Argumentation Semantics for Contextual Defeasible Logic

1. Antonis Bikakis University College London Based on the joint work with Grigoris Antoniou The London Argumentation Forum 2012, King’s College London Argumentation Semantics for Contextual Defeasible Logic

2. Overview Background Contextual Defeasible Logic Representation Model Argumentation Semantics Properties More about CDL Operational Semantics, Applications, Future Work…

3. Background • Context in AI • A partial and approximate theory of the world from some individual’s perspective (McCarthy, 1987) • A logical theory – a set of axioms and inference rules • Multi-Context Systems (Giunchiglia &Trento group) • Distributed context theories connected through mappings that enable information flow between different contexts • Mappings modeled as inference rules with premises and consequences in different contexts

4. Background • Nonmonotonic MCS • MCS enriched with nonmonotonic features to handle imperfections, e.g. incomplete knowledge, inconsistencies Context A k ¬k Context C Context B

5. Background • Nonmonotonic MCS (Vienna Group) • Bridge rules modeled as default rules • Diagnoses / Explanations to resolve inconsistency • Centralized inconsistency resolution (global monitoring) • Contextual Defeasible Logic • Bridge rules modeled as defeasible rules • Preference information on contexts to resolve inconsistency • Distributed inconsistency resolution (local view)

6. Overview Background Contextual Defeasible Logic Representation Model Argumentation Semantics Properties Future Steps

7. Representation Model A Defeasible MCS Cis a collection of distributed defeasible theories Ci Each context Ciis a tuple (Vi , Ri, Ti ) • Vi: vocabulary used by Ci • Ri: set of rules • Ti: preference ordering on C Vi: a set of positive literals and their negations

8. Representation Model Three types of rules inRi • Strict local rules ril : ai1 , ai2 ,…, ain-1→ ain • Defeasible local rules rid : ai1 , ai2 ,…, ain-1ain • Mapping rules rim : ai1 , aj2 ,…, akn-1 aln Tiis a partial preference ordering on C modeled as a Directed Acyclic Graph

9. Argumentation Semantics • Extends the argumentation semantics of Defeasible Logic • Distribution of available knowledge • Preference information • Main Features • Arguments with local range • Arguments made by different contexts associated through mapping rules • Partial preference preorder on the set of arguments • Variants • Blocking / Propagating Ambiguity • With / Without Team Defeat

10. Support Relation (SRC) • Nodes ofPTpi labeled by literals: • Root labeled by pi • For every node with label q • If q in Vi and a1, a2,…, an label the children of q then there is a rule ri in Ciwith body a1, a2,…, an and head q • If q does not belong to Vithen this is a leaf node, and there is a triple of the form (Cj, PTq, q) in SRC • Arcs ofPTpi labeled by the rules used to obtain them • Set of triples of the form (Ci, PTpi, pi) • Ci: context inC, • pi: literal inVi,, • PTpi: proof tree forpi

11. Argument • pi is the conclusion of A • Any literal labeling a node in A is a conclusion of A • A’ is a (proper) subargumentof A if its proof tree is a (proper) subtree of the proof tree of A • A is a local argument ofCi if it contains only literals from Vi– otherwise it is a mapping argument ofCi • Strict local arguments contain only strict local rules • Defeasible local arguments contain at least one defeasible local rule • ArgsCiis the set of all arguments in Ci • ArgsC is the set of all arguments in C An argument Aforpiis a triple(Ci , PTpi, pi)inSRC

12. Example 1 • Consider the following context theory C1 • r11l: a1 → x1 r15d: b1 • r12m: a2a1 r16l: d1→ ¬b1 • r13m: a3 , a4¬a1r17l: → d1 • r14d: b1x1 • Arguments inArgsC1 A1 B1 A3 A4 x1 x1 ¬a1 ¬b1 r13 r13 r14 r16 r11 a1 a3 a4 b1 d1 r12 r15 r17 a2

13. Preference • An argument Ais preferred to argument Bin context Ciiff one of the following conditions hold • Ais a strict local argument of CiandBis not • Ais a local argument of CiandBis not • Both arguments are mapping arguments of Ciand for all nodes labeled by a foreign literal akinA(akinVk≠Vi) there is a node labeled by a foreign literalblinB (blinVl≠Vi) such that akis preferred to blinCi • - akis preferred toblinCiiff there is a path fromCltoCkinTi Partial Order on Contexts => Partial Preorder on Arguments

14. Attack • An argument Aattacks an argumentBatpif • pis a conclusion of B, • ¬pis a conclusion of A, and • B’ is not preferred to A’

15. Example 1 (cont’d) A1 B1 A3 A4 x1 ¬a1 x1 ¬b1 r13 r13 r14 r16 r11 a1 a3 a4 b1 d1 r12 r15 r14 a2 AssumingT1 = {[C2, C4]} • A1attacksB1at¬a1 • B1does not attackA1ata1 • A4attacksA3atb1 • A3does not attack A4(strict local argument)

16. Argumentation Line • Head of argumentation lineALis the argument added in step 1 • pis called the conclusion of AL • ALis a finite argumentation line if the number of steps required to build it is finite • An argumentation line ALfor a literal pis a sequence of arguments constructed in steps as follows • In the first step add in ALone argument for p • In each next step, for each distinct literal qj labeling a leaf node of theproof trees of the arguments added in the previous step, add one argument with conclusionqj

17. Support - Undercut • An argumentAis supported by a set of arguments Sif • Every proper subargument of Ais inSand • There is a finite argumentation lineALwith head Asuch that every argument inAL – {A}is inS • An argument Ais undercut by a set of arguments Siffor every argumentation lineALwith headAthere is an argument Bs.t. • Bis supported by S and • Battacks a proper subargument of A or an argument inAL – {A}

18. Example 2 C1 C2 A1 A1 B1 A2 B2 x1 ¬a1 a2 ¬a2 A1’ a3 a4 a1 a5 a6 T1 = [C3, C2, C4] T2 = [C6, C5] a2 C3 C4 C5 C6 B3 B4 A5 B6 A6 a3 a4 a5 a6 ¬a6 • Argumentation lines:AL1={A1, A2, A5}, BL1={B1, B3, B4} , BL2={B2, B6} • Assuming thatS={A5, A6}, A2 supported by S, B2 undercut byS • Assuming that S={A5, A6 , B3 , B4 , A2}, B1, A1’supported byS, A1’not undercut byS

19. Acceptability - Justifiability • An argumentAis acceptablew.r.t. a set of arguments Sif • Ais a strict local argument or • Ais supported bySand every argument attacking Ais undercut byS • The set of justified argumentsis defined as JArgsC = UJiCwhere • J0C = {} • Ji+1C = {A | A is acceptable w.r.t.JiC} A literal piis justified if it is a conclusion of an argument inJArgsC

20. Refutability • An argumentAis rejected by a set of argumentsSwhen • Ais undercut bySor • Ais attacked by an argument that is supported by S rejected arguments(RArgsC): set of arguments rejected by JArgsC A literal pis rejected if there is no argument forpinArgsC-RArgsC

21. Example 2 (cont’d) C1 C2 A1 A1 B1 A2 B2 x1 ¬a1 a2 ¬a2 A1’ a3 a4 a1 a5 a6 T1 = [C2, C4] T2 = [C6, C5] a2 C3 C4 C5 C6 B3 B4 A5 B6 A6 a3 a4 a5 a6 ¬a6 • J0C={} • J1C={B3, B4, A5,A6} • J2C={B3, B4, A5,A6 ,A2} • J3C={B3, B4, A5,A6 ,A2 ,A1’} • J4C={B3, B4, A5,A6,A2 ,A1’,A1} = JArgsC • RArgsC={B6, B2, B1}

22. Properties of Argumentation System • The sequence JiCis monotonically increasing • No argument is both justified and rejected. • No literal is both justified and rejected • If the set of justified arguments JArgsCcontains two arguments with contradictory conclusions, then both are strict local arguments • Assuming consistency in the strict local rules of each context, the entire framework is consistent

23. More about CDL • Operational Semantics • Algorithms for distributed query evaluation • Alternative strategies for conflict resolution • Implemented in Logic Programming • Applications • Mobile Social Networks • Ambient Intelligence (Internet of Things) • Future Work • Relation with Abstract Argumentation Frameworks • Preference-based Afs, Context Argumentation Systems • Access Control Layer • Large-scale applications

24. Thank you for your attention! Questions? The London Argumentation Forum 2012, King’s College London Argumentation Semantics for Contextual Defeasible Logic