) that may attack and/or be attacked by other arguments Example (a simple AF) a a = Our friends will have great fun at our party on Saturday b = Saturday will rain (according to the weather forecasting service 1) b c = Saturday will be sunny (according to the weather forecasting service 2) c
Introduction Incremental Computation Experiments Conclusions and future work Motivation Argumentation Semantics Several semantics have been proposed to identify “reasonable” sets of arguments, called extensions Example (AF A 0 ) a c b Semantic S Set of extensions E S ( A 0 ) complete ( co ) {{ f , g } , { a , f , g } , { b , f , g }} preferred ( pr ) {{ a , f , g } , { b , f , g }} d e f stable ( st ) {{ b , f , g }} grounded ( gr ) {{ f , g }} g h Argumentation semantics can be also defined in terms of labelling Function L : A → { IN , OUT , UN } assigns a label (accepted, rejected, undecided) to each argument
Introduction Incremental Computation Experiments Conclusions and future work Motivation Argumentation Semantics Several semantics have been proposed to identify “reasonable” sets of arguments, called extensions Example (AF A 0 ) a c b Semantic S Set of extensions E S ( A 0 ) complete ( co ) {{ f , g } , { a , f , g } , { b , f , g }} preferred ( pr ) {{ a , f , g } , { b , f , g }} d e f stable ( st ) {{ b , f , g }} grounded ( gr ) { {f,g} } g h Argumentation semantics can be also defined in terms of labelling Function L : A → { IN , OUT , UN } assigns a label (accepted, rejected, undecided) to each argument
Introduction Incremental Computation Experiments Conclusions and future work Motivation Dynamic Abstract Argumentation Frameworks Most argumentation frameworks are dynamic systems, which are often updated by adding/removing arguments/attacks. For each semantics, extensions/labellings change if we update the initial AF by adding/removing arguments/attacks Example (Updated AF A = +( c , f )( A 0 ) ) a c b S E S ( A 0 ) E S ( A )) +( c, f ) {{ f , g } , { a , f , g } , { b , f , g }} ? co {{ a , f , g } , { b , f , g }} ? d e f pr {{ b , f , g }} ? st {{ f , g }} ? gr g h Should we recompute the semantics of updated AFs from scratch?
Introduction Incremental Computation Experiments Conclusions and future work Motivation Dynamic Abstract Argumentation Frameworks Most argumentation frameworks are dynamic systems, which are often updated by adding/removing arguments/attacks. For each semantics, extensions/labellings change if we update the initial AF by adding/removing arguments/attacks Example (Updated AF A = +( c , f )( A 0 ) ) a c b S E S ( A 0 ) E S ( A )) {{ f , g } , { a , f , g } , { b , f , g }} {{ g } , { a , g } , { b , f , g }} co {{ a , f , g } , { b , f , g }} {{ a , g } , { b , f , g }} pr d e f {{ b , f , g }} {{ b , f , g } } st {{ f , g }} { {g} } gr g h Should we recompute the semantics of updated AFs from scratch?
Introduction Incremental Computation Experiments Conclusions and future work Contributions Reduced AF We show that for four well-known semantics (i.e., grounded , complete , preferred , and stable ) an extension of the updated AF can be efficiently computed by looking only at a small part of the AF, called the Reduced AF, which is “influenced by” the update operation Example (From the updated AF to the Reduced AF ) a c b +( c, f ) ⇒ Reduced AF: e f d e f g h Once computed an extension for the reduced AF, it can be combined with the initial extension of the given AF to get an extension of the updated AF
Introduction Incremental Computation Experiments Conclusions and future work Contributions Incremental Algorithm We formally define the Reduced AF 1 Sub-AF consisting of the arguments whose status could change after an update It depends on both the update and the initial extension E 0 (and thus the semantics) We present an incremental algorithm for recomputing an extension of an 2 updated AF for the grounded , complete , preferred , and stable semantics It calls a non-incremental solver to compute an extension of the reduced AF It obtains the final extension by merging a portion of the initial extension with that computed for the reduced AF . A thorough experimental analysis showing the effectiveness of our 3 approach for all the four semantics Our technique outperforms the computation from scratch of the best solvers by two orders of magnitude
Introduction Incremental Computation Experiments Conclusions and future work Outline Introduction 1 Motivation Contributions Incremental Computation 2 Influenced Arguments Reduced Argumentation Framework Incremental Algorithm Experiments 3 Conclusions and future work 4 References
Introduction Incremental Computation Experiments Conclusions and future work Influenced Arguments Overview of the approach Given an initial AF A 0 , an extension E 0 , and an update u = ± ( a , b ) ERASE SOLVERS Three main steps/modules: Meta Solver 1) Identify a sub-AF A d = � A d , Σ d � , called reduced AF (R-AF) on the # " R-AF ! " OUTPUT basis of the updates in U and e f # e f additional information provided by a b c the initial extension E 0 Merger CoQuiAAS d e f R-AF Builder g h 2) Compute an S -extension E d of Cegartix the reduced AF A d by using an ! $ % # $ external (non-incremental) solver a c b c a b c +( c, f ) d e f d e f f 3) Merge E d with the portion g h g h INPUT ( E 0 \ A d ) of the initial extension Architecture of ERASE, our system for that does not change Efficiently Recomputing Argumentation SEmantics.
Introduction Incremental Computation Experiments Conclusions and future work Influenced Arguments Irrelevant updates (1/2) Updates preserving a given initial extension/labelling Cases for which E 0 is still an extension of the updated AF after a positive update update L 0 ( b ) +( a , b ) IN UN OUT co , pr , st , gr IN L 0 ( a ) co , gr co , pr , gr UN co , pr , st co , gr co , pr , st , gr OUT Example (For the update +( c , f ) the initial preferred extension E 0 = { b , f , g } is preserved, as L 0 ( c ) = OUT and L 0 ( f ) = IN ) initial labelling: updated labelling: a b c a b c +( c, f ) d e f d e f g h g h
Introduction Incremental Computation Experiments Conclusions and future work Influenced Arguments Irrelevant updates (2/2) Similar result for negative updates Cases for which E 0 is still an extension of the updated AF after a negative update update L 0 ( b ) − ( a , b ) IN UN OUT NA NA IN NA co , pr , gr UN L 0 ( a ) co , pr , st , gr co , pr , gr co , pr , st , gr OUT In these cases we do not need to recompute the semantics of the updated AF: just return the initial extension
Introduction Incremental Computation Experiments Conclusions and future work Influenced Arguments Influenced set: Intuition I ( u , A 0 , E 0 ) ) denotes the influenced set of u = ± ( a , b ) w.r.t. A 0 and E 0 1) I ( u , A 0 , E 0 ) = ∅ if u is irrelevant w.r.t. E 0 and the considered semantics. 2) The status of an argument can change only if it is reachable from b : I ( u , A 0 , E 0 ) ⊆ Reach A ( b ) 3) If argument z is not reachable from b and z ∈ E 0 , then also the status of the arguments attacked by z cannot change: their status remain OUT Example (Set of arguments influenced by an update operation) Update +( c , f ) is irrelevant w.r.t. the preferred a b c extension E 0 = { b , f , g } +( c, f ) ⇒ I (+( c , f ) , A 0 , { b , f , g } ) = ∅ d e f g h
Introduction Incremental Computation Experiments Conclusions and future work Influenced Arguments Influenced set: Intuition I ( u , A 0 , E 0 ) ) denotes the influenced set of u = ± ( a , b ) w.r.t. A 0 and E 0 1) I ( u , A 0 , E 0 ) = ∅ if u is irrelevant w.r.t. E 0 and the considered semantics. 2) The status of an argument can change only if it is reachable from b : I ( u , A 0 , E 0 ) ⊆ Reach A ( b ) 3) If argument z is not reachable from b and z ∈ E 0 , then also the status of the arguments attacked by z cannot change: their status remain OUT Example (Set of arguments influenced by an update operation) a b c I (+( c , f ) , A 0 , E 0 ) ⊆ Reach A ( f ) = { e , d , a , b , c } +( c, f ) ⇒ g , h �∈ I (+( c , f ) , A 0 , E 0 ) d e f g h
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