1 / 10

constraint-based problem solving constraint-based problems Generate&Test approach generate candidate solution, then

Dave Reed. constraint-based problem solving constraint-based problems Generate&Test approach generate candidate solution, then test constraints Test&Generate approach apply constraints first, then fill in blanks in candidate timing code: time , doReps. Constraint-based problems.

jebediah
Télécharger la présentation

constraint-based problem solving constraint-based problems Generate&Test approach generate candidate solution, then

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Dave Reed • constraint-based problem solving • constraint-based problems • Generate&Test approach generate candidate solution, then test constraints • Test&Generate approach apply constraints first, then fill in blanks in candidate timing code: time, doReps

  2. Constraint-based problems • some problems are not perfectly suited to search space representation • not really interested in a sequence of transitions from one state to another • instead, know the basic form of a solution, plus constraints on that solution Consider 4 people: Bob, Karen, Dan, Sue Constraints: Bob is older than Karen. Dan is the youngest. Sue is not the oldest. Problem: Given these constraints, find the relative ages of the 4 people. • In fact, this problem is under-constrained: • Solution 1 (by increasing ages): Dan, Sue, Karen, Bob • Solution 2 (by increasing ages): Dan Karen, Sue, Bob • constraint-based problems can still be reduced to search using a Generate&Test approach • generate a candidate solution (based on the known solution form) • test the candidate solution to see if it meets the constraints • if not, BACKTRACK to try a new candidate solution

  3. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%% ages1.pro Dave Reed 2/24/02 %%% %%% Test and Generate solution to ages problem: %%% Bob is older than Karen. %%% Dan is the youngest. %%% Sue is not the oldest. %%% This program represents the solution as a list, ordered by age. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% solve(X) :- candidate(X), constraints(X). candidate(Candidate) :- permutation(Candidate,[bob,karen,dan,sue]). constraints(Candidate) :- Candidate = [P1,P2,P3,P4], comes_before(karen,bob,Candidate), P1 = dan, member(sue,[P1,P2,P3]). %%%%%%%%%% Utilities %%%%%%%%%% comes_before(X,Y,[X|T]) :- member(Y,T). comes_before(X,Y,[_|T]) :- comes_before(X,Y,T). permutation([],[]). permutation([H|T],Perm) :- permutation(T,P), select(H, Perm, P). Age problem: Generate&Test • will represent a solution as a list of people, ordered by age • solve predicate: • generates a candidate solution (i.e., permutation of the names) • tests to see if that candidate solution meets the problem constraints ?- solve(X). X = [dan, karen, sue, bob] ; X = [dan, sue, karen, bob] ; HANGS!

  4. Generate&Test vs. Test&Generate • if the solution space is large, Generate&Test can be infeasible worst case: would have to generate every potential solution & test for ages problem: • 4! = 24 possible arrangements of the people  exhausting solution space is doable • if there were 8 people, 8! = 40,320 possible arrangements • if there were 10 people, 10! = 3,628,800 possible arrangements • instead, use a Test&Generate approach • first apply constraints to a solution template • then generate permutations to fill in the remaining blanks for ages problem: the constraint that Dan is the youngest eliminates 18 possibilities the constraint that Sue in not the oldest eliminates another 2 possibilities . . .

  5. Age problem: Test&Generate %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%% ages2.pro Dave Reed 2/24/02 %%% %%% Test and Generate solution to ages problem: %%% Bob is older than Karen. %%% Dan is the youngest. %%% Sue is not the oldest. %%% This program represents the solution as a list, ordered by age. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% solve(X) :- constraints(X), candidate(X). candidate(Candidate) :- permutation(Candidate,[bob,karen,dan,sue]). constraints(Candidate) :- Candidate = [P1,P2,P3,P4], comes_before(karen,bob,Candidate), P1 = dan, member(sue,[P1,P2,P3]). %%%%%%%%%% Utilities %%%%%%%%%% comes_before(X,Y,[X|T]) :- member(Y,T). comes_before(X,Y,[_|T]) :- comes_before(X,Y,T). permutation([],[]). permutation([H|T],Perm) :- permutation(T,P), select(H, Perm, P). • again, represent solution as a list of people, ordered by age • solve predicate: • apply constraints to an abstract solution (i.e., a list of variables) • fill in the remaining blanks using the candidate predicate ?- solve(X). X = [dan, karen, sue, bob] ; X = [dan, sue, karen, bob] ; No EXHAUSTS THE SEARCH

  6. time predicate to quantify how much better Test&Generate is, can make use of the built-in time predicate time(Goal): solves Goal and reports the # of inferences and cpu time required note: time does not allow backtracking to see alternate answers ?- consult(ages1). % ages1 compiled 0.06 sec, 0 bytes Yes ?- time(solve(X)). % 239 inferences in 0.00 seconds (Infinite Lips) X = [dan, karen, sue, bob] Yes ?- consult(ages2). % ages2 compiled 0.06 sec, 0 bytes Yes ?- time(solve(X)). % 44 inferences in 0.00 seconds (Infinite Lips) X = [dan, karen, sue, bob] Yes

  7. Better timings %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%% doreps.pro Dave Reed 2/24/02 %%% %%% A predicate for repeatedly solving a goal. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% doReps(Goal, 1) :- Goal. doReps(Goal, Reps) :- not(not(Goal)), NewReps is Reps-1, doReps(Goal, NewReps). • since time rounds to nearest 100th of a second, quick goals appear as 0 • can define our own predicate for repeating a goal some # of repetitions • note use of double negation to discard bindings -- makes sure each call to Goal uses the initial variables • by doing 1000 reps each, see that Test&Generate is 4-5 times faster ?- consult(ages1). % ages1 compiled 0.06 sec, 0 bytes Yes ?- time(doReps(solve(X), 1000)). % 242,997 inferences in 0.27 seconds (899989 Lips) X = [dan, karen, sue, bob] Yes ?- consult(ages2). % ages2 compiled 0.06 sec, 0 bytes Yes ?- time(doReps(solve(X), 1000)). % 47,997 inferences in 0.06 seconds (799950 Lips) X = [dan, karen, sue, bob] Yes

  8. Another example • Consider a more complex example: Three friends came in 1st, 2nd and 3rd in a programming contest. Each has a different name (Michael, Simon, Richard), nationality (American, Australian, Israeli), and favorite sport (basketball, tennis, cricket). The following is known: • Michael likes basketball, and did better than the American. • Simon, the Israeli, did better than the tennis player. • The cricket player came in first. Solution space: 6 ways to assign names to the 3 places 6 ways to assign nationalities to the 3 places 6 ways to assign sports to the 3 places  6*6*6 = 216 possible arrangements

  9. Competition problem: Test&Generate %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%% compete.pro Dave Reed 2/24/02 %%% %%% Test and Generate solution to the competition problem. %%% %%% This program represents the solution as a list, ordered by finish. %%% Each element of the list is of the form "Name::Nationality::Sport". %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% :- op(100,xfy,'::'). solve(X) :- constraints(X), candidate(X). candidate([P1::N1::S1, P2::N2::S2, P3::N3::S3]) :- permutation([P1,P2,P3],[michael,richard,simon]), permutation([N1,N2,N3],[american,australian,israeli]), permutation([S1,S2,S3],[basketball,cricket,tennis]). constraints(Candidate) :- Candidate = [X1,X2,X3], member(michael::_::basketball,Candidate), comes_before(michael::_::_,_::american::_,Candidate), member(simon::israeli::_,Candidate), comes_before(simon::_::_,_::_::tennis,Candidate), X1 = _::_::cricket. %%%%%%%%%% Utilities %%%%%%%%%% permutation([],[]). permutation([H|T],Perm) :- permutation(T,P), select(H, Perm, P). represent solution as a list of competitors, ordered by finish (1st place at HEAD) use '::' operator to combine info about each competitor

  10. Timing the Test&Generate solution ?- solve(X). X = [simon::israeli::cricket, michael::australian::basketball, richard::american::tennis] ; No ?- time(solve(X)). % 97 inferences in 0.00 seconds (Infinite Lips) X = [simon::israeli::cricket, michael::australian::basketball, richard::american::tennis] Yes ?- time(doReps(solve(X), 1000)). % 100,997 inferences in 0.11 seconds (918155 Lips) X = [simon::israeli::cricket, michael::australian::basketball, richard::american::tennis] Yes code finds the unique solution Generate&Test would require 1,191 inferences Generate&Test would take 1.54 seconds for 1000 repetitions

More Related