Belief dynamics and defeasible argumentation in rational agents
description
Transcript of Belief dynamics and defeasible argumentation in rational agents
![Page 1: Belief dynamics and defeasible argumentation in rational agents](https://reader036.fdocuments.us/reader036/viewer/2022062315/56814f19550346895dbca9f9/html5/thumbnails/1.jpg)
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
![Page 2: Belief dynamics and defeasible argumentation in rational agents](https://reader036.fdocuments.us/reader036/viewer/2022062315/56814f19550346895dbca9f9/html5/thumbnails/2.jpg)
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.
![Page 3: Belief dynamics and defeasible argumentation in rational agents](https://reader036.fdocuments.us/reader036/viewer/2022062315/56814f19550346895dbca9f9/html5/thumbnails/3.jpg)
Falappa, García & Simari International Workshop on Non-Monotonic Reasoning 3
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
![Page 4: Belief dynamics and defeasible argumentation in rational agents](https://reader036.fdocuments.us/reader036/viewer/2022062315/56814f19550346895dbca9f9/html5/thumbnails/4.jpg)
Falappa, García & Simari International Workshop on Non-Monotonic Reasoning 4
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)
![Page 5: Belief dynamics and defeasible argumentation in rational agents](https://reader036.fdocuments.us/reader036/viewer/2022062315/56814f19550346895dbca9f9/html5/thumbnails/5.jpg)
Falappa, García & Simari International Workshop on Non-Monotonic Reasoning 5
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
![Page 6: Belief dynamics and defeasible argumentation in rational agents](https://reader036.fdocuments.us/reader036/viewer/2022062315/56814f19550346895dbca9f9/html5/thumbnails/6.jpg)
Falappa, García & Simari International Workshop on Non-Monotonic Reasoning 6
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).
![Page 7: Belief dynamics and defeasible argumentation in rational agents](https://reader036.fdocuments.us/reader036/viewer/2022062315/56814f19550346895dbca9f9/html5/thumbnails/7.jpg)
Falappa, García & Simari International Workshop on Non-Monotonic Reasoning 7
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 }
![Page 8: Belief dynamics and defeasible argumentation in rational agents](https://reader036.fdocuments.us/reader036/viewer/2022062315/56814f19550346895dbca9f9/html5/thumbnails/8.jpg)
Falappa, García & Simari International Workshop on Non-Monotonic Reasoning 8
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.
![Page 9: Belief dynamics and defeasible argumentation in rational agents](https://reader036.fdocuments.us/reader036/viewer/2022062315/56814f19550346895dbca9f9/html5/thumbnails/9.jpg)
Falappa, García & Simari International Workshop on Non-Monotonic Reasoning 9
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
![Page 10: Belief dynamics and defeasible argumentation in rational agents](https://reader036.fdocuments.us/reader036/viewer/2022062315/56814f19550346895dbca9f9/html5/thumbnails/10.jpg)
Falappa, García & Simari International Workshop on Non-Monotonic Reasoning 10
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
![Page 11: Belief dynamics and defeasible argumentation in rational agents](https://reader036.fdocuments.us/reader036/viewer/2022062315/56814f19550346895dbca9f9/html5/thumbnails/11.jpg)
Falappa, García & Simari International Workshop on Non-Monotonic Reasoning 11
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.
![Page 12: Belief dynamics and defeasible argumentation in rational agents](https://reader036.fdocuments.us/reader036/viewer/2022062315/56814f19550346895dbca9f9/html5/thumbnails/12.jpg)
Falappa, García & Simari International Workshop on Non-Monotonic Reasoning 12
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.
![Page 13: Belief dynamics and defeasible argumentation in rational agents](https://reader036.fdocuments.us/reader036/viewer/2022062315/56814f19550346895dbca9f9/html5/thumbnails/13.jpg)
Falappa, García & Simari International Workshop on Non-Monotonic Reasoning 13
Informational Content
This postulate avoids the following cases:
• Self-explanation:
{ } be an explanation of
• Redundancy:
{ , } be an explanation of
![Page 14: Belief dynamics and defeasible argumentation in rational agents](https://reader036.fdocuments.us/reader036/viewer/2022062315/56814f19550346895dbca9f9/html5/thumbnails/14.jpg)
Falappa, García & Simari International Workshop on Non-Monotonic Reasoning 14
• 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
![Page 15: Belief dynamics and defeasible argumentation in rational agents](https://reader036.fdocuments.us/reader036/viewer/2022062315/56814f19550346895dbca9f9/html5/thumbnails/15.jpg)
Falappa, García & Simari International Workshop on Non-Monotonic Reasoning 15
Revision by a set of sentences
A Explanans for
A
( A)
Possiblyinconsistent
state
could not be accepted
![Page 16: Belief dynamics and defeasible argumentation in rational agents](https://reader036.fdocuments.us/reader036/viewer/2022062315/56814f19550346895dbca9f9/html5/thumbnails/16.jpg)
Falappa, García & Simari International Workshop on Non-Monotonic Reasoning 16
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.
![Page 17: Belief dynamics and defeasible argumentation in rational agents](https://reader036.fdocuments.us/reader036/viewer/2022062315/56814f19550346895dbca9f9/html5/thumbnails/17.jpg)
Falappa, García & Simari International Workshop on Non-Monotonic Reasoning 17
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 .
![Page 18: Belief dynamics and defeasible argumentation in rational agents](https://reader036.fdocuments.us/reader036/viewer/2022062315/56814f19550346895dbca9f9/html5/thumbnails/18.jpg)
Falappa, García & Simari International Workshop on Non-Monotonic Reasoning 18
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) )
![Page 19: Belief dynamics and defeasible argumentation in rational agents](https://reader036.fdocuments.us/reader036/viewer/2022062315/56814f19550346895dbca9f9/html5/thumbnails/19.jpg)
Falappa, García & Simari International Workshop on Non-Monotonic Reasoning 19
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)}
![Page 20: Belief dynamics and defeasible argumentation in rational agents](https://reader036.fdocuments.us/reader036/viewer/2022062315/56814f19550346895dbca9f9/html5/thumbnails/20.jpg)
Falappa, García & Simari International Workshop on Non-Monotonic Reasoning 20
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)
![Page 21: Belief dynamics and defeasible argumentation in rational agents](https://reader036.fdocuments.us/reader036/viewer/2022062315/56814f19550346895dbca9f9/html5/thumbnails/21.jpg)
Falappa, García & Simari International Workshop on Non-Monotonic Reasoning 21
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
![Page 22: Belief dynamics and defeasible argumentation in rational agents](https://reader036.fdocuments.us/reader036/viewer/2022062315/56814f19550346895dbca9f9/html5/thumbnails/22.jpg)
Falappa, García & Simari International Workshop on Non-Monotonic Reasoning 22
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) }
![Page 23: Belief dynamics and defeasible argumentation in rational agents](https://reader036.fdocuments.us/reader036/viewer/2022062315/56814f19550346895dbca9f9/html5/thumbnails/23.jpg)
Falappa, García & Simari International Workshop on Non-Monotonic Reasoning 23
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.