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Controlling Backtracking

Controlling Backtracking. Notes for Ch.5 of Bratko For CSCE 580 Sp03 Marco Valtorta. Preventing Backtracking. Prolog automatically backtracks in its attempts to satisfy a goal (equivalently, in its traversal of a goal tree) Sometimes, backtracking is a source of inefficiency

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Controlling Backtracking

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  1. Controlling Backtracking Notes for Ch.5 of Bratko For CSCE 580 Sp03 Marco Valtorta

  2. Preventing Backtracking • Prolog automatically backtracks in its attempts to satisfy a goal (equivalently, in its traversal of a goal tree) • Sometimes, backtracking is a source of inefficiency • For example, Prolog, when requested for additional answers to a goal, may try clauses that are known to lead, eventually, to failure

  3. A Double-step Function • if X < 3 then Y = 0 • if 3 =< X and X < 6 then Y = 2 • if 6 =< X then Y = 4 • In Prolog, f( X,0) :- X < 3. % Rule 1 f( X,2) :- 3 =< X, X < 6. % Rule 2 f( X,4) :- 6 =< X. % Rule 3 • Assume that X is instantiated to a number when the program is used • The program does not exploit the fact that the three rules are mutually exclusive

  4. Experiment 1: ch5_1.pl [trace] 6 ?- f( 1,Y), Y > 2. Call: (7) f(1, _G423) ? creep ^ Call: (8) 1<3 ? creep ^ Exit: (8) 1<3 ? creep Exit: (7) f(1, 0) ? creep No need to go beyond here! Redo: (7) f(1, _G423) ? creep But rules 2 and 3 ^ Call: (8) 3=<1 ? creep are tried anyway ^ Fail: (8) 3=<1 ? creep Redo: (7) f(1, _G423) ? creep ^ Call: (8) 6=<1 ? creep ^ Fail: (8) 6=<1 ? creep Fail: (7) f(1, _G423) ? creep No

  5. Using the cut: ch5_2.pl f( X,0) :- X < 3, !. % commit! f( X,2) :- 3 =< X, X < 6, !. % commit! f( X,4) :- 6 =< X. [trace] 9 ?- f( 1,Y), Y > 2. Call: (7) f(1, _G423) ? creep ^ Call: (8) 1<3 ? creep ^ Exit: (8) 1<3 ? creep Exit: (7) f(1, 0) ? creep Fail: (7) f(1, 0) ? creep since rule1 was “used”, No rules 2 and 3 are not tried • “used” means: the cut in rule 1 was reached

  6. Green and Red Cuts f( X,0) :- X < 3, !. % if X < 3 then Y = 0 f( X,2) :- X < 6, !. % otherwise, if X < 6 then Y = 2 f( _,4). % otherwise Y = 4 (ch5_3.pl) • But the following is incorrect f( X,0) :- X < 3. f( X,2) :- X < 6 . f( _,4). • Sometimes a cut changes the meaning of a program • Red cuts change the meaning of a program (as in the program above) • Green cuts do not change the meaning of a program (as in ch5_2.pl)

  7. Cut Mechanism • Let the parent goal be the one that matches the head of the clause containing the cut • When the cut is encountered as a goal, it succeeds immediately, but it commits the system to all choices made between the time the parent goal was invoked and the time the cut was encountered: all the remaining possibilities between the parent goal and the cut are discarded

  8. Examples Using Cut: ch5_4.pl • max/3 (with cut); max1/3 (without cut) • strange behavior when one of the arguments is a list of numbers: the first number in the list is used in the comparison. Cf. SWI-Prolog manual, section 4.6.1 • problem with max( 3,1,1): it succeeds! • max2/3 fixes this problem max2( X,Y,Max) :- X >= Y, !, Max = X. max2( _,Y,Max) :- Max = Y. • deterministic membership (member1/2)

  9. Examples Using Cut: ch5_4.pl, Ctd. • adding an element to a list without duplication • add/3 • classification into categories • class/2 • in both cases, the last argument should not be instantiated

  10. Negation as Failure • different( X,Y) :- X = Y, !, fail. • different( X,Y). Or: • different( X,Y) :- X = Y, !, fail; true. • not( P) :- P, !, fail ; true. • op( 900, fy, [not, ‘\+’]). • not P succeeds if P fails. • different( X,Y) :- not( X = Y).

  11. Problem with Negation As Failure • Cf. even/odd mutually recursive program (ch5_5.pl) • A positive query ?-p(X) is interpreted as a proof for “there exists X s.t. P(X) is true” • A negative query ?- not(p(X)) is interpreted as a proof for “for all X, P(X) is false)

  12. Problems with cut and negation • The main problem with cut is that we may lose the valuable correspondence between declarative and procedural meanings of a program • Prolog negation is based on the closed world assumption: “if something cannot be proven, it is false.” • This leads to especially bad results when the argument of not is a term containing unistantiated variables • Some Prolog dialects (e.g., NU-Prolog) try to delay not subgoals until all variables in them are instantiated, when possible

  13. Logical Negation • Logical negation cannot be handled using Prolog’s goal tree mechanism • See HoleInGoalTrees.ppt

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