Arguments and Misunderstandings: A Fuzzy Approach to Conflict Resolution in Open Systems

1. Arguments and Misunderstandings:A Fuzzy Approach to Conflict Resolution in Open Systems

2. Overview • Motivation: Expressive Knowledge Representation • Part I: Argumentation as LP semantics • Notions of attack and justified arguments • Hierarchy of semantics • Proof procedure • Part II: Fuzzy unification and argumentation • Fuzzy negation • Fuzzy argumentation • Fuzzy unification (work done together with David Gilbert)

3. Knowledge representation • Pete earns 500.000$ p.a. • earns(pete,500000). • Cross the street if there are no cars • cross  not car • cross   car • The fridge is quite cheap • cheap(fridge):70% • Does Mike live in Londn? • address(mike,london) = address(mike,londn): 95%

4. fdFB fdDB dDB dFB fDB fFB rDB rFB fuzzy deductive negation Knowledge System Cube r: relational f: fuzzy d: deductive DB: database FB: factbase

5. Part I:Argumentation as semantics for Extended Logic Programs fdFB fdDB r: relational f: fuzzy d: deductive DB: database FB: factbase dDB dFB fDB fFB rDB rFB fuzzy deductive negation

6. Extended Logic Programming • Logic Programming with 2 negations • Default negation: not p : true if all attempts to prove p fail. • Explicit negation: p : falsehood of a literal may be stated explicitly. • Coherence principle: p  not p

7. Argumentation • Interaction between agents in order to • gain knowledge • revise existing knowledge • convince the opponent • solve conflicts • Elegant way to define semantics for (extended) logic programming • Dung • Kowalski, Toni, Sadri • Prakken & Sartor • Etc.

8. Arguments • An argument is a partial proof, with implicitly negated literals as assumptions. • Formally: Argument for objective literal L: • sequence of rules [ r1, …, rn ] such that • L is the head of r1 ; • no two rules have the same head ; • for each objective literal L’ in the body of a rule ri, there is a rule rj (j > i) such that L’ is the head of rj.

9. Attacking arguments • Two fundamental kinds of attack: • A undercuts B = A invalidates premise of B • A rebuts B = A contradicts B • Derived notions of attack used in Literature: • A attacks B = A u B or A r B • A defeats B = A u B or (A r B and not B u A) • A strongly attacks B = A a B and not B u A • A strongly undercuts B = A u B and not B u A

10. Proposition: Hierarchy of attacks Attacks = a = u  r Defeats = d = u  ( r - u -1) Undercuts = u Strongly attacks = sa = (u  r ) - u -1 Strongly undercuts = su = u - u -1

11. Fixpoint Semantics • Argumentation: • game between proponent and opponent • argument A is acceptable if opponent’s x-attack is countered by proponent’s y-attack, which proponent already accepted earlier. • Acceptable • Let x,y be notions of attack. • An argument A is x,y-acceptable w.r.t. a set of arguments S iff • for every argument B, such that (B,A)  x, there is a C  S such that (C,B)  y • Fixpoint semantics • Fx/y (S) = { A | A is x,y-acceptable w.r.t. S } • x/y-justified arguments = Least Fixpoint of Fx/y. • x/y-overruledarguments = x-attacked by a justified argument. • x/y-defensible iff neither justified nor overruled

12. Theorem: Relationship of semantics Prakken and Sartor’ssemantics w/o priorities If opponent is allowed to attack,type of defense does not matter • Weakening opponent or strengthening proponent increases justified arguments • Different notions of acceptability give rise to different argumentation semantics If opponent is allowed defeat,type of defense does not matter Dung’s groundedargumentation semantics WFSX su/a=su/d If opponent is allowed undercut,defense with (a,u,sa) or without(su,u) rebut makes a difference su/u su/sa sa/u=sa/d=sa/a su/su u/a=u/d=u/sa sa/su=sa/sa u/su=u/u d/su=d/u=d/a=d/d=d/sa a/su=a/u=a/a=a/d=a/sa

13. Proof procedure • Dialogues: • x/y-dialogue is sequence of moves such that • Proponent and Opponent alternate • Players cannot repeat arguments • Opponent x-attacks Proponent’s last argument • Proponent y-attacks Opponent’s last argument • Player wins dialogue if other player cannot move • Argument A is provably justified if proponent wins all branches of dialogue tree with root A • Concrete implementation SLXA: • Since u/a=u/d=u/sa=WFSX  compute justified arguments with top-down proof procedure SLXA for WFSX [Alferes, Damasio, Pereira] • SLXA can be adapted for other notions

14. Part II:Fuzzy unification and argumentation fdFB fdDB r: relational f: fuzzy d: deductive DB: database FB: factbase dDB dFB fDB fFB rDB rFB fuzzy deductive negation

15. Classical Fuzzy Logic • Solution: • Truth values in [0,1] instead of {0,1}. • Assertions: • p:V (p a formula, V a truth value). • Conjunction: • p:V, q:W p q : min(V,W) • Disjunction: • p:V, q:W p q : max(V,W) • Inference: • p  q1, …, qn ; q1:V1, …, qn:Vnp : min(V1, …, Vn)

16. Fuzzy Negation • Classical fuzzy negation: • L:V L: 1-V (Zadeh) • Our setting (fuzzy adaptation of WFSX): • L:V and L:V’ with V’  1-V possible • L and L not directly related.

17. Fuzzy Coherence Principle • If L:V and V > 0, and not L:V’, then V’ > V. • “If there is some explicit evidence that L is false, then there is at least the same evidence that L is false by default.” •  If L:V and V > 0, then not L: 1.

18. p  p :V V > 0 possible Contradictory programs! not p p : V V > 0 possible By coherence principle! Contradiction removal not p  p : V V > 0 p  p : V V = 0 possible p is unknown Law of excluded... ...contradiction ...middle

19. Strength of an argument • Strength st of an argument: • st ( L:V ) = V • st ( L  L1,…,Ln ) = min { st (L1 ),…,st (Ln ) } • st ( [r1,…,rn] ) = st ( r1 ) • Least fuzzy value of the facts contributing to the argument.

20. Theorems • Theorem (Soundness and Completeness) There is a justified argument of strength V for L iff There is a successful T-tree of truth value V for L • Theorem (Conservative Extension) Argumentation semantics is a conservative extension of WFSX.

21. Application: Fuzzy unification • Open systems: • knowledge and ontologies may not match • interaction with humans • “Does Mike live in Londn?” • Approach: • address(mike,london) = address(mike,londn): 95% • adapt unification algorithm(normalised edit distance over trees net) • embed into argumentation framework

22. Finding Mismatches: Edit distance • Edit distance between strings A and B: • minimal number of delete, add, replace operations to convert A into B. • efficient implementation with dynamic programming • Example: • e(address,adresse)=2, e(007,aa7)=2 • Normalise: • ne(A,B) = e(A,B) / max{ |A|, |B| } • Trees: • net = sum of all mismatches divided by sum of all max lengths

23. Fuzzy unification and arguments • net is conservative extension of MGU (most general unifier) • net(t,t’)  ne(t,t’) • V-argument: for all L in a body, there is L’ in head such that net(L,L’)  1-V • A V-undercuts B if A contains not L and B’s head is L’ and net(L,L’)  1-V • A V-rebuts B if A’s head is L and B’s head is L’ and net(L,L’)  1-V • Adapt previous definitions accordingly

24. Comparison: Argumentation • Our framework allows us to relate existing and new argumentation semantics: • Dung= a/su=a/u=a/a=a/d=a/sa • Prakken&Sartor = d/su=d/u=d/a=d/d=d/sa • WFSX = u/a = u/d = u/sa • Dung  Prakken&Sartor  WFSX • Proof Theory and Top-down Proof Procedure adapted from Alferes, Damasio, Pereira’s SLXA

25. Comparison: Fuzzy Argumentation • Wagner: • Scale: -1 to +1 • Unlike WFSX, he relates F and F:  F: -V iff F:V • We adopted his interpretation for not:not F:1 if F:V, V>0 • Relates his work to stable models, but there is no top-down proof procedure for stable models [Alferes&Pereira] • Our approach conservatively extends WFSX, hence we can adapt proof procedure SLXA

26. Comparison: Fuzzy unification • Arcelli, Formato, Gerla • define abstract fuzzy unification/resolution framework • cannot deal with missing parameters (common problem [Fung et al.]) • no conservative extension of classical unification • we use concrete distance: edit distance • Evaluated idea on bioinfo DB

27. Conclusion • “A database needs two kinds of negation” (Wagner) • Argumentation is an elegant way of defining semantics • Our framework allows classification of various new and existing semantics • Efficient top-down proof procedure for justified arguments • Argumentation as basis for belief revision (REVISE) • We cover the whole knowledge system cube including fuzzy argumentation • Defined fuzzy unification, which is useful in open systems

28. Other relevant work • ACA: Arguing and Cooperating Agents • multi-agent argumentation engine • Demo online at www.soi.city.ac.uk/~msch

29. Other relevant work(together with Gerd Wagner) • Vivid Agents • action and reaction rules • executable specifications • Implemented in simulation and with PVM-Prolog • Demo online at www.soi.city.ac.uk/~msch

30. Other relevant work(together with Carlos Damasio, Luis Pereira) • REVISE • Contradiction removal • Applied to many domains like circuit diagnosis, information integration in bioinformatics, alarm correlation in cellular phone network • Demo online at www.soi.city.ac.uk/~msch