1. cs774 (Prasad)L12CLP1 Constraint Logic Programming t.k.prasad@wright.edu http://www.knoesis.org/tkprasad/

2. Ordinary Logic Programming ?- p(a,X) = p(Y,b). Y=a X=b ?- p(X,X) = p(a,b). *fails* Unification solves equality constraints (under unique name hypothesis) cs774 (Prasad)L12CLP2

3. Constraint Logic Programming ?- X + 2 = 5. *fails in ordinary LP* In CLP, succeeds with: X = 3 Generalizes to constraint satisfaction problem, interpreting “+” as the arithmetic addition operator cs774 (Prasad)L12CLP3

4. Constraint Satisfaction Problem Given a set of variables, their types (domains of values and applicable operations), and constraints on them, determine assignment of values to variables so that the constraints are satisfied. Benefit: Embodiment of Declarative Programming Challenge: Designing tractable constraint languages cs774 (Prasad)L12CLP4

5. Applications Scheduling and Resource Management in production and transportation teachers and courses to classes rooms machines to jobs crews to planes Linear and Non-linear programming problems Mortgage Calculations cs774 (Prasad)L12CLP5

6. CLP : Motivation fib(0,1). fib(1,1). fib(N,X) :- N1 is N-1, N2 is N-2, fib(N1,X1), fib(N2,X2), X is X1 + X2. ?- fib(4,5). Succeeds. ?- fib(X,5). Fails. Treatment of arithmetic expression looks unnatural, as it does not smoothly integrate with the logic programming paradigm (e.g., invertibility destroyed) cs774 (Prasad)L12CLP6

7. Introducing the Theory of Arithmetic fib(0,1). fib(1,1). fib(N, X1 + X2) :- N > 1, fib(N - 1,X1), fib(N - 2,X2). Interpret arithmetic operators in the domain of reals –improves readability –improves invertibility ?- fib(N, 13). N = 6. ?- fib(N, X), 10 =< X, X =< 25. N = 6. X = 13. N = 7. X = 21. cs774 (Prasad)L12CLP7

8. Solving Simultaneous Equations ?- X + Y = 12, 2*X + 4*Y = 34. X = 7. Y = 5. Complex Multiplication zmul(c(R1,I1),c(R2,I2),c(R3,I3)) :- R3 = R1 * R2 - I1 * I2, I3 = R1 * I2 + R2 * I1. ?- zmul(c(2,2),Ans,c(0,16)) Unique solution: Ans = C(4,4) ?- zmul(A,B,c(0,16)) *Set of constraints* Infinite solution: *Set of constraints* cs774 (Prasad)L12CLP8

9. Prolog vs CLP(R) ?- 5 + 2 = X + Y. X = 5 Y = 2 ?- 5 + 2 = X + 3. *fail* Equality constraints over terms ?- 5 + 2 = X + Y. X = 7 - Y ?- 5 + 2 = X + 3. X = 4 General constraints over arithmetic expressions; Equality constraints over terms cs774 (Prasad)L12CLP9

10. Prolog vs CLP(R) Uninterpreted function symbols Unification algorithm Backtrack when terms fail to unify Arithmetic operators + uninterpreted term trees General constraint solvers – E.g., Simplex algorithm for linear constraints Backtrack when constraints violated cs774 (Prasad)L12CLP10

11. CLP : Linear Programming light_meal(A,M,D) :- appetizer(A,I), main_course(M,J), dessert(D,K), I >= 0, J >= 0, K >= 0, I + J + K <= 12. appetizer(soup,1). appetizer(nachos,6). main_course(sphagetti,3). main_course(alfredo,7). dessert(fruit,2). dessert(ice_cream,6). cs774 (Prasad)L12CLP11

12. ?- light_meal(App,Main,Dess). Dess = fruit Main = sphagetti App = soup *** Retry? y Dess = ice_cream Main = sphagetti App = soup *** Retry? y Dess = fruit Main = alfredo App = soup *** Retry? y Dess = fruit Main = sphagetti App = nachos cs774 (Prasad)L12CLP12

13. CLP : Mortgage Payment Calculator %mortgage(Principal,Time_Months, Interest_Rate,Monthly_Payment, Balance) mortgage(P,0,_,_,P). mortgage(P,1,_,_,B) :- B = P * (1 + (I / 1200)) - MP. mortgage(P, T, I, MP, B) :- mortgage( (P * (1 + I / 1200)) - MP, T – 1, I, MP, B). cs774 (Prasad)L12CLP13

14. CLP : Mortgage Payment Queries %mortgage(Principal,Time_Months, Interest_Rate,Monthly_Payment, Balance) %Customer: What will be the monthly payment? ?- mortgage(148000,180,8,M,0). M = 1414.37 %Lender: How much loan does one qualify for, given monthly payment limit? ?- mortgage(P,180,8,1200,0). P = 125569 cs774 (Prasad)L12CLP14

15. (cont’d) %mortgage(Principal,Time_Months, Interest_Rate,Monthly_Payment, Balance) %Customer/Lender: What is the remaining balance? ?- mortgage(148000,180,8.375,1124.91,B). B = 115088 %Customer: How long will it take to clear the debt? ?- mortgage(148000,L,8.5,1400,0). L = 195.731 cs774 (Prasad)L12CLP15

16. (cont’d) %mortgage(Principal,Time_Months, Interest_Rate,Monthly_Payment, Balance) %Customer: What is the contribution to the principal in the first month (in a 15 yr loan at 8 or a 30 yr loan at 8.375)? %Customer: Building equity. ?- mortgage(148000,1,8,1414,148000-CP). CP = 427 ?- mortgage(148000,1,8.375,1124,148000-CP). CP = 92 cs774 (Prasad)L12CLP16

17. Non-linear constraints : CLP(R) Fails Here %mortgage(Principal,Time_Months, Interest_Rate,Monthly_Payment, Balance) %Customer: What is the maximum interest rate for the given principal and monthly payment? %Customer: Affordability ?- mortgage(148000,180,I,1400,0). *lots of constraints output* cs774 (Prasad)L12CLP17

18. CLP : Annuity Calculator %annuity(Time_Months, Interest_Rate,Monthly_Payment, Initial_Principal, Total_Value) annuity(0,_,_,IP,IP). annuity(T, I, MP, IP, TV) :- annuity(T-1, I, MP, IP, V), TV = MP + (V * (1 + I / 1200)). cs774 (Prasad)L12CLP18

19. CLP : Annuity Queries %annuity(Time_Months, Interest_Rate,Monthly_Payment, Initial_Principal, Total_Value) %Customer: What will be the final investment value? ?- annuity(180,5,225,0,FV). FV = 60140 %Customer: What is the net gain? ?- annuity(180,5,225,0,225*180 + G). G = 19640 cs774 (Prasad)L12CLP19

20. CLP : Limitations ?- (Cows + Pigs + Sheeps) = 100, (10*Cows + 3*Pigs + Sheeps/2) = 100, Cows >= 1, Pigs >= 1, Sheeps >= 1. Pigs = -1.35714*Sheeps + 128.571 Cows = 0.357143*Sheeps - 28.5714 Sheeps <= 94 * Lots of constraints * CLP(R) is unable to determine the unique solution: Cows = 5, Pigs = 1, Sheeps = 94 cs774 (Prasad)L12CLP20