Adaptive Reasoning for Cooperative Agents

Luís Moniz Pereira Alexandre Pinto

Centre for Artificial Intelligence – CENTRIAUniversidade Nova de Lisboa

INAP’09, 5-7 November Évora, Portugal

Summary - 1

• Explicit affirmation and negation plus a 3rd logic value of undefined, is useful in situations where decisions must be taken based of scarce, ambiguous, or contradictory information

• In a 3-valued setting, we consider agents that learn definitions for a target concept and its opposite, taking both positive and negative examples as instances of the two classes

Summary - 2

• A single agent exploring an environment can gather only so much information, which may not suffice to find good explanations

• A cooperative multi-agent strategy, where each agent explores part of the environment and shares its findings, provides better results

• We employ a distributed genetic algorithm framework, enhanced by a Lamarckian belief revision operator

• Agents communicate explanations—coded as belief chromosomes—by sharing them in a common pool

Summary - 3

• Another way of interpreting this communication is via agent argumentation

• When taking in all arguments to find a common ground or consensus, we may have to revise assumptions in each argument

• A collaborative viewpoint results: Arguments are put together to find 2-valued consensus on conflicting learnt concepts, within an evolving genetic pool, so as to identify “best” joint explanations to observations

Learning Positive and Negative Knowledge

– Autonomous agent: acquisition of information by means of experiments

– Experiment:• execution of an action• evaluation of the results with respect to the

achievement of a goal• positive and negative results

– Learning general rules on actions:• distinction among actions with a positive, negative or

unknown or undefined outcome

2-valued vs. 3-valued Learning

– Two values:• bottom-up generalization from instances• top-down refinement from general classes

– Three values:• learning a definition both for the concept and its

opposite

+++

+ +++ +

- - ---- --

+++

+ +++ +

- - ---- --

+++

+ +++ +

- - ---- --

a cb

Learning in a 3-valued Setting

– Extended Logic Programs (ELP): with explicit negation “ A”

– Clauses of the formL0 L1, ... , Lm, not Lm+1, ... , not Lm+n

where Li can be either A or A

– Explicit representation of negative information fly(X) penguin(X).

– Three logical values: true, false, unknown or undefined

Problem definition

Given• a set P of possible ELP programs (bias)• a set E+ of positive examples• a set E- of negative examples• a consistent extended logic program

(background knowledge)

Use learning to find an ELP, P P such that

P E+ P E- (completeness)P L- P L+ L E+ E- (consistency)

Intersection of definitions

E+ E-

pp

Exceptions to the positive concept: negative examples

Exceptions to the negative concept: positive examples

Unseen atoms

Unseen atoms

– Unseen atoms which are both true and false are classified as unknown or undefined:

p(X) p+(X), not p(X). p(X) p-(X), not p(X).

– If the concept is true and its opposite undefined, then it is classified as true:

p(X) p+(X), undefined( p(X) ). p(X) p-(X), undefined( p(X) ).

Training set atoms

– They must be classified according to the training set

– Default literals, representing non-abnormality conditions, are added to rules:

p(X) p+(X), not abp (X), not p(X).

p(X) p-(X), not abp (X), not p(X).

Example: knowledge

B: bird(a). has_wings(a).jet(b). has_wings(b).angel(c). has_wings(c). has_limbs(c).penguin(d). has_wings(d). has_limbs(d).dog(e). has_limbs(e).cat(f). has_limbs(f).

E+ = { flies(a) } E- = { flies(d), flies(e) }

flies+ E+

a b c fd e

flies-E-

flies+(X) has_wings(X). flies-(X) has_limbs(X).

flies(X) flies+(X), not abflies+(X), not flies(X).

flies(X) flies-(X), not flies(X). abflies+(d).

flies(X) flies+(X), undefined( flies(X) ).flies(X) flies-(X), undefined( flies(X) ).

Generalizing exceptions we obtain:

abflies+(X) penguin(X).

Example: learned theory

Least General vs. Most General Solutions

– Bottom-up methods: • Search from specific to general:

Least General Solution (LGS)

• GOLEM (RLGG), CIGOL (Inverse Resolution)

– Top-down methods:• Search from general to specific:

Most General Solution (MGS)

• FOIL, Progol

Criteria for chosing the generality

– Risk that can derive from a classification error

high risk LGSlow risk MGS

– Confidence in the set of negative examples

high confidence MGSlow confidence LGS

Generality of Solutions

B: bird(X) sparrow(X).

mammal(X) cat(X).

sparrow(a). cat(b). bird(c). mammal(d).

E+ = { flies(a) } E– = { flies(b) }

fliesMGS(X) bird(X).

fliesLGS(X) sparrow(X).

fliesMGS(X) cat(X).

fliesLGS(X) mammal(X).

beggar2 beggar1attacker1

attacker2

Of Beggars and Attackers

Example: Mixing LGS and MGS (1)

– Concept of attacker maximize the concept and minimize its opposite:

attacker1(X) attacker +MGS(X), not attacker1(X).

attacker1(X) attacker –LGS(X), not attacker1(X).

– Concept of beggar —give only to those appearing to need it minimize the concept and maximize its opposite:

beggar1(X) beggar +LGS(X), not beggar1(X).

beggar1(X) beggar –MGS(X), not beggar1(X).

– However, rejected beggars, may turn into attackers maximize the concept and minimize its opposite:

beggar2(X) beggar +MGS(X), not beggar2(X).

beggar2(X) beggar –LGS(X), not beggar2(X).

– Concepts can be used to minimize the risk when carrying a lot of money:

run lot_of_money, attacker1(Y), not beggar2(Y). run give_money. give_money beggar1(Y). give_money attacker1(Y), beggar2(Y).

Example: Mixing LGS and MGS (2)

– When carrying little money, one may prefer to risk being beaten up.

Therefore one wants to relax attacker1 but not so much as to use attacker –

MGS :

run little_money, attacker2(Y).

attacker2(X) attacker +LGS(X), not attacker2(X).

attacker2(X) attacker –LGS(X), not attacker2(X).

Example: Mixing LGS and MGS (3)

Abduction

• Consider rule: a => b

• Deduction: from a conclude b

• Abduction: knowing or observing b, assume a as its hypothetical explanation

• From theory + observations find abductive models —the explanations for observations

Distributing Observations

• Code observations as Integrity Constraints (ICs):

<-- not some_observation

• Find abductive explanations for observations

• Create several agents, give each the same base theory and a subset of the observations

• Each agent comes up with alternative abductive explanations for its own ICs. These need not be minimal sets of hypotheses

Choosing the Best Explanation

• “Brainstorming” is used for solving complex problems

• Each participant contributes by adding ideas (abducibles) to a common idea pool, shared by all

• Ideas are mixed, crossed, mutated, and selected• Solutions arise from pool by iterating this

evolutionary process• Our work is inspired on the evolution of alternative

ideas and arguments to find collaborative solutions

Lamarckian Evolution - 1

• Lamarckian evolution = meme evolution

• “meme” is cognitive equivalent of gene

• In genetic programming Lamarckian evolution has proven a powerful concept

• There are GAs that include, additionally, a logic-based Lamarckian belief revision operator, where assumptions are coded as memes

Lamarckian Evolution - 2

• Lamarckian operator (L-op) Darwinian ones (D-ops)• L-op modifies chromosomes coding beliefs to improve

fitness with experience, rather than randomly• L-op and D-ops play distinct roles• L-op employed to bring chromosomes closer to a

solution, by belief revision • D-ops randomly produce alternative belief

chromosomes to deal with unencountered situations, by interchanging memes

Specific Belief Evolution Method

• In traditional multi-agent problem solving, agents benefit from others’ knowledge & experience by message-passing

• Our new method: knowledge & experience coded as memes, and exchanged by crossover

• Crucial: logic-based belief revision is used to modify belief assumptions (memes) based on individual agent experience

Fitness Functions

• Various fitness functions can be used in belief revision. The simplest is:

Fitness( ci ) = (ni /n) / (1 + NC )

where- ni is number of ICs satisfied by chromosome

ci

- n is total number of ICs- NC is number of contradictions

depending on chromosome ci

Assumptions & Argumentation - 1

• Assumptions are coded as abducible literals in LPs

• Abducibles are packed together in chromosomes

• Evolutionary operators —crossover, mutation, revision, selection— are applied to chromosomes

• This setting provides means to search for evolutionary consensus from initial assumptions

Assumptions & Argumentation - 2

• 3-valued contradiction removal presented before (with undefined value) is superficial

• Removes contradiction p(X) & ¬p(X) by forcing a 3-valued semantics, not looking into reasons why both hold

• Improvement relies on principles of argumentation: find arguments supporting p(X) — or ¬p(X) —

and change some of their assumptions

Collaborative Opposition

• Challenging environment of Semantic Web is a ‘place’ for future intelligent systems to float in

• Learning in 2- or in 3-values are open possibilities

• Knowledge & Reasoning shared and distributed

• Opposing arguments will surface from agents

• Need to know how to reconcile opposing arguments

• Find 2-valued consensus as much as possible

• Least commitment 3-valued consensus are not enough

Argumentation & Consensus - 1

• The non-trivial problem we addressed was that of defining 2-valued complete models, consistent with the 3-valued preferred maximal scenarios of Dung

• The resulting semantics —Approved Models— is a conservative extension to the well-known Stable Models semantics, in that every SM is an Approved Model

Argumentation & Consensus - 2

• Approved Models are guaranteed to exist for every Normal Logic Program (NLP), whereas SMs are not

• Examples show NLPs with no SMs can usefully model knowledge

• The guarantee is crucial in program composition of knowledge from diverse sources

• Model existence warrant also crucial after external- or self-updating NLPs

Argumentation & Consensus - 3

• Start by merging all opposing abductive arguments• Draw conclusions from program + single merged

argument• If contradictions arise: non-deterministically choose

one assumption of single argument and revise its truth value

• Iteration finds the non-contradictory arguments• Evolutionary method presented implements yet

another mechanism to find consensual non-contradictory arguments

Thank you for your attention!