e + : boss(mary,john). boss(phil,mary).boss(phil,john). e - :

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1 E + : boss(mary,john). boss(phil,mary).boss(phil,john). E - : boss(john,mary). boss(mary,phil). boss(john,phil). BK: employee(john, ibm). employee(mary,ibm). employee(phil,ibm). reports_to_imm(john,mary). reports_to_imm(mary,phil). h: boss(X,Y):- employee(X,O), employee(Y,O),reports_to(Y, X). reports_to(X,Y):-reports_to_imm(X,Z), reports_to(Z,Y). reports_to(X,X). Another example of recursive learning:

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Another example of recursive learning:. E + : boss(mary,john). boss(phil,mary).boss(phil,john). E - : boss(john,mary). boss(mary,phil). boss(john,phil). BK: employee(john, ibm). employee(mary,ibm). employee(phil,ibm). reports_to_imm(john,mary). reports_to_imm(mary,phil). - PowerPoint PPT Presentation

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Page 1: E + : boss(mary,john). boss(phil,mary).boss(phil,john). E - :

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E+:boss(mary,john). boss(phil,mary).boss(phil,john).

E-:boss(john,mary). boss(mary,phil). boss(john,phil).

BK:employee(john, ibm). employee(mary,ibm). employee(phil,ibm).reports_to_imm(john,mary). reports_to_imm(mary,phil).

h: boss(X,Y):- employee(X,O), employee(Y,O),reports_to(Y, X).

reports_to(X,Y):-reports_to_imm(X,Z), reports_to(Z,Y).reports_to(X,X).

Another example of recursive learning:

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How is learning done: covering algorithm

Initialize the training set T to all k-tuples of constants while the global training set contains + tuples: find a clause that describes part of relationship Q remove the +tuples covered by this clause

Finding a clause:

initialize the clause to Q(V1,…Vk) :- while T contains –tuples find a literal L to add to the right-hand side of the clause

Finding a literal : greedy search

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• ‘Find a clause’ loop describes search – bottom up or top down –

• Need to structure the search space – generality – semantic and syntactic

• since logical generality is not decidable, a stronger property of -subsumption

• then search from general to specific (refinement)

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Refinement

boss(X,Y):-

boss(X,Y):-X=Y

… boss(X,Y):-reports_to(X,Y).

boss(X,Y):-empl(X,O).

boss(X,Y):-empl(X,O),empl(Y,O1).

boss(X,Y):-empl(X,O),empl(Y,O).

boss(X,Y):-empl(X,O),empl(Y,O),rep_to(Y,X).

boss(X,Y):-empl(X,O),empl(Y,O),rep_to(X,Y).

Heuristics: link to head

new variables

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How is learning done: covering algorithm

• Inner loop describes search – bottom up and top down - we do the latter

• Need to structure the search space – generality – semantic and syntactic – theta subs.

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Constructive learning

• Do we really learn something new?• Hypotheses are in the same language as

examples• constructive induction• How do we learn multiplication from

examples? We need to invent plus –we have shown [IJCAI93] that true constructivism requires recursion, i.e. in mult(X,s(Y),Z) :- mult(X,Y,T), newp(T,Y,Z)mult(X,0) :- 0.

• Newp – plus - must be recursive.

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Philosophical motivation

• Constructive induction is analogical to “revolution” in the methodology of science

• Kuhn’s Structure of Scientific Revolution: normal science -> crisis -> revolution -> normal science• Normal science = learning a “theory” in a fixed

language• Crisis = failure to cope with anomalies observed,

due to inadequate language• Revolution = introduction of new terms into the

language (cannot be done in AV)

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Example: predicting colour in flowers

• Language: r, y; a is any red flower, b is any yellow flower; col(X,Y) X is of colour Y; ch(X,Y) = result of breeding of X and Y

• Observations (that Czech monk and his peas…)1. col(a,r) % Adam and Eve

2. col(b,y).

3. col(ch(a,a),r). % first generation

4. col(ch(a,b),r).

5. col(ch(b,b),b).

6. col(ch(a,ch(b,b),r).%original and 1st

7. …

8. col(ch(ch(a,b)ch(a,b),y). 1st and 1st

9. ….

10.:-col(ch(a,a),y).

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col(ch(a,X),r).

col(ch(X,Y),a) :- col(X,r), col(Y,r).

col(ch(b,b),y).

col(ch(X,Y), y) :- col(X,y),col(Y,y).

• But in some generations y and r produce r, and in some – y

• We need either infinitely many clauses, or infinitely long clauses

• A revolution is necessary

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A new necessary predicate is invented

• n00 represents purebred flowers with recessive character, n11 – with dominant, and n10 – hybrid with dominant

• In fact, the invented predicates represent the concept of a gene!

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Success story: mutagenicity

• heterogeneous chemical compounds – their structure requires relational representation

• BK: properties of specific atoms and bonds between them (relation!) and generic organic chemistry info (e.g. structure of benzene rings, etc.)

• Regression-unfriendly A learned rule has been published in

Science

conjugated double bond in a five-member ring

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problems

• Expressivity – efficiency• Dimensionality reduction• Therefore, interest in feature

selection