3. Argumentation Frameworks

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1 3. Argumenttion Frmeworks Argumenttion current hot topic in AI. Historiclly more recent thn other pproches discussed here. Bsic ide: to construct cceptble set(s) of beliefs from given KB: 1 construct rguments (beliefs with ssocited resons), 2 determine jointly cceptble rguments (extensions), 3 ccept their conclusions. Assumption: step 2 cn be done independently nd bstrctly. Dung s Abstrct Argumenttion Frmeworks widely used tool. Brewk/Woltrn () Nonmonotonic Resoning Mrch / 23

2 Argumenttion Frmeworks, ctd. Abstrct Argumenttion Arguments re tomic, their structure irrelevnt. All tht mtters re ttcks mong rguments. Argumenttion frmeworks (AFs) represent ttck reltions. Semntics formlize different intuitions bout how to solve conflicts nd how to pick cceptble rguments. Semntics mp n AF to subsets of its rguments (extensions). Nonmonotonic: new rgument my throw out wht ws ccepted. Brewk/Woltrn () Nonmonotonic Resoning Mrch / 23

3 Definition Argumenttion Frmeworks An rgumenttion frmework (AF) is pir (A, R) where A is set of rguments, R A A is reltion representing ttcks. ( defets ) Brewk/Woltrn () Nonmonotonic Resoning Mrch / 23

4 Definition Argumenttion Frmeworks An rgumenttion frmework (AF) is pir (A, R) where A is set of rguments, R A A is reltion representing ttcks. ( defets ) Exmple Brewk/Woltrn () Nonmonotonic Resoning Mrch / 23

5 Semntics: miniml requirement no conflicts Conflict-Free Set Given n AF F = (A, R). A set S A is conflict-free in F, if, for ech, b S, (, b) / R. Exmple cf (F) = { {, c}, Brewk/Woltrn () Nonmonotonic Resoning Mrch / 23

6 Semntics: miniml requirement no conflicts Conflict-Free Set Given n AF F = (A, R). A set S A is conflict-free in F, if, for ech, b S, (, b) / R. Exmple cf (F) = { {, c}, {, d}, Brewk/Woltrn () Nonmonotonic Resoning Mrch / 23

7 Semntics: miniml requirement no conflicts Conflict-Free Set Given n AF F = (A, R). A set S A is conflict-free in F, if, for ech, b S, (, b) / R. Exmple cf (F) = { {, c}, {, d}, {b, d}, Brewk/Woltrn () Nonmonotonic Resoning Mrch / 23

8 Semntics: miniml requirement no conflicts Conflict-Free Set Given n AF F = (A, R). A set S A is conflict-free in F, if, for ech, b S, (, b) / R. Exmple cf (F) = { {, c}, {, d}, {b, d}, {}, {b}, {c}, {d}, } Brewk/Woltrn () Nonmonotonic Resoning Mrch / 23

9 No undefended ttcked rguments Admissible Set Given n AF F = (A, R). A set S A is dmissible in F, if S is conflict-free in F ech S is defended by S in F, A is defended by S in F, if for ech b A with (b, ) R, there exists c S, such tht (c, b) R. Exmple dm(f) = { {, c}, Brewk/Woltrn () Nonmonotonic Resoning Mrch / 23

10 No undefended ttcked rguments Admissible Set Given n AF F = (A, R). A set S A is dmissible in F, if S is conflict-free in F ech S is defended by S in F, A is defended by S in F, if for ech b A with (b, ) R, there exists c S, such tht (c, b) R. Exmple dm(f) = { {, c}, {, d}, Brewk/Woltrn () Nonmonotonic Resoning Mrch / 23

11 No undefended ttcked rguments Admissible Set Given n AF F = (A, R). A set S A is dmissible in F, if S is conflict-free in F ech S is defended by S in F A is defended by S in F, if for ech b A with (b, ) R, there exists c S, such tht (c, b) R. Exmple dm(f) = { {, c}, {, d}, {b, d}, Brewk/Woltrn () Nonmonotonic Resoning Mrch / 23

12 No undefended ttcked rguments Admissible Set Given n AF F = (A, R). A set S A is dmissible in F, if S is conflict-free in F ech S is defended by S in F, A is defended by S in F, if for ech b A with (b, ) R, there exists c S, such tht (c, b) R. Exmple dm(f) = { {, c}, {, d}, {b, d},{}, {b}, {c}, {d}, } Brewk/Woltrn () Nonmonotonic Resoning Mrch / 23

13 Wnt ll defended rguments Complete Set Given n AF F = (A, R). A set S A is complete in F, if S is dmissible in F ech A defended by S in F is contined in S A is defended by S in F, if for ech b A with (b, ) R, there exists c S, such tht (c, b) R. Exmple comp(f) = { {, c}, {, d}, {}, {c}, {d}, } Brewk/Woltrn () Nonmonotonic Resoning Mrch / 23

14 A skepticl pproch Grounded Extension Given n AF F = (A, R). A set S A is grounded in F, if S is complete in F for ech T A complete in F, T S Proposition [Dung 95]: The grounded extension of n AF F = (A, R) is given by the lest fix-point of the opertor Γ F : 2 A 2 A, defined s Γ F (S) = { A is defended by S in F} Exmple ground(f) = { {, c}, {, d},{} } Brewk/Woltrn () Nonmonotonic Resoning Mrch / 23

15 A credulous pproch Stble Extension Given n AF F = (A, R). A set S A is stble in F, if S is conflict-free in F for ech A \ S, there exists b S, such tht (b, ) R. Exmple stble(f) = { {, c}, Brewk/Woltrn () Nonmonotonic Resoning Mrch / 23

16 A credulous pproch Stble Extension Given n AF F = (A, R). A set S A is stble in F, if S is conflict-free in F for ech A \ S, there exists b S, such tht (b, ) R. Exmple stble(f) = { {, c},{, d}, Brewk/Woltrn () Nonmonotonic Resoning Mrch / 23

17 A credulous pproch Stble Extension Given n AF F = (A, R). A set S A is stble in F, if S is conflict-free in F for ech A \ S, there exists b S, such tht (b, ) R. Exmple stble(f) = { {, c},{, d}, {b, d}, Brewk/Woltrn () Nonmonotonic Resoning Mrch / 23

18 A credulous pproch Stble Extension Given n AF F = (A, R). A set S A is stble in F, if S is conflict-free in F for ech A \ S, there exists b S, such tht (b, ) R. Exmple stble(f) = { {, c},{, d}, {b, d}, {}, {b}, {c}, {d}, } Brewk/Woltrn () Nonmonotonic Resoning Mrch / 23

19 Gurnteeing existence of extensions Preferred Extension Given n AF F = (A, R). A set S A is preferred in F, if S is dmissible in F for ech T A dmissible in T, S T Exmple pref (F) = { {, c}, {, d}, {}, {c}, {d}, } Brewk/Woltrn () Nonmonotonic Resoning Mrch / 23

20 Complexity Reltion between Semntics stble pref ground compl dm Complexity stble dm pref comp ground Cred NP-c NP-c NP-c NP-c in P Skept conp-c (trivil) Π P 2 -c in P in P [Dimopoulos & Torres 96; Dunne & Bench-Cpon 02; Coste-Mrquis et l. 05] Brewk/Woltrn () Nonmonotonic Resoning Mrch / 23

21 Complexity Reltion between Semntics stble pref ground compl dm Complexity stble dm pref comp ground Cred NP-c NP-c NP-c NP-c in P Skept conp-c (trivil) Π P 2 -c in P in P [Dimopoulos & Torres 96; Dunne & Bench-Cpon 02; Coste-Mrquis et l. 05] Brewk/Woltrn () Nonmonotonic Resoning Mrch / 23

22 Further results nd conclusions AFs: simple grph representtion of rgumenttion scenrios. Semntics mp AFs to collection of sets of rguments. grounded: (1) ccept unttcked rgs, (2) delete rgs ttcked by ccepted rgs, (3) goto 1, stop when fixpoint reched. preferred: mximl conflict-free sets ttcking ll their ttckers. stble: conflict free sets ttcking ll unccepted rgs. Grounded lwys unique, others my produce multiple extensions. Unlike stble extensions preferred extensions lwys exist. Grounded extension subset of ech preferred (nd thus ech stble) extension. Extending n AF my chnge extensions nonmonotoniclly. Mny other semntics hve been defined. Brewk/Woltrn () Nonmonotonic Resoning Mrch / 23

23 Further results nd conclusions AFs: simple grph representtion of rgumenttion scenrios. Semntics mp AFs to collection of sets of rguments. grounded: (1) ccept unttcked rgs, (2) delete rgs ttcked by ccepted rgs, (3) goto 1, stop when fixpoint reched. preferred: mximl conflict-free sets ttcking ll their ttckers. stble: conflict free sets ttcking ll unccepted rgs. Grounded lwys unique, others my produce multiple extensions. Unlike stble extensions preferred extensions lwys exist. Grounded extension subset of ech preferred (nd thus ech stble) extension. Extending n AF my chnge extensions nonmonotoniclly. Mny other semntics hve been defined. Brewk/Woltrn () Nonmonotonic Resoning Mrch / 23

24 Further results nd conclusions AFs: simple grph representtion of rgumenttion scenrios. Semntics mp AFs to collection of sets of rguments. grounded: (1) ccept unttcked rgs, (2) delete rgs ttcked by ccepted rgs, (3) goto 1, stop when fixpoint reched. preferred: mximl conflict-free sets ttcking ll their ttckers. stble: conflict free sets ttcking ll unccepted rgs. Grounded lwys unique, others my produce multiple extensions. Unlike stble extensions preferred extensions lwys exist. Grounded extension subset of ech preferred (nd thus ech stble) extension. Extending n AF my chnge extensions nonmonotoniclly. Mny other semntics hve been defined. Brewk/Woltrn () Nonmonotonic Resoning Mrch / 23

25 Further results nd conclusions AFs: simple grph representtion of rgumenttion scenrios. Semntics mp AFs to collection of sets of rguments. grounded: (1) ccept unttcked rgs, (2) delete rgs ttcked by ccepted rgs, (3) goto 1, stop when fixpoint reched. preferred: mximl conflict-free sets ttcking ll their ttckers. stble: conflict free sets ttcking ll unccepted rgs. Grounded lwys unique, others my produce multiple extensions. Unlike stble extensions preferred extensions lwys exist. Grounded extension subset of ech preferred (nd thus ech stble) extension. Extending n AF my chnge extensions nonmonotoniclly. Mny other semntics hve been defined. Brewk/Woltrn () Nonmonotonic Resoning Mrch / 23

26 Further results nd conclusions AFs: simple grph representtion of rgumenttion scenrios. Semntics mp AFs to collection of sets of rguments. grounded: (1) ccept unttcked rgs, (2) delete rgs ttcked by ccepted rgs, (3) goto 1, stop when fixpoint reched. preferred: mximl conflict-free sets ttcking ll their ttckers. stble: conflict free sets ttcking ll unccepted rgs. Grounded lwys unique, others my produce multiple extensions. Unlike stble extensions preferred extensions lwys exist. Grounded extension subset of ech preferred (nd thus ech stble) extension. Extending n AF my chnge extensions nonmonotoniclly. Mny other semntics hve been defined. Brewk/Woltrn () Nonmonotonic Resoning Mrch / 23

27 Further results nd conclusions AFs: simple grph representtion of rgumenttion scenrios. Semntics mp AFs to collection of sets of rguments. grounded: (1) ccept unttcked rgs, (2) delete rgs ttcked by ccepted rgs, (3) goto 1, stop when fixpoint reched. preferred: mximl conflict-free sets ttcking ll their ttckers. stble: conflict free sets ttcking ll unccepted rgs. Grounded lwys unique, others my produce multiple extensions. Unlike stble extensions preferred extensions lwys exist. Grounded extension subset of ech preferred (nd thus ech stble) extension. Extending n AF my chnge extensions nonmonotoniclly. Mny other semntics hve been defined. Brewk/Woltrn () Nonmonotonic Resoning Mrch / 23

28 Further results nd conclusions AFs: simple grph representtion of rgumenttion scenrios. Semntics mp AFs to collection of sets of rguments. grounded: (1) ccept unttcked rgs, (2) delete rgs ttcked by ccepted rgs, (3) goto 1, stop when fixpoint reched. preferred: mximl conflict-free sets ttcking ll their ttckers. stble: conflict free sets ttcking ll unccepted rgs. Grounded lwys unique, others my produce multiple extensions. Unlike stble extensions preferred extensions lwys exist. Grounded extension subset of ech preferred (nd thus ech stble) extension. Extending n AF my chnge extensions nonmonotoniclly. Mny other semntics hve been defined. Brewk/Woltrn () Nonmonotonic Resoning Mrch / 23

29 Restrictions of AFs Exmple Fixed mening of links: ttck. Fixed cceptnce condition for rgs: no prent ccepted. Wnt more flexibility: 1 Links supporting rguments/positions, 2 Nodes not ccepted unless supported, 3 Flexible mens of combining ttck nd support. From clculus of opposition to clculus of support nd opposition. Current work in our group: generlize to Dilecticl Frmeworks where ech node hs its own cceptnce condition. Brewk/Woltrn () Nonmonotonic Resoning Mrch / 23

30 Argumenttion in AI (ctd.) Literture C. Chesnevr, A. Mguitmn, R. Loui: Logicl models of rgument. ACM Comput. Surv. 32(4): (2000) T. Bench-Cpon, P. Dunne: Argumenttion in Artificil Intelligence. Artif. Intell. 171(10-15): (2007) P. Besnrd, A. Hunter: Elements of Argumenttion. The MIT Press (2008). G. Simri, I. Rhwn (eds.): Argumenttion in Artificil Intelligence. Springer, Brewk/Woltrn () Nonmonotonic Resoning Mrch / 23

A Short Introduction to Abstract Argumentation Frameworks

A Short Introduction to Abstract Argumentation Frameworks A Short Introduction to Abstrct Argumenttion Frmeworks Stefn Woltrn Technische Universität Wien Gerhrd Brewk Universität Leipzig November 2009 () Abstrct Argumenttion Frmeworks November 2009 1 / 20 Argumenttion