Belief dynamics and defeasible argumentation in rational agents

1. 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

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. 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 strictrules. • 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 defeasiblerules denoted as: L0L1, L2, …, Ln International Workshop on Non-Monotonic Reasoning

4. Defeasible rules • A defeasible rule is denoted as: L0L1, L2,…, Ln L0is 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) International Workshop on Non-Monotonic Reasoning

5. Strict Rules  Facts DefeasibleRules  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) International Workshop on Non-Monotonic Reasoning

6. Defeasible Argumentation Definition: Let Lbe a literal and (, ) be a program. , L is an argument for L, if  is a set of rules in  such that: • There exists a defeasible derivation from that supports L. • The set    is non contradictory; •  is minimal, that is, there is no proper subset  of such that satisfies 1) and 2). International Workshop on Non-Monotonic Reasoning

7. Arguments: some examples From: file_for_printing high_quality use(inkjet) use(laser) use(laser) use(inkjet) use(inkjet) file_for_printing use(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 } 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. International Workshop on Non-Monotonic Reasoning

9. h0 A0 B2 B3 B1 B4 C3 C2 C4 C1 D3 Marked Dialectical Tree and pruning U: Undefeated D: Defeated D U D D D U D U U U International Workshop on Non-Monotonic Reasoning

10. Belief Revision Which is the motivation of belief revision? To model the dynamic of knowledge How can we do that? Classical Logic + Selection Mechanism _________________________________________ Non-classical Logic 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. International Workshop on Non-Monotonic Reasoning

12. Explanations An 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 . • InformationalContent: It is not the case that  A. 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  International Workshop on Non-Monotonic Reasoning

14. Revision by a set of sentences • 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. International Workshop on Non-Monotonic Reasoning

15. A Explanans for  Revision by a set of sentences   A Possibly inconsistent state ( A)   could not be accepted International Workshop on Non-Monotonic Reasoning

16. Main ways of contraction Partial 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. 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 . 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) ) 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)} 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) International Workshop on Non-Monotonic Reasoning

21. Revision on DeLP: an example In 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) 1 metal(hg) pressure(normal) solid(X) metal(X) liquid(hg)  metal(hg), pressure(normal) solid(X) liquid(X) 2 International Workshop on Non-Monotonic Reasoning

22. Revision on DeLP: an example 1 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) } 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. International Workshop on Non-Monotonic Reasoning