cs774 (Prasad)L6Backtracking1 Controlling Backtracking : Cuts

Motivation Efficiency by preventing needless backtracking –Cf. CFL Parsing Lookaheads vs Backtracking vs Dynamic programming Avoiding duplicates or resatisfaction Prolog as programming language –Tailoring control strategy cs774 (Prasad)L6Backtracking2

Efficiency Cut Preventing backtracking when faced with mutually disjoint cases sign(X, positive) :- X > 0. sign(X, negative) :- X < 0. sign(X, zero). cs774 (Prasad)L6Backtracking3

(cont’d) Good use of cut if declarative reading preserved In general, reordering of clauses may not preserve the declarative meaning. sign(X, positive) :- X > 0, !. sign(X, negative) :- X < 0, !. sign(X, zero). cs774 (Prasad)L6Backtracking4

Green Cut vs Red Cut A use of a cut which improves only efficiency of a program, without altering its declarative meaning, is referred to as a green cut. –In practice, prunes away irrelevant proofs, or proofs bound to fail. –In practice, use them when mutually exclusive cases occur, or if non-determinism can be minimized. cs774 (Prasad)L6Backtracking5

Green Cut vs Red Cut A use of a cut that alters the declarative meaning of a program is referred to as a red cut. –In practice, eliminates unwanted logical solutions Note that mature Prolog implementations index clauses for a predicate on the principal functor of the first argument in the head of a clause thereby reducing non-determinism. pf p(f(_),_,_) :- q1(…), …, qn(…). cs774 (Prasad)L6Backtracking6

Implementing Conditional “p(x) :- if c(x) then r(x) else q(x)” p(X) :- c(X), !, r(X). p(X) :- q(X). In this case, reordering of clauses changes the meaning. cs774 (Prasad)L6Backtracking7

Avoiding Duplicates member(X,[X|L]) :-!. member(X,[_|L]) :- member(X,L). Without cut, the goal ?- member(a, [a, a, a]) will succeed thrice. Note that with cut member -predicate cannot be used to generate/enumerate elements in a list. That is, invertibility is destroyed. cs774 (Prasad)L6Backtracking8

Bugs with Cuts min(X, Y, X) :- X =< Y,!. min(X, Y, Y). Flaw: the goal ?- min(3,6,6) does not fail. Reason: min(3,6,6) does not unify with min(X,Y,X). Fix : change first rule to: min(X, Y, Z) :- X =< Y,!,Z = X. cs774 (Prasad)L6Backtracking9

Effect of Cut Matching Rule: H :- B1, B2, …, Bm,!, …, Bn. Goal: ?-G. When “!” is encountered, the bindings for variables in goals B1, …,Bm, and G are frozen. That is, alternate solutions to B1, …, Bm, and remaining alternatives for G are discarded. Note that the ordering of body literals and clauses becomes significant in the presence of cut. cs774 (Prasad)L6Backtracking10

Effect of Cut p(X) :- q(X), r(X), !, s(X). p(d). Declarative Reading (ignoring cut): { p(d)} Procedural reading (using cut): { p(d)} ?- p(X). -> one solution ?- p(a). -> no ?- p(d). -> yes cs774 (Prasad)L6Backtracking11

Effect of Cut p(X) :- q(X), r(X), !, s(X). p(d). Additional Facts: {q(a),r(a),s(a)} Declarative Reading : { p(a), p(d)} Procedural reading : { p(a)} ?- p(X). -> X = a ?- p(a). -> yes ?- p(d). -> yes cs774 (Prasad)L6Backtracking12

(cont’d) p(X) :- q(X), r(X), !, s(X). p(d). Additional Facts: {q(b),s(b)} Declarative Reading (ignoring cut): { p(d)} Procedural reading (using cut): { p(d)} ?- p(X). -> X = d ?- p(a). -> no ?- p(d). -> yes cs774 (Prasad)L6Backtracking13

(cont’d) p(X) :- q(X), r(X), !, s(X). p(d). Additional Facts: {q(c),r(c)} Declarative Reading : { p(d)} Procedural reading : { } ?- p(X). -> no ?- p(c). -> no ?- p(d). -> yes cs774 (Prasad)L6Backtracking14

(cont’d) p(X) :- q(X), r(X), !, s(X). p(d). Additional Facts: {q(a),r(a),s(a), q(b),r(b),s(b)} ?- p(X). Declarative Reading (ignoring cut): { p(a), p(b), p(d)} Procedural reading (using cut): { p(a)} cs774 (Prasad)L6Backtracking15

Disadvantages of Cut Destroys the declarative reading : Need to know the behavior of the interpreter to understand the meaning (“side-effects”) p :- a, b. p :- c. p :- a, !, b. p :- c. p :- a, !, b. cs774 (Prasad)L6Backtracking16

Conditional vs Cut tp(X,Y) :- q(X) -> r(Y) ; s(Y). tp(m,m). tp(n,n). is not equivalent to sp(X,Y) :- q(X),!,r(Y). sp(X,Y) :- s(Y). sp(m,m). sp(n,n). q(a). q(c). r(e). r(f). s(g). s(h). cs774 (Prasad)L6Backtracking17

(cont’d) ?- tp(X,Y). X = a Y = e X = a Y = f X = m Y = m X = n Y = n –Local cut ?- sp(X,Y). X = a Y = e X = a Y = f –sp-goal is not resatisfied for sp-facts because of the cut. cs774 (Prasad)L6Backtracking18

Other uses of Cut Implementing default or otherwise clause. a(X,Y) :- p1(X), !, q1(X,Y). … a(X,Y) :- pn(X), !, qn(X,Y). a(X,Y) :- r(X,Y). Cut-Fail combination to formalize conditions under which failure is guaranteed. a(X) :- p(X), !, fail. cs774 (Prasad)L6Backtracking19