logic & Prolog

1
Dave Reed

• constraint-based problem solving
• constraint-based problems
• GenerateTest approach
• generate candidate solution, then test
  constraints
• TestGenerate 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 Soluti
on 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 GenerateTest 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
Age problem GenerateTest

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.
solv
e(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,XT) -
member(Y,T). comes_before(X,Y,_T) -
comes_before(X,Y,T). permutation(,). permutat
ion(HT,Perm) - permutation(T,P), select(H,
Perm, P).

• 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
GenerateTest vs. TestGenerate

• if the solution space is large, GenerateTest 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 TestGenerate 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 TestGenerate

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.
solv
e(X) - constraints(X),
candidate(X). candidate(Candidate) -
permutation(Candidate,bob,karen,dan,sue). const
raints(Candidate) - Candidate
P1,P2,P3,P4, comes_before(karen,bob,Candidat
e), P1 dan, member(sue,P1,P2,P3).
Utilities comes_before(X,Y,
XT) - member(Y,T). comes_before(X,Y,_T) -
comes_before(X,Y,T). permutation(,). permutat
ion(HT,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 TestGenerate 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 TestGenerate
  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
• ? 666 216 possible arrangements

9
Competition problem TestGenerate

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 "NameNationalitySport".

- op(100,xfy,''). solve(X)
- constraints(X), candidate(X). candidate(P1
N1S1, P2N2S2, P3N3S3)
- permutation(P1,P2,P3,michael,richard,simon
), permutation(N1,N2,N3,american,australian,is
raeli), permutation(S1,S2,S3,basketball,crick
et,tennis). constraints(Candidate)
- Candidate X1,X2,X3, member(michael_ba
sketball,Candidate), comes_before(michael__,_
american_,Candidate), member(simonisraeli
_,Candidate), comes_before(simon__,__ten
nis,Candidate), X1 __cricket.
Utilities permutation(,). permut
ation(HT,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 TestGenerate solution
?- solve(X). X simonisraelicricket,
michaelaustralianbasketball,
richardamericantennis No ?-
time(solve(X)). 97 inferences in 0.00 seconds
(Infinite Lips) X simonisraelicricket,
michaelaustralianbasketball,
richardamericantennis Yes ?-
time(doReps(solve(X), 1000)). 100,997
inferences in 0.11 seconds (918155 Lips) X
simonisraelicricket, michaelaustralianbas
ketball, richardamericantennis Yes
code finds the unique solution GenerateTest
would require 1,191 inferences GenerateTest
would take 1.54 seconds for 1000 repetitions