CO-INDUCTIVE LOGIC PROGRAMMING AND ITS

APPLICATIONS

Overview

• Induction vs. co-induction• Co-inductive logic programming• A goal-directed approach for Answer Set

Computing

Induction

Inductive definitions have 3 components:• Initiality (e.g., [] is a list)• Iteration (e.g., [H|T] is a list if T is a list, and H

is a number)• Minimality (e.g., nothing else is a list)Inductive definitions correspond to least fixed point interpretations of recursive definitions.

Co-induction

• Eliminate the initiality condition• Replaces the minimality condition with maximality

• Iteration: [H|T] is a list if T is a list and H is a number• Maximality: the set of lists is the maximal set of

such lists

Co-induction corresponds to the greatest point interpretation of recursive definition

Example - list

list([]).list([1|T]) :- list(T).• What is its inductive semantics?• What is its co-inductive semantics?

Co-inductive logic programming

• Operational semantics relies on a co-inductive hypothesis set (CHS);

• During execution, if the current resolvent R contains a call C’ that unifies with a call C encountered earlier, then the call C’ succeeds; the new resolvent is R’ where = mgu(C, C’) and R’ is obtained by deleting C’ from R.

Example - list

:- coinductive list/1.list([]).list([1|T]) :- list(T).?- list(X)X = [1|X]

list(X)

list(T)

X = [1|T]

Example - list

:- coinductive list/1.list([]).list([1|T]) :- list(T).?- list(X)X = []X = [1|X]X = [1]X = [1, 1]…

list(X)

list(T)

X = [1|T]X = []

T = [] T = [1] … …

Example - Stream

:- coinductive stream/1.stream([H|T]) :- number(H), stream(T).number(0).number(s(N)) :- number(N).?- stream([0, s(0), s(s(0)) | T]).

Example – list membership

(1) member(H, [H|_]).(2) member(H, [_|T]) :- member(H, T).the desired element is the last element of some prefix of the listmembera(X, L) :- drop(X, L, _).drop(H, [H|T], T).drop(H, [_|T], T1) :- drop(H, T, T1).

Example - comember

:- coinductive comember/2.comember(X, L) :- drop(X, L, L1), comember(X, L1).?- X = [1, 2, 3 | X], comember(2, X).?- X = [1, 2, 3, 1, 2, 3], comember(2, X).?- X = [1, 2, 3 | X], comember(Y, X).

comember/2 is true if and only if the desired element does occur in an infinite number of times in the list.

APPLICATION ON COMPUTING ANSWER SETS (STABLE MODELS)

Why does a stable model matter?

p.r :- p, q.s :- p, not q.

With negation as failure

p q r s

T F F T

p.r :- p, q.s :- p, not q.

Another model!!

p q r s

T T T F

What makes the left model so special?

Stable model

• [Gelfond and Lifschitz, 1988]• For a program P not containing any negation, the stable

model is unique, defined as its least fixed point.• E.g., a model I = {p, s}• the reduct of P relative to I is the set of rules without

negation obtained from P by GL-transformation:– dropping each rule s.t. C in I and ‘not C’ in the body of the rule– dropping all the rest negative atom ‘not C’ from the bodies of

the remaining rules• I is a stable model of P if I is the stable model of the reduct

of P relative to I.

Is a Stable Model

p.r :- p, q.s :- p, not q.

I = {p, s}

p q r s

T F F T

The reductp.r :- p, q.s :- p.

I is a stable model

NOT a Stable Model

p.r :- p, q.s :- p, not q.

I = {p, q, r}

The reductp.r :- p, q.

I is NOT a stable modelp q r s

T T T F

Non-monotonic reasoning (nmr)

q. p :- q, not p.

Is {q} a stable model?Is {p, q} a stable model? r :- not s.s :- not r. p :- s, not p.

// {q} is a stable model

// {r} or {s} is a stable model

Goal-directed ASP

• Ordinary Rules– all non-cyclical rules– Cyclical rules which when used to expand a call to

a subgoal G lead to a recursive call to G through an even (but non-zero) number of negations. E.g., (1) p :- not q.(2) q :- not p.(3) r. (4) s :- r.

:- p CHS = {}

:- not q CHS = {p}

:- not not p CHS = {p, not q}

:- p CHS = {p, not q}

Goal-directed ASP

• Odd Loops Over Negation (OLON)– Cyclical rules which when used to expand a call to

subgoal G lead to a recursive call to G that is in the scope of an odd number of negations.E.g., (1) p :- q, not r.(2) r :- not p.(3) q :- t, not p.

OLON rules

• p :- q, r, not p.• If p is true through other parts of the

program, then it is useless.• If p is not true through the rest of the

program, then q or r has to be false.chk_p :- p.chk_p :- not q.chk_p :- not r.

Goal-directed executionp :- q, not r. (od & olon)r :- not p. (od)q :- t, not p. (olon)q. (od)

p :- q, not r.r :- not p.q.

chk_p :- p.chk_p :- not q.chk_p :- r.chk_q :- q.chk_q :- not t.

nmr_chk :- chk_p, chk_q.

:- p, nmr_chk. {}:- q, not r, nmr_chk. {p, q}:- not r, nmr_chk. {p, q}:- not not p, nmr_chk {p, q, not r}:- p, nmr_chk {p, q, not r}:- nmr_chk {p, q, not r}

Issues

• Identifying OLON and ordinary rules– Through a graph travel algorithm in O(|P| * n)

• Partial answer set• If cyclical rules not through any number of

negations, then the recursive call fails.E.g., p :- p.