Constraint Logic Programming

1
Constraint Logic Programming

• t.k.prasad_at_wright.edu
• http//www.knoesis.org/tkprasad/

2
Ordinary Logic Programming

• ?- p(a,X) p(Y,b).
• Ya Xb
• ?- p(X,X) p(a,b).
• fails
• Unification solves equality constraints (under
  unique name hypothesis)

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

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

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

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)

7
Introducing the Theory of Arithmetic

• fib(0,1).
• fib(1,1).
• fib(N, X1 X2) - N gt 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 lt X, X lt 25.
  
• N 6. X 13.
• N 7. X 21.

8
Solving Simultaneous Equations

• ?- X Y 12, 2X 4Y 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))
• Infinite solution Set of constraints

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

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

11
CLP Linear Programming

• light_meal(A,M,D) - appetizer(A,I),
• main_course(M,J),
  dessert(D,K),
• I gt 0, J gt 0, K gt 0,
• I J K lt 12.
• appetizer(soup,1).
• appetizer(nachos,6).
• main_course(sphagetti,3).
• main_course(alfredo,7).
• dessert(fruit,2).
• dessert(ice_cream,6).

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

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).

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

15
(contd)

• 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

16
(contd)

• 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

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

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)).

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(180,5,225,0,225180 G).
• G 19640

20
CLP Limitations

• ?- (Cows Pigs Sheeps) 100,
• (10Cows 3Pigs Sheeps/2) 100,
• Cows gt 1, Pigs gt 1, Sheeps gt 1.
• Pigs -1.35714Sheeps 128.571
• Cows 0.357143Sheeps - 28.5714
• Sheeps lt 94
• Lots of constraints
• CLP(R) is unable to determine the unique
  solution
• Cows 5, Pigs 1, Sheeps 94