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Preferential Defeasibility: Utility in Defeasible Logic Programming

Preferential Defeasibility: Utility in Defeasible Logic Programming. Fernando A. Tohmé Dept. of Economics Guillermo R. Simari Dept. of Computer Science and Engineering U NIVERSIDAD N ACIONAL DEL S UR ARGENTINA. Outline. Motivation The Argumentation Framework Comparison Criteria

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Preferential Defeasibility: Utility in Defeasible Logic Programming

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  1. Preferential Defeasibility: Utility in Defeasible Logic Programming Fernando A. Tohmé Dept. of Economics Guillermo R. Simari Dept. of Computer Science and Engineering UNIVERSIDAD NACIONAL DEL SUR ARGENTINA

  2. Outline • Motivation • The Argumentation Framework • Comparison Criteria • Example and Results • Conclusions

  3. Deafeasible Logic Programming: DeLP Strict Rules  Facts Defeasible Rules  A Defeasible Logic Program (dlp) is a set of facts, strict and defeasible rules denoted  = (, ) bird(X) chicken(X) chicken(tina) bird (X) penguin(X) penguin(opus) flies(X) penguin(X) scared(tina) flies(X)  bird(X) flies(X) chicken(X) flies(X) chicken (X), scared(X)

  4. Defeasible Argumentation Def: Let L be a literal and (, ) be a program. , L is an argument, for L, if  is a set of rules in  such that: • There exists a defeasible derivation of L from   ; • The set    is non contradictory; and •  is minimal, that is, there is no proper subset  of such that satisfies 1) and 2).

  5. buy_shares(X) good_price(X) buy_shares (X) good_price(X), risky(X) risky(X)  in_fusion(X, Y) risky(X) in_debt(X) risky(X) in_fusion(X, Y), strong(Y) good_price(acme) in_fusion(acme, estron) strong(estron) buy_shares(acme) good_price(acme) risky(acme) good_price(acme) in_fusion(acme, enron) in_fusion(acme, enron) {buy_shares(acme)  good_price(acme), risky(acme)., risky(acme)  in_fusion(acme, enron).}, buy_shares(acme)

  6. buy_shares(acme) good_price(acme) risky(acme) good_price(acme) in_fusion(acme, enron) in_fusion(acme, enron) , Q is a subargument of, L if is an argument forQ and   = {risky(acme)  in_fusion(acme, enron). }  = {buy_shares(acme) good_price(acme), risky(acme)., risky(acme)  in_fusion(acme, enron). }

  7. Counter-argument buy_shares(acme) good_price(acme) risky(acme) risky(acme) good_price(acme) in_fusion(acme,estron) in_fusion(acme,estron) strong(estron) in_fusion(acme,estron) in_fusion(acme,estron) strong(estron)   { risky(acme), risky(acme)} is a contradictory set

  8. Argument Comparison: Generalized Specificity Def:Let = (, ) be a program, let G be the set of strict rules in  and let F be the set of all literals that can be defeasibly derived from . Let 1, L1and 2, L2be two arguments built from , where L1, L2F. Then 1, L1isstrictly more specific than2, L2if: • For all H F, if there exists a defeasible derivation GH  1L1whileGHL1then GH  1L2, and • There existsH F such that there exists a defeasible derivation GH 2 L2andGH L2 but GH 1 L1 (Poole, David L. (1985). On the Comparison of Theories: Preferring the Most Specific Explanation. pages 144—147 Proceedings of 9th IJCAI.)

  9. Defeaters P L Q    An argument , Pis a defeater for , L if , P is a counter-argument , L that atacks a subargument , Q de , L and one of the following conditions holds: (a), Pis better than, Q (proper defeater), or (b), Pis not comparable to, Q (blocking defeater)

  10. Argumentation Line L4 L1 L2 L3 L0 1 2 3 4 0 Given = (, ), and 0, L0an argument obtained from. An argumentation linefor0, L0is a sequence of arguments obtained from , denoted  = [0, L0, 1, L1, …] where each element in the sequence i, hi, i > 0is a defeater fori-1, hi-1.

  11. Argumentation Line L1 L4 L2 L3 L0 1 2 3 4 0 Given an argumentation line = [0, L0, 1, L1, …], the subsequence S= [0, L0, 2, L2, …]contains supporting arguments and I= [1, L1, 3, L3, …]are interfering arguments. S

  12. Argumentation Line L1 L4 L2 L3 L0 1 2 3 4 0 Given an argumentation line = [0, L0, 1, L1, …], the subsequence S= [0, L0, 2, L2, …]contains supporting arguments and I= [1, L1, 3, L3, …]are interfering arguments. I

  13. Acceptable Argumentation Line Given a program= (, ), an argumentation line  = [0, L0, 1, L1, …] will beacceptable if: •  is a finite sequence. • The sets Sof supporting arguments is concordant, and the set Iof interfering arguments is concordant. • There is no argument k, Lk in  that is a subargument of a preceeding argument i, Li,i < k. • For all i, such that i, Liis a blocking defeater for i-1, Li-1, if there exists i+1, Li+1 then i+1, Li+1is a proper defeater for , Li (i.e.,, Li could not be blocked).

  14. 0 1 2 4 4 5 3 2 1 3 4 2 Dialectical Tree Given a program= (, ), a literal Lwill be warranted if there is an argument , L built from , and that argument has a dialectical tree whose root node is marked U. That is, argument , L is an argument for which all the possible defeaters have been defeated. We will say that is a warrant for L. 2 3 1 , L 3

  15. U D D D U U U U D D U U Marking of a Dialectical Tree *, L

  16. Answers in DeLP Given a program= (, ), and a query forL the posible answers are: • YES, if L is warranted. • NO, if L is warranted. • UNDECIDED, if neither Lnor L are warranted. • UNKNOWN,ifL is not in the language of the program.

  17. A Comparison Criterion • A key element for the warrant procedure is the defeat relation. • Generalized specificity is a purely syntactic comparison criterion and it is introduced as a choice among other possible comparison criteria for comparing arguments. • Here, we will offer an extension of generalized specificity using pragmatic considerations.

  18. A Comparison Criterion • We will allow utility values for facts and rules. • Decision-Theoretic Defeasible Logic Programming will be represented as = (, , , B), where  and  are as before, B is a Boolean algebra with top  and bottom, and  is defined :     B. • Band ()are used to represent the explicit preferences of the user in the sense that given two pieces of information 1, 2 in   , if 1 is strictly more preferred than 2 then (1) B(2) where Bis the order of B. • The elements  of    which are most preferred receive the label () = . • From the preferences over   , we can find preferential valuesover defeasible derivations.

  19. A Comparison Criterion • Given a defeasible derivation of Lfrom   , L1, L2, …,Ln, let D be the set { L1, L2, …, Ln } and {1, 2, …, n } a set such that iyieldsLi .Then, that derivation yields for its conclusion L a valueV(L, )i=1..nV(Li, i). Inductively: • V(L, ) () if Lis a fact, or • V(L, )  () k=1..mV(Bk, k)if  is a rule with head Land body B1, B2, …,Bm and k is a rule used to derive Bk. • The intuition is that a conclusion is as strongly preferred as the weakest of either its premises or the rule used in the derivation.

  20. A Comparison Criterion • By extension, an argument , L gives a value for its conclusion V(L, )  DV(L, D), where D is a derivation that uses all the defeasible rules in A and only those defeasible rules. • Note that there could be many different derivations D that contain the defeasible rules in . • In that manner, DV(L, D), will obtain the lowest value among the defeasible derivations of Lthat use the defeasible rules in .

  21. A Comparison Criterion • Let F be the set of all literals that can have a defeasible derivation from   . Any subset H F be has a value V(H) L HDV(L, D) • This means that H is as valuable as the most valuable of its elements, which in turn is as valuable as the weakest of its derivations. • We can use this notion to redefine specificity obtaining a relation of preferential specificity.

  22. Preferential Comparison Def:Let = (, , , B) be a program, let G be the set of strict rules in  and let F be the set of all literals that can be defeasible derived from . Let 1, L1and 2, L2be two arguments built from , where L1, L2F. Then 1, L1isstrictly more preferentialy specific than2, L2if: • For all H F, if there exists a defeasible derivation GH  1L1whileGH L1then GH  1L2, and • There existsH F such there exists a defeasible derivation GH 2L2andGH  L2 but GH 1L1 • For evey H verifying (1) andHverifying (2) holds V(H) BV(H)

  23. Example • Consider a classical example in defeasible argumentation where preferences are defined for B = { 0, 1 }, with 0  1:  { bird(X) penguin(X) (1), penguin(tweety) (0), bird(tweety) (1) }  { flies(X)  penguin(X) (1), flies(X) bird(X) (1) } • Notice that bird(tweety) yields two values: V(bird(tweety), {penguin(tweety), bird(tweety)})  min(0,1)  0 and V(bird(tweety), )  1, because the fact thattweety is a penguinhas a preference of 0 while the rule used to derive that it is a bird has a preference of 1.

  24. Example Now, consider the two arguments: {flies(X)  penguin(X)}, flies(X) and {flies(X)  bird(X)}, flies(X) then if we consider H  { penguin(tweety) } and H { bird(tweety) } we have that H { bird(X)  penguin(X) }  flies(tweety), but H { bird(X)  penguin(X) }  {flies(X)  penguin(X) } flies(tweety) H { bird(X)  penguin(X) }  { flies(X) bird(X) } flies(tweety) On the other hand H { bird(X)  penguin(X) }  flies(tweety), but H { bird(X)  penguin(X) }  { flies(X) bird(X) } flies(tweety) H { bird(X)  penguin(X) }  {flies(X) penguin(X) } flies(tweety) Therefore { flies(X)  penguin(X) }, flies(X) is strictly more specific than { flies(X)  bird(X) }, flies(X)

  25. Example • We found that { flies(X)  penguin(X). }, flies(X) is strictly more specific than { flies(X)  bird(X). }, flies(X) but is not strictly more preferentially specific since we have that V(H )  max(V(bird(tweety), ), V(bird(tweety), {penguin(tweety), bird(tweety)})  max(1, 0)  1 while V(H) V( penguin(tweety), )  0

  26. Results Proposition: If 1, L1is strictly more preferentially specific than2, L2 then 1, L1is strictly more specific than2, L2. Proposition: The relation strictly-more-preferentially-specific-than in program  (, , , B) is equivalent (i.e., yields the same subset of ARGARG where ARG is the class of argument structures) to the relation strictly-more-specific-than in program  (, ) if and only if for every pair of argument structures 1, L1, 2, L2 ARG, 1, L1 is strictly-more-specific-than2, L2 and for every pair of their corresponding activation sets H, HF, V(H) BV(H) .

  27. Results Proposition: Given a query Q in the preferential defeasible logic program  (, , , B), and an argument structure A, Q, its tagged dialectical tree is identical to T*A, Qin  (, ) iff the relation strictly-more-preferentially-specific-than for program  is equivalent to the relation strictly-more-specific-than in program  over ARGQ, where ARGQ is the class of all arguments that are either labels of the dialectical tree TA, Q or subarguments of them.

  28. Results Corollary: Given a query Q and an argument structure A, Q, the answer to Q in the preferential defeasible logic program  (, , , B) is identical to its answer in  (, ) iff the relation strictly-more-preferentially-specific-than for  is equivalent to the relation strictly-more-specific-than in  over ARGQ.

  29. Questions?

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