Preferential Defeasibility: Utility in Defeasible Logic Programming

Fernando A. TohméDept. of Economics

Guillermo R. SimariDept. of Computer Science and Engineering

UNIVERSIDAD NACIONAL DEL SUR

ARGENTINA

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Outline

Motivation

The Argumentation Framework

Comparison Criteria

Example and Results

Conclusions

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Deafeasible Logic Programming: DeLP

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)

Strict Rules Facts

Defeasible Rules

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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:

1) There exists a defeasible derivation of L from ;

2) The set is non contradictory; and

3) is minimal, that is, there is no proper subset of such that satisfies 1) and 2).

buy_shares(acme)

good_price(acme) risky(acme)

good_price(acme) in_fusion(acme, enron)

in_fusion(acme, enron)

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)., risky(acme) in_fusion(acme, enron).}, buy_shares(acme)

, Q is a subargument of , L if is an argument for Q and

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). }

= { risky(acme) in_fusion(acme, enron). }

Counter-argument

risky(acme)

in_fusion(acme,estron) strong(estron)

in_fusion(acme,estron) strong(estron)

buy_shares(acme)

good_price(acme) risky(acme)

good_price(acme) in_fusion(acme,estron)

in_fusion(acme,estron)

{ risky(acme), risky(acme) } is a contradictory set

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Argument Comparison: Generalized Specificity

Def: Let = (, ) be a program, let G be the set of strict rules in and let 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 . Then 1, L1 is strictly more specific than 2, L2 if:

1. For all , if there exists a defeasible derivationG 1 L1 while G L1 then

G 1 L2, and

2. There exists such that there exists a defeasible derivation G 2 L2 and G L2

but G 1 L1

(Poole, David L. (1985). On the Comparison of Theories: Preferring the Most Specific Explanation. pages 144—147 Proceedings of 9th IJCAI.)

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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)

L

P

Q

Defeaters

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Given = (, ), and 0, L0 an argument obtained from . An

argumentation line for 0, L0 is a sequence of arguments

obtained from , denoted = [0, L0, 1, L1, …] where each

element in the sequence i, hi, i > 0 is a defeater for i-1, hi-1.

0

L0

1

L1

Argumentation Line

2

L2

3

L3

4

L4

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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.

Argumentation Line

0

L0

1

L1

2

L2

3

L3

4

L4

S

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Argumentation Line

0

L0

1

L1

2

L2

3

L3

4

L4

I

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.

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Acceptable Argumentation Line

Given a program = (, ), an argumentation line = [0, L0, 1, L1, …] will be acceptable if:

1. is a finite sequence.

2. The sets S of supporting arguments is concordant, and

the set I of interfering arguments is concordant.

3. There is no argument k, Lk in that is a subargument of

a preceeding argument i, Li, i < k.

4. 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).

0

1

2

3 2

2

3

4 1

3

4

5 3

1

2 4

Dialectical Tree

Given a program = (, ), a literal L will 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.

, L

*, L

Marking of a Dialectical Tree

U

U

D U

U U

U

U

D

D D

D

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Answers in DeLP

Given a program = (, ), and a query for L the posible answers are:

• YES, if L is warranted.

• NO, if L is warranted.

• UNDECIDED, if neither L nor L are warranted.

• UNKNOWN, if L is not in the language of the program.

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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.

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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.

B and () 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 B is the order of B. The elements of which are most preferred receive

the label () = . From the preferences over , we can find preferential

values over defeasible derivations.

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A Comparison Criterion

Given a defeasible derivation of L from , L1, L2 , …, Ln,

let be the set { L1, L2 , …, Ln } and { 1, 2 , …, n } a set

such that i yields Li . Then, that derivation yields for its

conclusion L a valueV(L, ) i =1..nV(Li, i).

Inductively:

• V(L, ) () if L is a fact, or

• V(L, ) () k=1..mV(Bk, k) if is a rule with head L and 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.

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A Comparison Criterion

By extension, an argument , L gives a value for its conclusion

V(L, ) V(L, ),

where is a derivation that uses all the defeasible rules in and only those defeasible rules.

Note that there could be many different derivations that contain the defeasible rules in .

In that manner, V(L, ), will obtain the lowest value among the defeasible derivations of L that use the defeasible rules in .

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A Comparison Criterion

Let be the set of all literals that can have a defeasible derivation from .

Any subset be has a value

V() L V(L, )

This means that 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.

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Preferential Comparison

Def: Let = (, , , B) be a program, let G be the set of strict rules in and let 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 . Then 1, L1 is strictly more preferentialy specific than 2, L2 if:

1.For all , if there exists a defeasible derivation

G 1 L1 while G L1 then

G 1 L2 , and

2.There exists such there exists a defeasible

derivation G 2 L2 and G L2

but G 1 L1

3.For evey verifying (1) and verifying (2) holds V() B V( )

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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 that tweety is a penguin has a preference of 0 while the rule used to derive that it is a bird has a preference of 1.

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Example

Now, consider the two arguments:

{flies(X) penguin(X)}, flies(X) and {flies(X) bird(X)}, flies(X) then if we consider { penguin(tweety) } and { bird(tweety) }

we have that

{ bird(X) penguin(X) } flies(tweety), but

{ bird(X) penguin(X) } {flies(X) penguin(X) } flies(tweety)

{ bird(X) penguin(X) } { flies(X) bird(X) } flies(tweety)

On the other hand

{ bird(X) penguin(X) } flies(tweety), but

{ bird(X) penguin(X) } { flies(X) bird(X) } flies(tweety)

{ 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)

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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( ) max(V(bird(tweety), ),

V(bird(tweety), {penguin(tweety),

bird(tweety)})

max(1, 0) 1

while

V() V( penguin(tweety), ) 0

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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 where 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 , 1, L1 is strictly-more-specific-than 2, L2 and for

every pair of their corresponding activation sets , , V() B V() .

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Results

Proposition: Given a query Q in the preferential defeasible logic program (, , , B), and an argument structure , Q, its tagged dialectical tree is identical to *

, Q in (, ) iff the relation strictly-more-preferentially-specific-than for program is equivalent to the relation strictly-more-specific-than in program over Q, where Q is the class of all arguments that are either labels of the dialectical tree , Q or subarguments of them.

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Results

Corollary: Given a query Q and an argument structure , 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 Q.

Questions?