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
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  = (, )

Strict
Rules

Defeasible
Rules
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bird(X)  chicken(X)
bird (X)  penguin(X)
flies(X)  penguin(X)
chicken(tina)
penguin(opus)
scared(tina)
Facts
flies(X)  bird(X)
flies(X)  chicken(X)
flies(X)  chicken (X), scared(X)
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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).
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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)
buy_shares(acme)
strong(estron)
good_price(acme)
good_price(acme)
risky(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)
, Q is a subargument of , L if  is an argument for Q and   
buy_shares(acme)
good_price(acme)
good_price(acme)
risky(acme)
in_fusion(acme, enron)
in_fusion(acme, enron)
 = { risky(acme)  in_fusion(acme, enron). }
 = {buy_shares(acme)  good_price(acme), risky(acme).,
risky(acme)  in_fusion(acme, enron). }
Counter-argument
buy_shares(acme)
good_price(acme)
good_price(acme)
risky(acme)
in_fusion(acme,estron)
in_fusion(acme,estron)
  { risky(acme), risky(acme) }
is a contradictory set
risky(acme)
in_fusion(acme,estron)
in_fusion(acme,estron)
strong(estron)
strong(estron)
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 is strictly more specific than 2, L2 if:
1. For all H  F, if there exists a defeasible derivation
G  H  1 L1 while G  H  L1 then
G  H  1 L2, and
2. There exists H  F such that there exists a defeasible
derivation G  H  2 L2 and 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.)
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Defeaters
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)
P
L
Q


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
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Argumentation Line
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.
L1
L0
0
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1
L2
2
L3
3
L4
4
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Argumentation Line
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
L1
L0
0
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1
L2
2
L3
3
L4
4
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Argumentation Line
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
L1
L0
0
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1
L2
2
L3
3
L4
4
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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).
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Dialectical Tree
0
1
1
2
2
2
3
3
4
2
1
That is, argument , L is
an argument for which all the
possible defeaters have been
defeated.
3
4

, L
5
4
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.
We will say that  is a
warrant for L.
3

Marking of a
Dialectical Tree
U
D
U
D
D
U
U
D
U
U
 *, L
D
U
Answers in DeLP
Given a program  = (, ), and a query for L the
posible answers are:
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•
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 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  V(L, )
 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.
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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 is
strictly more preferentialy specific than 2, L2 if:
1. For all H  F, if there exists a defeasible derivation
G  H  1 L1 while G  H  L1 then
G  H  1 L2 , and
2. There exists H  F such there exists a defeasible
derivation G  H  2 L2 and G  H  L2
but G  H  1 L1
3. For evey H verifying (1) and H verifying (2) holds
V(H) B V(H )
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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 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)
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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(H )  max(V(bird(tweety), ),
V(bird(tweety), {penguin(tweety),
bird(tweety)})
 max(1, 0)  1
while
V(H) 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-preferentiallyspecific-than in program   (, , , B) is equivalent
(i.e., yields the same subset of RGRG where
RG 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  RG, 1, L1 is strictly-more-specific-than
2, L2 and for every pair of their corresponding
activation sets H, H  F, V(H) B V(H) .
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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-morepreferentially-specific-than for program  is equivalent
to the relation strictly-more-specific-than in program 
over RGQ, where RGQ 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-morepreferentially-specific-than for  is equivalent to the
relation strictly-more-specific-than in  over RGQ.
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Questions?