Belief dynamics and defeasible argumentation
in rational agents
M. A. Falappa - A. J. García
G. R. Simari
Artificial Intelligence Research and Development Laboratory
Department of Computer Science and Engineering
Universidad Nacional del Sur - Argentina
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Motivation
• Use a kind of non-prioritized revision on defeasible logic programming (DeLP).
• Apply this kind of operator on the beliefs of an BDI agent.
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Knowledge representation• The knowledge of an agent will be represented
by a defeasible logic program =(,). is a set of facts and strict rules.
– Facts are ground literals that could be negated by the use of strong negation “”.
– Strict rules are denoted as:
L0 L1, L2, …, Ln
where Li are ground literals.
is a set of defeasible rules denoted as:
L0 L1, L2, …, Ln
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Defeasible rules• A defeasible rule is denoted as:
L0 L1, L2 ,…, Ln
L0 is a ground literal called the head and L1, …, Ln
are ground literals that form the body of the rule.
• This kind of rule is used to represent tentative information:
“Reasons to believe in L1, L2 ,…, Ln
are reasons to believe in L0”
• Example:good_weather(today) low_pressure(today), high(humidity)
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Deafeasible Logic Program
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
Definition: 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 from that supports L.
2) The set is non contradictory;
3) is minimal, that is, there is no proper subset of such that satisfies 1) and 2).
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Arguments: some examplesFrom: file_for_printing high_quality use(inkjet) use(laser)
use(laser) use(inkjet)use(inkjet) file_for_printinguse(laser) file_for_printing, high_quality
Possible arguments: , use(inkjet) where:
= { use(inkjet) file_for_printing }
, use(inkjet) where: = { use(laser) file_for_printing, high_quality }
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Defeasible Argumentation in DeLP
• Counterargument of , L: is an argument , L that “contradicts” ,L.
• Defeater of , L: is an counterargument of , L “better” than it.
• Dialectical tree: a tree of arguments with , L as root where each node is a defeater for its parent node.
• Warranted Literal L: there exists an argument , L such that its dialectical tree has its root undefeated.
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C3
B2B1
Marked Dialectical Tree and pruning
A0
h0
B3 B4
C2C1 C4
D3
U
D
D
D
U U
U
U DD
U: Undefeated
D: Defeated
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Belief RevisionWhich is the motivation of belief revision?
To model the dynamic of knowledge
How can we do that?
Classical Logic
+ Selection Mechanism_________________________________________
Non-classical Logic
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Belief Bases
There are two kinds of beliefs:• Explicit Beliefs: all the sentences in the belief
base.• Implicit Beliefs: all sentences derived from the
belief base.
The implicit beliefs are “explained” from more basic beliefs.
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ExplanationsAn explanans justifies an explanandum.
Set of sentences A sentence
Properties [FKS02]:
• Deduction: A .• Consistency: It is not the case that A .• Minimality: There is no set A A such that A .• Informational Content: It is not the case that A.
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Informational Content
This postulate avoids the following cases:
• Self-explanation:
{ } be an explanation of
• Redundancy:
{ , } be an explanation of
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• We will define operators for revision with respect to an explanans (a set of sentences).
• The idea is the following:
– Instead of incorporating a sentence , call for an explanans A for .
– Add A to .– Eliminate all posible inconsistencies from
the result.
Revision by a set of sentences
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Revision by a set of sentences
A Explanans for
A
( A)
Possiblyinconsistent
state
could not be accepted
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Main ways of contractionPartial meet mode [AGM85]:
• Let be a set of sentences and be a sentence.
• Find all maximally subsets of failing to imply (-remainders), noted as .
• Select the “best” -remainders by a selection function .
• Intersect them.
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Main ways of contraction
Kernel mode [Hansson94]:
• Let be a set of sentences and be a sentence.
• Find all minimally subsets of implying (-kernels), noted as .
• Cut the -kernels by an incision function .
• Give up the cut sentences from .
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Revision by a Set of Sentences
Definition: Let and A be set of sentences, “” an external selection function for . The operator “” of partial meet revision by a set of sentences is defined as:
A = (( A) )
Definition: Let and A be set of sentences, “” an external incision function for . The operator “” of kernel revision by a set of sentences is defined as:
A = ( A) \ (( A) )
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Revision on DeLP: definition
T+( ) = (positive transformation)
T– ( ) = (negative transformation)
Definition: The composed revision of (,) with respect to A is defined as (,)A= (,) such that = A and = where:
= {T+(): \ (A)} {T–(): \ (A)}
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Revision on DeLP: an example
metal(hg)
metal(fe)
solid(X) metal(X)
liquid(X) solid(X)
solid(X) liquid(X)
= = { }
Then, we receive the following explanation for liquid(hg):
liquid(hg) metal(hg), pressure(normal) metal(hg)pressure(normal)
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Revision on DeLP: an exampleIn kernel revision by a set of sentences, it is necessary to remove any inconsistency from the following sets:
metal(hg)pressure(normal)solid(X) metal(X)liquid(hg) metal(hg), pressure(normal)liquid(X) solid(X)
metal(hg)pressure(normal)solid(X) metal(X)liquid(hg) metal(hg), pressure(normal)solid(X) liquid(X)
1
2
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Revision on DeLP: an example1 and 2 represent the minimally inconsistent subsets of A.
A possible result of (,)A= (,):
metal(hg)metal(fe)liquid(hg) metal(hg),pressure(normal)liquid(X) solid(X)solid(X) liquid(X)
=
= { solid(X) metal(X), metal(X) solid(X) }
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Conclusions and future work
• We apply a non-prioritized revision operator for changing the agent’s beliefs.
• We use a defeasible logic program (DeLP) for representing the beliefs of an agent.
• The combination of belief revision and DeLP is used for reasoning about beliefs.
• We will explore the properties of this operator on DeLP and develop multi-agent applications.