belief dynamics and defeasible argumentation in rational agents

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

Falappa, García & Simari International Workshop on Non-Monotonic Reasoning 2

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.


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


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)


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


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).


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 }


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.


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


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


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.


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.


Informational Content

This postulate avoids the following cases:

• Self-explanation:

{ } be an explanation of

• Redundancy:

{ , } be an explanation of


• 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


Revision by a set of sentences

A Explanans for

A

( A)

Possiblyinconsistent

state

could not be accepted


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.


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 .


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) )


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)}


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)


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


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) }


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.

belief dynamics and defeasible argumentation in rational agents

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