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S. Bistarelli1 ;2, D. Pirolandi1, F. Santini1;2

Solving Weighted Argumentation Frameworks with Soft Constraints @cilc2010, ( @ecai2010 , @RAC2009 , @soft2010 ). S. Bistarelli1 ;2, D. Pirolandi1, F. Santini1;2 1 Dipartimento di Matematica e Informatica, Università di Perugia [bista,pirolandi,francesco.santini]@dmi.unipg.it

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S. Bistarelli1 ;2, D. Pirolandi1, F. Santini1;2

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  1. SolvingWeightedArgumentationFrameworkswith Soft Constraints@cilc2010, (@ecai2010,@RAC2009, @soft2010) S. Bistarelli1;2, D. Pirolandi1, F. Santini1;2 1 Dipartimento di Matematica e Informatica, Università di Perugia [bista,pirolandi,francesco.santini]@dmi.unipg.it 2 Istituto di Informatica e Telematica (CNR), Pisa [stefano.bistarelli,francesco.santini@]iit.cnr.it

  2. Abstract 3 • WeightedArgumentationframework (based on semiring). • Preferencevaluesassociatedtoarguments • more informative • can beusedtoprefer a given set ofargumentsoverothers • A mappingfromweightedAFsto Soft ConstraintSatisfactionProblems (SCSPs) • computationofDungsemantics (e.g. admissible and stable) bysolving the related SCSP. • ImplementwithJaCoP, a Java constraint solver. 1 2 5 4 4 3 (S)CSP 1

  3. Outline • Background • Dung Argumentation • Soft CSPs • Weighted (semiring) Argumentation • (S)CSP mapping • Sketch of • From node to edge weight • SCSP mapping • Bipolar semiring Afs • Conclusions and FW

  4. Background • Dung Argumentation systems • Soft CSPs Argument Attack in AI (Dung)

  5. Background • Dung Argumentation systems • Soft CSPs Conflict free

  6. Background • Dung Argumentation systems • Soft CSPs admissible

  7. Background • Dung Argumentation systems • Soft CSPs stable

  8. Background • Dung Argumentation systems • Soft CSPs Complete, ground, preferred …

  9. Background • Dung Argumentation systems • Soft CSPs WeightedArgumentation Weighted AF: A forecast example • two individuals P and Q exchanging arguments A and B about the weather forecast: • P: Today will be dry in London since BBC forecast sunshine • Q: Today will be wet in London since CNN forecast rain

  10. Background • Dung Argumentation systems • Soft CSPs WeightedArgumentation A forecast example • A and Bclaimcontradictoryconclusions and so attackeachother • twodifferentadmissibleextensions: the sets{A} and {B}.. • However, onemightreasonthatA ispreferredtoBbecause the BBC are deemed more trustworthythan CNN.

  11. Background • Dung Argumentation systems • Soft CSPs WeightedArgumentation Weighted conflict free Sem. 2 3 4 1 5 2 1 0 6 4 4 5 3

  12. Background • Dung Argumentation systems • Soft CSPs WeightedArgumentation Weighted adm., stable, complete …

  13. Background • Dung Argumentation systems • Soft CSPs WeightedArgumentation (S)CSP mapping Mapping Afs to SCSPs (S)CSP

  14. C-semiring <A,+,´,0,1>: h i d A S A 0 1 + £ 2 ´ a a n ; ; ; ; ; Weighted <+,min,+,+,0> Probabilistic <[0,1],max,,0,1> Fuzzy <[0,1],max,min,0,1> Classical <{false,true},,,false,true> • Background • Dung Argumentation systems • Soft CSPs Soft Constraints • C-semiring: • A is the set of costs/preferences • + defines a partial order ≤S over S • x is used to compose the costs • 0 and 1 are the bot and top elements of A

  15. Background • Dung Argumentation systems • Soft CSPs WeightedArgumentation (S)CSP mapping From weight AFs to SCSPs • Preference constraints • Conflict-fre constraints • Admissible constraints • Complete constraints • Stable constraints

  16. Preference constraints • The weight function W(a) = s (s element of the semiring) can be modeled with the unary constraints c where • c(a = 1) = s, (a is selected in the set with cost s) • c(a = 0) = 1 (a is not in the extension and we do not care of the cost (in the weighted extension the cost is 0)) 2 3 4 1 5 <1>=3 <0>=0 <1>=1 <0>=0 a a b b c c d d e e <1>=2 <0>=0

  17. Conflict-free constraints • If R(ai; aj) is in the graph (ai attacks aj) we need to prevent the solution to include both ai and aj in the considered extension: • c(ai = 1; aj = 1) = 1 (that is we permit them with cost 0 in the weighted semiring). • c(ai = 1; aj = 0) = c(ai = 0; aj = 1) = c(ai = 0; aj = 0) = 0 (corresponding to infty in the weighted semiring) 2 3 4 1 5 <1,1>=0 <0,1>=\infty <1,0>=\infty <0,0>=\infty <1,1>=0 <0,1>=\infty <1,0>=\infty <0,0>=\infty a a b b c c d d e e

  18. Background • Dung Argumentation systems • Soft CSPs WeightedArgumentation (S)CSP mapping And the other… (see the paper) • Preference constraints • Conflict-fre constraints • Admissible constraints • Complete constraints • Stable constraints

  19. Background • Dung Argumentation systems • Soft CSPs WeightedArgumentation (S)CSP mapping Solution equivalence

  20. Background • Dung Argumentation systems • Soft CSPs WeightedArgumentation (S)CSP mapping Solving with JaCoP • The Java Constraint Programming solver (JaCoP) is a Java library, which provides Java user with Finite Domain Constraint Programming paradigm. • arithmetical constraints, equalities and inequalities, logical, reified and conditional constraints, combinatorial (global) constraints. • pruning events, multiple constraint queues, special data structures to handle efficiently backtracking, iterative constraint processing, and many more. K. Kuchcinski and R. Szymanek. Jacop - java constraint programming solver, 2001. http://jacop.osolpro.com/.

  21. Background • Dung Argumentation systems • Soft CSPs WeightedArgumentation (S)CSP mapping • Sketch of • From node to edge weight • SCSP mapping • Bipolar Afs Outline • Background • Dung Argumentation • Soft CSPs • Weighted (semiring) Argumentation • (S)CSP mapping • Sketch of • From node to edge weight • SCSP mapping • Bipolar semiring Afs (soft2010@CP, RAC2009) • Conclusions and FW

  22. Background • Dung Argumentation systems • Soft CSPs WeightedArgumentation (S)CSP mapping • Sketch of • From node to edge weight • SCSP mapping • Bipolar Afs FW1: bipolar • Bycomposingattack and supportvalues, itis • alsopossibletoquantitativelystudybipolarargumentationframeworks

  23. Background • Dung Argumentation systems • Soft CSPs WeightedArgumentation (S)CSP mapping • Sketch of • From node to edge weight • SCSP mapping • Bipolar Afs FW2: From node to edge weight • trustworthiness of argument CNN-rain can be computed as a function of the values associated with the attack towards it, i.e. (0.9 + 0.5)/2 = 0.7.

  24. Background • Dung Argumentation systems • Soft CSPs WeightedArgumentation (S)CSP mapping • Sketch of • From node to edge weight • SCSP mapping • Bipolar Afs FW2: Semantics when weight on edges 3 6 1 5 2 4 a b c d e

  25. Background • Dung Argumentation systems • Soft CSPs WeightedArgumentation (S)CSP mapping • Sketch of • From node to edge weight • SCSP mapping • Bipolar Afs FW2: Semantics when weight on edges 3 6 1 5 2 4 a b c d e

  26. Background • Dung Argumentation systems • Soft CSPs WeightedArgumentation (S)CSP mapping • Sketch of • From node to edge weight • SCSP mapping • Bipolar Afs FW2: Semantics when weight on edges 3 6 1 5 2 4 a b c d e

  27. conclusions • Probabilistic semiring: An argument can be seen as a chain of possible events that makes the hypothesis true. The credibility of a hypothesis can then be measured by the total probability that it is supported by arguments. • Fuzzy semiring: allows to represent the relative strength of the attack relationships between arguments, as well as the degree to which arguments are accepted. • Weighted semiring: can model the (e.g. money) cost of the attack: for example, during an electoral campaign, a candidate could be interested in how many efforts or resources he should spend to counteract an argument of the opposing party. • Boolean semiring: cast the classic AFs originally defined by Dung.

  28. Conclusions • Wehaveproposedanunifyingcomputationalframeworkwith strong mathematicalfoundations and solvingtechniques, wherebyonlyparametricallychanging the semiring we can deal withdifferentWeightedAFs. • Probabilistic semiring: An argument can beseenas a chainofpossibleeventsthatmakes the hypothesistrue. The credibilityof a hypothesis can thenbemeasuredby the total probabilitythatitissupportedbyarguments. • Fuzzy semiring: allowstorepresent the relative strengthof the attackrelationshipsbetweenarguments, aswellas the degreetowhicharguments are accepted. • Weighted semiring: can model the (e.g. money) costof the attack: forexample, duringanelectoralcampaign, a candidate couldbeinterested in howmanyefforts or resourcesheshouldspendtocounteractanargumentof the opposing party. • Boolean semiring: cast the classicAFsoriginallydefinedbyDung. • wehavepresented a mappingfromSCSPstoAFs and solved the obtained SCSP withJaCoP, a Java ConstraintProgramming solver.

  29. More Conclusions … • Wehavealso • Investigated the mappingbetweenweight on arguments or on the attacksrelationships • Given a generalsemantics (semiring based) and show how the semantcspreserveclassicalDungtheorems on equivalence and inclusionof the Semantics • Extended the semiring frameworktobipolarsemirings

  30. Background • Dung Argumentation systems • Soft CSPs WeightedArgumentation (S)CSP mapping • Sketch of • From node to edge weight • SCSP mapping • Bipolar Afs FW1: small world benchmark test • At last, we want to generate a small-world network, for example with the Java Universal Network/Graph Framework (JUNG) in order to test social network graphs (See also sci-dpt@unich: TR no. R-2009-003, December 2, 2009)

  31. FW2: cluster of arguments • We want to deal with team formation using argument clustering • according to their coherence • still using soft constraints as the framework to obtain the solution. • This can be useful to check the discrepancies/likeness during a negotiation process, inside different interviews to the same political candidate or during discussions in general. • As an example, • “We do not want immigrants with the right to vote" is clearly closer to “Immigration must be stopped", than to “We need a multicultural and open society in order to enrich the life of everyone and boost our economy", and should belong to the same cluster. • See also ICTCS2010 (accepted), ICTAI2010 (submitted)

  32. SolvingWeightedArgumentationFrameworkswith Soft Constraints@cilc2010, (@ecai2010,@RAC2009, @soft2010) S. Bistarelli1;2, D. Pirolandi1, F. Santini1;2 1 Dipartimento di Matematica e Informatica, Università di Perugia [bista,pirolandi,francesco.santini]@dmi.unipg.it 2 Istituto di Informatica e Telematica (CNR), Pisa [stefano.bistarelli,francesco.santini@]iit.cnr.it

  33. Related Work • Notice that in [23, 3, 2] the preference among arguments is given in a quali- • tative way, that is argument a is better than argument b, which is better than • argument c; in this section we study the problem from a quantitative point of • view, with scores associated with arguments.

  34. Properties of Attacks • Notice that arguments can attack each other. As shown on the figure, A6 attacks A7 and A7 also attacks A6. An example is the following pair of arguments. • Richard is a Quaker and Quakers are pacifists, so he is a pacifist. • Richard is a Republican and Republicans are not pacifists, so he is a not a pacifist. • In Dung’s system, the notion of argument attack is an undefined primitive, but the system can be used to model criteria of argument acceptability. • One such criterion is the view that an argument should be accepted only if every attack on it is attacked by an acceptable argument.

  35. Dung Argumentation • Dung Argumentation Framework (AF) is a directed graph consisting of a set of arguments and a binary conflict based attack relation among them.

  36. 2 3 1 5 4 a b c d e

  37. R.W. • P. E.Dunne, A. Hunter, P. McBurney, S. Parsons, and M. Wooldridge. Inconsistency tolerance in weighted argument systems. In AAMAS '09: Proceedings of The 8th International Conference on Autonomous Agents and Multiagent Systems, pages 851{858. International Foundation for Autonomous Agents and Multiagent Systems, 2009. • no solving mechanism is proposed to solve the problems. Moreover, they do not directly redefine all the Dung’s extensions according to the alpha-consistency budget, and do not provide a computational framework where to solve these weighted problems. • H. Jung, M. Tambe, and S. Kulkarni. Argumentation as distributed constraint satisfaction: applications and results. In Conference on Autonomous agents (AGENTS), pages 324{331, New York, NY, USA, 2001. ACM. • crisp constraint have been used to model argumentation as constraint propagation in Distributed Constraint Satisfaction Problem (DSCP). The paper shows the appropriateness of constraints in solving large-scale argumentation systems. However, it only solve classical problems, (i.e. no qualitative or quantitative extensions).

  38. R.W. • In [18], the authors have developed the notion of fuzzy unification and incorporated it into a novel fuzzy argumentation framework for extended logic programming: the attacks are associated to a fuzzy strength value, i.e. a V -attack. As well, a V -argument A is V -acceptable w.r.t. the set Args of V -arguments if each argument V -attacked A is V -attacked by an argument in Args. • In [2], AFs have been extended to Value Based Argumentation Frameworks (VAF) where V is a generic nonempty set of values and Val is a function mapping from elements of Args to elements of V . amount of attacks inside the set. What we do is instead a complete revision of the Dung’s semantics (also admissible, complete and stable ones), by introducing also the notion of the weight of attack between a set and a single argument (i.e. in Def. 6) used for example in the new concept of defense (see Def. 8) • The work in [1] concerns the acceptability of arguments in preference-based AFs. Preferences are represented with a preordering relationships (partial or total) that resembles the ordering defined by the + operator of semirings (see Sec. 3). • Probabilistic Argumentation [13]. This theory is an alternative approach for non-monotonic reasoning under uncertainty. It allows to judge open questions (hypotheses) about the unknown or future world in the light of the given knowledge. From a qualitative point of view, the problem is to derive arguments in favor and against the hypothesis of interest. • To compare these works with our solution, we need to say that in [12, 2] the authors redefine only the concept of conflict-free sets by tolerating a certain In [7] the authors solved AFs with weights on arguments instead of attacks. As already said, preferences on arguments are used in many works in literature in [15, 2]. Notice that we consider this kind of weights also by adding Preference constraints: the unary weight function W(ai) = s (s 2 A) of an AFS can be modeled with the unary constraints cai (ai = 1) = s, otherwise, when ai is assigned to 0, the argument is not taken in the considered extension an so its cost must not be computed. • Notice that in [15, 2, 1] the preference among arguments is given in a qualitative way, that is argument a is better than argument b, which is better than argument c; in this paper we study the problem from a quantitative point of view, with scores associated with arguments. We suggest the algebraic semiring structure (see Sec. 3) as a mean to parametrically represent and solve all the “weighted” Afs presented in literature (see Sec. 6), i.e. to represent the scores. • In [16] the authors define the “collective defeat”, where we have a set B of nodes each of which is defeated by other members of the set and none of which is defeated by undefeated nodes outside the set. Compared to Def. 8, we do not require that each member in B is defeated by other members of B and the defeat relationship is weighted.

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