Controlling Backtracking : Cuts

1
Controlling Backtracking Cuts

• t.k.prasad_at_wright.edu
• http//www.knoesis.org/tkprasad/

2
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

3
Efficiency Cut

• Preventing backtracking when faced with mutually
  disjoint cases
• sign(X, positive) - X gt 0.
• sign(X, negative) - X lt 0.
• sign(X, zero).

4
(contd)

• Good use of cut if declarative reading preserved
• In general, reordering of clauses may not
  preserve the declarative meaning.
• sign(X, positive) - X gt 0, !.
• sign(X, negative) - X lt 0, !.
• sign(X, zero).

5
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.

6
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.
• p(f(_),_,_) - q1(), , qn().

7
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.

8
Avoiding Duplicates

• member(X,XL) - !.
• 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.

9
Bugs with Cuts

• min(X, Y, X) - X lt 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 lt Y,!,Z X.

10
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.

11
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). -gt one solution
• ?- p(a). -gt no
• ?- p(d). -gt yes

12
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). -gt X a
• ?- p(a). -gt yes
• ?- p(d). -gt yes

13
(contd)

• 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). -gt X d
• ?- p(a). -gt no
• ?- p(d). -gt yes

14
(contd)

• p(X) - q(X), r(X), !, s(X).
• p(d).
• Additional Facts q(c),r(c)
• Declarative Reading p(d)
• Procedural reading
• ?- p(X). -gt no
• ?- p(c). -gt no
• ?- p(d). -gt yes

15
(contd)

• 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)

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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 - c.
• p - a, !, b.

17
Conditional vs Cut

• tp(X,Y) - q(X) -gt 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).

18
(contd)

• ?- 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.

19
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.