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Constraint Logic Programming (Ch.14)

Constraint Logic Programming (Ch.14). Constraint: a requirement for, or limitation on, a variable Constraint satisfaction: finding combinations of variables that satisfy a set of constraints C onstraint programming: algorithms that implement constraint satisfaction

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Constraint Logic Programming (Ch.14)

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  1. Constraint Logic Programming (Ch.14) COSC 2P93 Prolog: CLP • Constraint: a requirement for, or limitation on, a variable • Constraint satisfaction: finding combinations of variables that satisfy a set of constraints • Constraint programming: algorithms that implement constraint satisfaction • Constraint Logic Programming (CLP): merging CP with LP • A constraint satisfaction problem: • set of variables • domains for the variables (float, integer, ...) • constraints the variables have to satisfy • Problem: find an assignment to the variables that satisfies constraints • Constraint solver: an algorithm which, given the above, tries to efficiently find a solution • CLP is implemented by adding a constraint solver to Prolog

  2. CLP COSC 2P93 Prolog: CLP • Types of CLP: • clpr:variables take values over reals (decimal points) • clpq: rationals(fractions) • clpfd: integers, finite domains (sets) and others • clpb: booleans • Good for solving problems involving: • scheduling • linear equations and optimization • logistics • combinatorial problems (puzzles) • To run with SICStus: ?- use_module(library(clpq)).

  3. Example 1: arithmetic COSC 2P93 Prolog: CLP Standard Prolog... ?- 1+X = 5. % unification no ?- 1+X is 5. % Prolog’s “is” no CLPQ... ?- use_module(library(clpq)). % rationals ?- {1+X = 5}. % CLP: enclose constraint in { }’s X = 4

  4. Example 2: Arith COSC 2P93 Prolog: CLP Standard Prolog... convert(Cent, Fahren) :- % v1: Faren to Cent only Cent is (Fahren - 32)*5/9. convert2(Cent, Fahren) :-% v2: F to C, or C to F, but must use ‘var’ var(Cent), Cent is (Fahren - 32)*5/9. convert2(Cent, Fahren) :- var(Fahren), Fahren is 32 + (Cent * 9/5).

  5. Ex 2 (cont) COSC 2P93 Prolog: CLP convert3(Cent, Fahren) :- % v3: uses clpr (reals) {Cent = (Fahren - 32)*5/9}. ?- convert3(35, F). F = 95 ?- convert3(C, 95). C = 35 ?- convert3(C, F). ?- convert3(C,F). % using rationals {F = 32.0 + 1.8*C} {F=32+9/5*C}

  6. Example: integers and Fibonacci COSC 2P93 Prolog: CLP % Regular Prolog... fib(0, 1). fib(1, 1). fib(N, F) :- N > 1, % *** N1 is N-1, fib(N1, F1), N2 is N-2, fib(N2, F2), F is F1+F2. ?- fib(6, F). F = 13 ?- fib(N, 13). % program assumes N has value... see *** ERROR

  7. Example: Integers and Fibonacci COSC 2P93 Prolog: CLP % CLP... fib(N, F) :- {N=0, F=1}. fib(N, F) :- {N=1, F=1}. fib(N, F) :- { N>1, F=F1+F2, N1 = N-1, N2 = N-2 }, fib(N1, F1), fib(N2, F2). ?- fib(N, 13). N = 6 ?- fib(N, 4). INF LOOP -> stack overflow (no fibonacci number 4)

  8. Fibonacci % CLP... fib(N, F) :- {N=0, F=1}. fib(N, F) :- {N=1, F=1}. fib(N, F) :- {N>1, F=F1+F2, N1 = N-1, N2 = N-2, F1 >= N1, % new constraints F2 >= N2}, fib(N1, F1), fib(N2, F2). COSC 2P93 Prolog: CLP • Infinite loop happens because constraint solver doesn’t know that there is no fib number = 4; keeps recursing, finding larger and larger possibilities • Solution: add more constraints • Fib(N) >= N • then it won’t search for any N > F

  9. Example: money puzzle! COSC 2P93 Prolog: CLP • Module clpfd (finite domains) uses its own operators, utilities • Note: no { }’s like clpq, clpr #= is constraint ‘=‘ (equality) all_different/1: indicates all variables in argument must differ domain/3: indicates integer domain of variables labeling/2: does the actual inference (backtracking, etc.)

  10. Money puzzle COSC 2P93 Prolog: CLP ?- use_module(library(clpfd)). % step 1: declare domains of the variables % step 2: post the problem constraints % step 3: money([S,E,N,D,M,O,R,Y], Type) :- domain([S,E,N,D,M,O,R,Y], 0, 9), % step 1 S#>0, M#>0, all_different([S,E,N,D,M,O,R,Y]), % step 2a sum(S,E,N,D,M,O,R,Y), % step 2b labeling(Type, [S,E,N,D,M,O,R,Y]). % step 3 sum(S,E,N,D,M,O,R,Y) :- 1000*S + 100*E + 10*N + D + 1000*M + 100*O + 10*R + E #= 10000*M + 1000*O + 100*N + 10*E + Y.

  11. More clpfd features COSC 2P93 Prolog: CLP Expr1 #= Expr2 : Expr1 equals Expr Expr1 #> Expr2 : Expr1 is strictly larger than Expr2 P #\/ Q : True iff either P or Q ?- X #> 3. X in 4..sup. ?- X #\= 20. X in inf..19\/21..sup. ?- X*X #= 144. X in -12\/12. ... And many more!

  12. Conclusion COSC 2P93 Prolog: CLP • CLP is an advanced and powerful extension of logic programming • gives Prolog mathematical intelligence about arithmetic, sets, numbers • Prolog’s search still used; constraints help guide search, and are used to determine values and ranges • Many nontrivial problems are solvable with CLP, which would be difficult for plain Prolog.

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