Epistemological Framework for Reuse

Problem SolvingMethod

Generic Task

Multi-Functional Domain

MappingKnowledge

Application-specificProblem-Solving Knowledge

Application Configuration

Domain Ontology

Task Ontology Method Ontology

Mapping Ontology

Application Model

A Library of Components for Classification Problem Solving

Main Goals

• To carry out a knowledge-level analysis of classification

• To develop a practical resource to support the development of classification applications

• To provide a concrete set of components to act as a test case for IBROW brokering system and IRS

Classification

Classification can be seen as the problem of finding the solution (class), which best explains a set of known facts (observables), according to some criterion

Observables

Candidate Sols.Classification Solution

Solution Criterion

Example

Observables

Candidate Sols.

Criterion

Classification Solution

{background=green; area=china...}

Complete-coverage-criterion(every observable has to be explained)

{chinese-granny, dutch-granny, etc..}

{chinese-granny}

Observables

Observables = set_of (Observable);Observable = {feature, value}.

Well defined Observables (obs):

({f1, v1} obs {f1, v2} obs) -> v1 = v2

({f1, v1} obs) -> legal_feature_value (f1, v1 )

Solutions

Solution = set_of (Feature_Spec);Feature_Spec = {Feature, Feature_value_spec}Feature_value_spec = Unary_Relation

Well defined Solution (sol):{f1, s1} sol holds (s1, v1 ) ->

legal_feature_value (f1, v1 )

Matching

Observable={f1, v1} matches Solution=sol iff:

{f1, c} sol holds (c, v1 )

Matching Sets of Obs to a Solution

Sol: {{fsol1, c1}...{fsolm, cm}}; Obs: {{fob1, v1}...{fobn, vn}}

Four possible cases: {fj, cj} sol {fj, vj} obs holds (cj, vj)

-> Explained (fj)

{fj, cj} sol {fj, vj} obs not holds (cj, vj) -> Inconsistent(fj)

{fj, vj} obs {fj, cj} sol -> Unexplained (fj)

{fj, vj} obs {fj, cj} sol -> Missing (fj)

Default Match Criterion

Match Score:Vector: <I, E, U, M>

Match Comparison RelationS1 = (i1, e1, u1, m1); S2 = (i2, e2, u2, m2)

S1 better_score than S2 iff:

(i1 < i2)

(i2 = i1 e2 < e1) (i2 = i1 e2 = e1 u1 < u2) (i2 = i1 e2 = e1 u2 = u1 m1 < m2)

Other Match Criteria

• SUBSET-BASED-MATCH-CRITERION– Uses subset instead of < to determine best

match

• ABSTRACTION-AWARE-MATCH-CRITERION – Matching mechanism able to handle both

observables and abstracted data

• EXPLANATION-CENTRED-MATCH-CRITERION– Focuses on explanation power

• EQUAL-RATING-MATCH-CRITERION– Very stupid one, which gives every solution the

same score

Possible Solution Criteria

• Positive Coverage– Some feature is explained and none is

inconsistent

• Complete Coverage– All features are explained and none is

inconsistent

• No missing features – No inconsistent features and no missing

features

Hierarchy of Criteria

Solution Criterion

Match Criterion

Match Score Comparison Rel

Macro Score MechanismFeature Score Mechanism

Match Score Mechanism

Observables

(def-class observables (set) ?obs "This is simply a set of observables. An important constraint is that there cannot be two values for the same

feature in a set of observables" :iff-def (every ?obs observable) :constraint (not (exists (?ob1 ?ob2) (and (member ?ob1 ?obs) (member ?ob2 ?obs) (has-observable-feature ?ob1 ?f) (has-observable-feature ?ob2 ?f) (has-observable-value ?ob1 ?v1) (has-observable-value ?ob2 ?v2) (not (= ?v1 ?v2))))))

Solutions

(def-class solution () ?x "A solution is a set of feature definitions" :iff-def (every ?x feature-definition))

(def-class feature-definition () ?x ((has-feature-name :type feature) (has-feature-value-spec :type unary-relation)) :constraint (=> (and (has-feature-name ?x ?f) (has-feature-value-spec ?x ?spec)) (=> (holds ?spec ?v) (legal-feature-value ?f ?v))))

Solution Criterion

(def-class solution-admissibility-criterion () ?c ((applies-to-match-score-type :type match-score-type) (has-solution-admissibility-relation :type unary-relation)) :constraint (=> (and (solution-admissibility-criterion ?c) (has-solution-admissibility-relation ?c ?r) (domain ?r ?d)) (subclass-of ?d match-score)))

Monotonicity of Admissibile Solutions

(def-axiom admissibility-is-monotonic "This axiom states that the admissibility criterion is monotonic. That is, if a

solution, ?sol, is admissible, then any solution which is better than ?sol will also be admissible"

(forall (?sol1 ?sol2 ?obs ?criterion) (=> (and (admissible-solution ?sol1 (apply-match-criterion

?criterion ?obs ?sol1) ?criterion)

(better-match-than ?sol2 ?sol1 ?obs ?criterion)) (admissible-solution ?sol2 (apply-match-criterion

?criterion ?obs ?sol2) ?criterion))))

Complete Coverage

(def-instance complete-coverage-admissibility-criterion solution-admissibility-criterion ((applies-to-match-score-type default-match-score) (has-solution-admissibility-relation complete-coverage-admissibility-relation)))

(def-relation complete-coverage-admissibility-relation (?score) "a solution should be consistent and explain all features" :constraint (default-match-score ?score) :iff-def (and (= (length (first ?score)) 0) ;;no inconsistency (= (length (third ?score)) 0))) ;;no unexplained

Classification Task Ontology

• 70 Definitions• Provides both a theory of classification and a

vocabulary to describe classification problems• Ontology is separated from task specifications

Generic Classification Task

• Input roles– Candidate Solutions, Match Criterion, Solution

Criterion, Observables

• Precondition– Both observables and candidate solutions have

to be provided

• Goal– To find a solution from the candidate solutions

which is admissible with respect to the given observables, solution criterion and match criterion

(def-class classification-task (goal-specification-task) ?task "Classification is defined here as finding one or more admissible solutions out of a predefined solution space, which explain the features of a given set of observables, in accordance with a given match criterion and solution admissibility criterion. Because different variants of the goal can be formulated, the goal of the task is given here only as a default" ((has-input-role :value has-candidate-solutions :value has-observables :value has-match-criterion :value has-solution-admissibility-criterion) (has-output-role :value has-solutions) (has-candidate-solutions :type solution-space) (has-observables :type observables) (has-match-criterion :type match-criterion :default-value default-match-criterion) (has-solution-admissibility-criterion :type solution-admissibility-criterion :default-value default-solution-admissibility-criterion) (has-solutions :type solution-space) ……………….))

(has-precondition :value (kappa (?task) (exists (?x ?y) (and (member ?x (role-value ?task 'has-observables)) (member ?y (role-value

?task 'has-candidate-solutions))))))(has-goal-expression :default-value (kappa (?task ?sols) (forall ?sol (=> (member ?sol (role-value ?task 'has-solutions)) (admissible-solution ?sol (apply-match-criterion (role-value ?task 'has-match-criterion) (role-value ?task 'has-observables) ?sol) (role-value ?task 'has-solution-admissibility-criterion))))))

Specific Classification Tasks

• Single-Solution Classification Task– Single-solution assumption

• Optimal Classification Tasks– Goal requires optimality

Problem Solving Library

• Based on heuristic classification model• Includes both data-directed and solution-

directed methods• Supported by a method ontology

Method Ontology: Main Concepts

• Abstractors– Mechanism for performing abstraction on

observables– Abstractor: Obs* -> Obs

• Refiners– Mechanism for specialising a solution– Refiner: Sol -> Sol*

• Candidate Exclusion Criterion– A criterion which is used to decide when a

search path is a dead-end– Default criterion rules out inconsistent

solutions

Monotonicity of Exclusion Criterion

(def-axiom exclusion-is-monotonic (forall (?sol1 ?sol2 ?obs ?criterion) (=> (and (ruled-out-solution ?sol1 (the-match-score ?sol1) ?criterion) (not (better-match-than ?sol2 ?sol1 ?obs ?criterion))) (ruled-out-solution ?sol2 (the-match-score ?sol2)?criterion))))

Axiom of Congruence

(def-axiom CONGRUENT-ADMISSIBILITY-AND-EXCLUSION-CRITERIA (forall (?sol ?task) (=> (member ?sol (the-solution-space ?task)) (not (and (admissible-solution ?sol (the-match-score ?sol) (role-value ?task 'has-solution-admissibility-criterion)) (ruled-out-solution ?sol (the-match-score ?sol)

(role-value ?psm

'has-solution-exclusion-criterion)))))))

Three Heuristic Classification PSMs

• Two Data-directed– Admissible Solution Classifier

• Finds one admissible solution according to the given criteria• Uses backtracking hill climbing

– Optimal Classifier• Performs complete search looking for optimal solution• Uses best-first strategy• Uses candidate exclusion criterion to prune search space

• One Solution-directed– Goes down the solution hierarchy, acquiring

observables as needed– Ask for observables with max discrimination power

Four Assumptions in Main PSMs

• No cycles in abstraction hierarchy• No cycles in refinement hierarchy• At least one class in the solution space is an

admissible solution• The solution refinement hierarchy is consistent

with the candidate exclusion criterion. That is if sol is ruled out, all refinements of sol can also be ruled out

Task-Method Hierarchy

abstraction

heuristic-classification-psm

classification

rank-solutions refinement

basic-heuristic-matchselect-abstractor one-step-abstraction collect-refiners apply-refiners

abstraction-psm refinement-psmrank-solutions-psm

Example

• Apple Domain– Originally developed in Amsterdam

• Solutions = Apple Types = {granny, noble, delicious...}

• Hierarchy of Apple Types• Features = {bkg-colour, fg-colour, rusty....}• Pretty trivial really!

Classification TaskOntology

Heuristic ClassificationOntology

Apple Heuristic ClassificationApplication

Classification TaskSpecification

Classification-to-Class-RepresentationMapping Ontology

AppleDomain Model

Heuristic ClassificationPSMs

Mapping Solutions and Obs to Apples

(def-relation-mapping solution :up ((solution ?x) if (or (= ?x apple) (subclass-of ?x apple))))

(def-relation-mapping observable :up ((observable ?x) if (or (== ?X (?f ?v ?obs)) (== ?x (?f ?v)))))

More Relation Mappings

(def-relation-mapping has-observable-feature :up ((has-observable-feature ?x ?f) if (or (== ?X (?f ?v ?obs)) (== ?x (?f ?v)))))

(def-relation-mapping has-observable-value :up ((has-observable-value ?x ?v) if (or (== ?X (?f ?v ?obs)) (== ?x (?f ?v)))))

(def-relation-mapping directly-abstracts-from :up ((directly-abstracts-from ?ob ?obs) if (== ?ob (?f ?v ?obs))))

Sample Abstractor

(def-instance sugar-abstractor abstractor ((has-body '(lambda (?obs) (in-environment ((?v . (observables-feature-value ?obs 'sugar))) (cond ((>= ?v 70) (list-of 'sweet-level 'high (list-of (list-of 'sugar ?v)))) ((and (< ?v 70) (> ?v 40)) (list-of 'sweet-level 'medium (list-of (list-of 'sugar ?v)))) ((<= ?v 40) (list-of 'sweet-level 'low (list-of (list-of 'sugar ?v)))))))) (applicability-condition (kappa (?obs) (member 'sugar (all-features-in-observables ?obs))))))

Generic (reusable) Refiner

(def-instance refinement-through-subclass-of-links refiner "If the solution space is specified by means of classes arranged in a

subclass-of hierarchy, then this is a good refiner to use" ((has-body '(lambda (?sol) (setofall ?sub (direct-subclass-of ?sub ?sol)))) (applicability-condition (kappa (?sol) (and (class ?sol) (exists ?sub (direct-subclass-of ?sub ?sol)))))))

Evaluation/Results

• Library provides an analytical tool to understand classification problem solving

• Library tested on a number of domains– ECAI Paper classification– Paleontology– Medical Diagnosis

• No changes the organization of library needed– Only adding new components, e.g. new match

criteria

• Used by van Harmelen and Ten Teije to study automatic PSM adaptation