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 & SimariInternational Workshop on Non-Monotonic Reasoning2 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.

Falappa, García & SimariInternational Workshop on Non-Monotonic Reasoning3 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: L 0  L 1, L 2, …, L n where L i are ground literals.  is a set of defeasible rules denoted as: L 0  L 1, L 2, …, L n

Falappa, García & SimariInternational Workshop on Non-Monotonic Reasoning4 Defeasible rules A defeasible rule is denoted as: L 0  L 1, L 2,…, L n L 0 is a ground literal called the head and L 1, …, L n are ground literals that form the body of the rule. This kind of rule is used to represent tentative information: “Reasons to believe in L 1, L 2,…, L n are reasons to believe in L 0 ” Example:  good_weather(today)  low_pressure(today), high(humidity)

Falappa, García & SimariInternational Workshop on Non-Monotonic Reasoning5 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  

Falappa, García & SimariInternational Workshop on Non-Monotonic Reasoning6 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).

Falappa, García & SimariInternational Workshop on Non-Monotonic Reasoning7 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 }

Falappa, García & SimariInternational Workshop on Non-Monotonic Reasoning8 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.

Falappa, García & SimariInternational Workshop on Non-Monotonic Reasoning9 C3C3 B2B2 B1B1 Marked Dialectical Tree and pruning A0A0 h0h0 B3B3 B4B4 C2C2 C1C1 C4C4 D3D3 U D D D U U U U D D U: Undefeated D: Defeated

Falappa, García & SimariInternational Workshop on Non-Monotonic Reasoning10 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

Falappa, García & SimariInternational Workshop on Non-Monotonic Reasoning11 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.

Falappa, García & SimariInternational Workshop on Non-Monotonic Reasoning12 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  . Informational Content: It is not the case that   A.

Falappa, García & SimariInternational Workshop on Non-Monotonic Reasoning13 Informational Content This postulate avoids the following cases: Self-explanation: {  } be an explanation of  Redundancy: {   ,     } be an explanation of 

Falappa, García & SimariInternational Workshop on Non-Monotonic Reasoning14 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

Falappa, García & SimariInternational Workshop on Non-Monotonic Reasoning15 Revision by a set of sentences  A Explanans for    A  A (   A)  Possibly inconsistent state  could not be accepted

Falappa, García & SimariInternational Workshop on Non-Monotonic Reasoning16 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.

Falappa, García & SimariInternational Workshop on Non-Monotonic Reasoning17 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 .

Falappa, García & SimariInternational Workshop on Non-Monotonic Reasoning18 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 )   )

Falappa, García & SimariInternational Workshop on Non-Monotonic Reasoning19 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 )}

Falappa, García & SimariInternational Workshop on Non-Monotonic Reasoning20 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)

Falappa, García & SimariInternational Workshop on Non-Monotonic Reasoning21 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) metal(hg) pressure(normal) solid(X)  metal(X) liquid(hg)  metal(hg), pressure(normal)  solid(X)  liquid(X) 11 22

Falappa, García & SimariInternational Workshop on Non-Monotonic Reasoning22 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) }

Falappa, García & SimariInternational Workshop on Non-Monotonic Reasoning23 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.