620 lines
14 KiB
Plaintext
620 lines
14 KiB
Plaintext
% examples-thom1.math --------------------------------------------------------
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% thom fruehwirth 1991-93
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% examples for *math* constraint handlers compiled from various sources
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% results shown are from old obsolete versions
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:- op(1200,fx,'example').
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:- dynamic (example)/1.
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% Equation Examples
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example X1+X2=<4, 2*X1+3*X2>=18, X1>=0,X2>=0.
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%NO
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example Y1>=0,Y2>=0,X1>=0,X2>=0,Y1=:=X1-X2,Y2=:=X2-X1.
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%YES Y1 = 0 , Y2 = 0,
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% X2 =:= slack(_2),
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% X1 =:= slack(_2),
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% slack(_1) =:= slack(_2)
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example 3*X+2*Y-4*(3+Z)=:=2*(X-3)+(Y+Z)*7,
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2*(X+Y+Z)=:=3*(X-Y-Z),
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5*(X+Y)-7*X-Z=:=(2+1+X)*6.
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%YES Y = 0.657143 , Z = -1 , X = -1.71429
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example 3*X+2*Y-4*(3+Z)=:=2*(X-3)+(Y+Z)*7,
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2*(X+Y+Z)=:=3*(X-Y-Z).
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%YES Z = -1,
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% X =:= 5 * -1 + 5 * Y
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example 3*X+2*Y-4*(3+Z)=:=2*(X-3)+(Y+Z)*7,
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2*(X+Y+Z)=:=3*(X-Y-Z),
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5*(X+Y)-7*X-Z=:=(2+1+X)*6 ,
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2*(X-Y+Z)=:=Y+X-7.
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%Failure, test = 0.0799999
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%NO
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example 3*X+2*Y-4*(3+Z)=:=2*(X-3)+(Y+Z)*7,
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2*(X+Y+Z)=:=3*(X-Y-Z),
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5*(X+Y)-7*X-Z >= (2+1+X)*6 ,
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2*(X-Y+Z)=:=Y+X-7.
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%Success, test = 0.0666666
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%YES Z = -1 , Y = 0 , X = -5
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example X>=Y,Y>=X.
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%Success, test = 0.0216667
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%YES ,
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% X =:= slack(0) + Y
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example X>=Y+1,Y>=X+1.
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%Failure, test = 0.0133333
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%NO
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example X>=Y+1,Y>=X-1.
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example X>=Y+1,Y>=X-2.
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%Success, test = 0.0199999
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%YES ,
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% X =:= slack(0) + Y + 1
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example X*Y=:=6,X+Y=:=5,X-Y=:=1.
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%Success, test = 0.0200001
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%YES X = 3.0 , Y = 2.0
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example X*Y=:=6,X+Y=:=5,X>=Y.
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%Success, test = 0.04
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%YES ,
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% 6 =:= X * Y,
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% X =:= 0.5 * slack(_1) + 2.5,
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% Y =:= -0.5 * slack(_1) + 2.5
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example X>=Y+Z,Z>=X+1,Y>=Z.
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%Success, test = 0.0883333
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%YES ,
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% Z =:= -(slack(_1)) - slack(_3) - slack(_2) - 1,
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% X =:= -(slack(_1)) - 2 * slack(_3) - slack(_2) - 2,
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% Y =:= -(slack(_3)) - slack(_2) - 1
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% men_and_horses
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mh(Men,Horses,Heads,Legs):-
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Men >= 0, Horses >= 0,
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Heads =:= Men + Horses,
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Legs =:= 2*Men + 4*Horses.
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% fibonacci
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% loops for first argument var !
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% works if multi-headed rules are evaluated eagerly as soon as a var is bound
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fib(N, X):-
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N =:= 0, X =:= 1.
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fib(N, X):-
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N =:= 1, X =:= 1.
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fib(N, X):-
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N >= 2, X >= N,
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X =:= X1 + X2,
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%N1 =:= N - 1,
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%N2 =:= N - 2,
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fib(N-1, X1),
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fib(N-2, X2).
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% prove of paralellogram in arbitrary polygon with 4 corners
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:- op(31,xfx,#).
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mid(AX#AY,BX#BY,CX#CY):-
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AX+CX =:= 2*BX,
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AY+CY =:= 2*BY.
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para(AX#AY,BX#BY,CX#CY,DX#DY):- % nonlinear part
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(AX-BX)*(CY-DY) =:= (AY-BY)*(CX-DX).
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pp(P0,P1,P2,P3,[P4,P5,P6,P7]):-
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mid(P0,P4,P1),
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mid(P1,P5,P2),
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mid(P2,P6,P3),
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mid(P3,P7,P0),
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para(P4,P5,P7,P6),
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para(P4,P7,P5,P6).
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% for solution, 4 points must be given
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example pp(1#5,2#3,3#1,5#2,L).
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%Success, test = 0.025
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%YES L = [1.5 # 4.0, 2.5 # 2.0, 4.0 # 1.5, 3.0 # 3.5]
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example pp(A,B,C,D,[1.5 # 4.0, 2.5 # 2.0, 4.0 # 1.5, 3.0 # 3.5]).
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/*
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YES A = _2 # _4 , B = _1 # _3 , C = _5 # _7 , D = _6 # _8,
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_1 =:= _6 - 3.0,
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_2 =:= -(_6) + 6.0,
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_3 =:= _8 + 1.0,
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_4 =:= -(_8) + 7.0,
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_5 =:= -(_6) + 8.0,
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_7 =:= -(_8) + 3.0
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*/
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example L = [1.5 # 4.0, 2.5 # 2.0, 4.0 # 1.5, 3.0 # 3.5],pp(A,B,1#1,D,L).
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%Success, test = 0.0333328
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%YES A = -1 # 5.0 , B = 4.0 # 3.0 , D = 7.0 # 2.0 , L = [1.5 # 4.0, 2.5 # 2.0, 4.0 # 1.5, 3.0 # 3.5]
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% CLP(R) Version 1.0 - Example Programs
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% Standard mortgage relationship between:
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% P: Principal
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% T: Life of loan in months
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% I: Fixed (but compounded) monthly interest rate
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% B: Outstanding balance at the end
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% M: Monthly payment
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% doesn't run in CHIP because of nonlinear constraints ?
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mg(P, T, I, B, MP) :-
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T =:= 1,
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B =:= P + P*I - MP.
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mg(P, T, I, B, MP) :-
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T > 1,
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T1 =:= T - 1,
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P1 =:= P + P*I - MP,
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mg(P1, T1, I, B, MP).
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mg1(P, T, I, B, MP) :-
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T =:= 1,
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B =:= P + P*I - MP.
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mg1(P, T, I, B, MP) :-
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T > 1,
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mg1(P + P*I - MP, T-1, I, B, MP).
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% code in CLP9R) language and system ACM TPLS paper 1992
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mg2(P, T, I, B, MP) :-
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T>0,T=<1,
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Int =:= T*(P*I/1200),
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B =:= P + Int - (T*MP).
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mg2(P, T, I, B, MP) :-
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T > 1,
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Int =:= P*I/1200,
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mg2(P+Int-MP, T-1, I, B, MP).
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mg1(M):- mg(999999, 6, 0.01, 0, M). % 6 was 360
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mg2(P,B,M):- mg(P, 6, 0.01, B, M). % 6 was 720
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example mg(999999, 6, 0.01, 0, M).
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%YES M = 172548.2
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example mg(P, 6, 0.01, B, M).
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/*
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YES ,
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_1 =:= -0.990099 * B + 1.9901 * _5,
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M =:= -(B) + 1.01 * _5,
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P =:= -4.85343 * B + 5.85343 * _5,
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_2 =:= -2.94098 * B + 3.94098 * _5,
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_3 =:= -3.90197 * B + 4.90196 * _5,
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_4 =:= -1.97039 * B + 2.97039 * _5
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*/
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example B=0, P=999999,
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_1 =:= -0.990099 * B + 1.9901 * _5,
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M =:= -(B) + 1.01 * _5,
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P =:= -4.85343 * B + 5.85343 * _5,
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_2 =:= -2.94098 * B + 3.94098 * _5,
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_3 =:= -3.90197 * B + 4.90196 * _5,
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_4 =:= -1.97039 * B + 2.97039 * _5.
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%Success, test = 0.0199997
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%YES _1 = 339988.4 , M = 172548.2 , P = 999999 , _2 = 673276.4 , _3 = 837450.1 , _4 = 507461.0 , B = 0 , _5 = 170839.8
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example B=0, P=999999,
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M =:= -(B) + 1.01 * _5,
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P =:= -4.85343 * B + 5.85343 * _5.
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%Success, test = 0.0116669
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%YES M = 172548.2 , P = 999999 , B = 0 , _5 = 170839.8
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example mg(P, 6,I,B,M).
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/*
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YES ,
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_1 =:= P * I,
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M =:= _11 - B + _9,
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P =:= -(_3) + _4 + 2 * M - _1,
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_10 =:= _6 * I,
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_11 =:= _9 * I,
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_2 =:= 2 * _8 - 2 * _6 - _5 + 3 * _7 - _3,
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_3 =:= _2 * I,
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_4 =:= 2 * _10 - 2 * _9 - _8 + 3 * _6 - _5,
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_5 =:= _4 * I,
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_6 =:= _11 - B - _10 + 2 * _9,
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_7 =:= 2 * _11 - 2 * B - _10 + 3 * _9 - _8,
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_8 =:= _7 * I
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*/
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% CLP(R) Version 1.0 - Example Programs
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% (Slow implementation of) The classic cryptarithmetic puzzle:
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%
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% S E N D
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% + M O R E
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% ---------
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% M O N E Y
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% works, but very slow
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example sendmory([9, 5, 6, 7, 1, 0, 8, 2]).
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%Success, test = 0.00999908
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%YES (was no before because floats don't unify bit/1)
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sendmory([S, E, N, D, M, O, R, Y]) :-
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constraints([S, E, N, D, M, O, R, Y]),
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gen_diff_digits([S, E, N, D, M, O, R, Y]).
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constraints([S, E, N, D, M, O, R, Y]) :-
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% S >= 0, E >= 0, N >= 0, D >= 0, M >= 0, O >= 0, R >= 0, Y >= 0,
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% S =< 9, E =< 9, N =< 9, D =< 9, M =< 9, O =< 9, R =< 9, Y =< 9,
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% S >= 1,
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M >= 1,
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% C1 >= 0, C2 >= 0, C3 >= 0, C4 >= 0,
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% C1 =< 1, C2 =< 1, C3 =< 1, C4 =< 1,
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M = C1,
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S + M + C2 =:= O + 10 * C1,
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E + O + C3 =:= N + 10 * C2,
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N + R + C4 =:= E + 10 * C3,
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D + E =:= Y + 10 * C4,
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bit(C1), bit(C2), bit(C3), bit(C4).
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bit(0).
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bit(1).
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gen_diff_digits(L) :-
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gen_diff_digits(L, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9]).
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gen_diff_digits([], _).
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gen_diff_digits([H | T], L) :-
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delete(H, L, L2), gen_diff_digits(T, L2).
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% CLP(R) Version 1.0 - Example Programs
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% Algebraic combinations of options transactions
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% very slow if Stockprice not given, because of
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% backtracking caused by h/3, r/3
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% heaviside function
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h(X, Y, Z) :- Y < X, Z =:= 0.
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h(X, Y, Z) :- Y >= X, Z =:= 1.
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% ramp function
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r(X, Y, Z) :- Y < X , Z =:= 0.
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r(X, Y, Z) :- Y >= X, Z =:= Y - X.
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% option valuation
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% changed order of subgoals
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value(Type,Buy_or_Sell,S,C,P,I,X,B,Value) :-
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lookup_option(Type,S,C,P,I,X,B,B1,B2,H1,H2,R1,R2),
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check_param(S,C,P,I,X,B),
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get_sign(Buy_or_Sell,Sign),
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h(B1,S,T1),h(B2,S,T2),r(B1,S,T3),r(B2,S,T4),
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Value =:= Sign*(H1*T1 + H2*T2 + R1*T3 + R2*T4).
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% safety check
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check_param(S,C,P,I,X,B) :-
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S >= 0, C >= 0, P >= 0,
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I >= 0, X >= 0, B >= 0 .
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% Buy or sell are just opposite
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get_sign(buy,(-1)).
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get_sign(sell,1).
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% lookup option vector
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lookup_option(Type,S,C,P,I,X,B,B1,B2,H1,H2,R1,R2) :-
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table(Type,S,C,P,I,X,B,B1,B2,H1,H2,R1,R2).
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% table(Type,S,C,P,I,X,B,B11,B21,H11,H21,R11,R21),
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% B1 =:= B11, B2 =:= B21, H1 =:= H11, H2 =:= H21, R1 =:= R11, R2 =:= R21.
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% Table of values for B1,B2,H1,H2,R1,R2
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% generic format - lookup_table(Type,Pos_neg,S,C,P,I,X,B,B1,B2,H1,H2,R1,R2).
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% where K to R2 are obtained from the table
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% M is a multiplier which is -1 or 1 depending on whether one
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% is buying or selling the option
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table( stock, S, C, P, I, X, B, 0, 0, S*(1+I), 0, -1, 0).
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table( call, S, C, P, I, X, B, 0, X, C*(1+I), 0, 0, -1).
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table( put, S, C, P, I, X, B, 0, X, P*(1+I)-X, 0, 1, -1).
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table( bond, S, C, P, I, X, B, 0, 0, B*(1+I), 0, 0, 0).
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stocks1(Wealth, Stockprice) :-
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Wealth =:= Wealth1 + Wealth2,
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X = 99,
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P = 10, C = 10,
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I = 0,
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value(put, buy, Stockprice, _, P, I, X, _, Wealth1),
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value(call, buy, Stockprice, C, _, I, X, _, Wealth2).
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stocks2(Wealth, Stockprice) :-
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I = 0.1, P1 = 10, X1 = 20,
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value(put, sell, Stockprice, _, P1, I, X1, _, Wealth1),
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P2 = 18, X2 = 40,
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value(put, buy, Stockprice, _, P2, I, X2, _, Wealth2),
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C3 = 15, X3 = 60,
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value(call, buy, Stockprice, C3, _, I, X3, _, Wealth3),
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C4 = 10, X4 = 80,
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value(call, sell, Stockprice, C4, _, I, X4, _, Wealth4),
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Wealth =:= Wealth1 + Wealth2 + Wealth3 + Wealth4.
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example stocks1(W,10).
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/*
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Success, test = 0.075
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YES W = 69,
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_1 =:= slack(_2),
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_3 =:= slack(_4),
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_5 =:= slack(_6),
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_7 =:= slack(_8)
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*/
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example stocks2(W,10).
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/*
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YES W = 5.7,
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_1 =:= slack(_2),
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_11 =:= slack(_12),
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_13 =:= slack(_14),
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_15 =:= slack(_16),
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_3 =:= slack(_4),
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_5 =:= slack(_6),
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_7 =:= slack(_8),
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_9 =:= slack(_10)
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*/
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% Hoon Hong EXAMPLES
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% electric circuit (from clp(r))
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resistor(V,I,R):-
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V =:= I*R, R>0.
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par_circuit(V,I,R1,R2):-
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I=:=I1+I2,
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resistor(V,I1,R1),
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resistor(V,I2,R2).
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example par_circuit(A,B,2,2).
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%Success, test = 0.0566666
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%YES ,
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% _1 =:= 0.5 * A,
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% B =:= A,
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% _2 =:= 0.5 * A
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example par_circuit(2,A,2,B).
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%Success, test = 0.0300003
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%YES ,
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% nonzero(B - 0),
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% 2 =:= _1 * B,
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% A =:= _1 + 1,
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% B =:= slack(_2)
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% complex number arithmetic ((from clp(r))
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zmul(R1#I1,R2#I2,R3#I3):-
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R3 =:= R1*R2-I1*I2,
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I3 =:= R1*I2+R2*I1.
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example zmul(1#2,3#4,C).
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%Success, test = 0.00333405
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%YES C = -5 # 10
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example zmul(A,B,(-5)#10).
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/*
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Success, test = 0.0483337
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YES A = _1 # _4 , B = _3 # _2,
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_5 =:= _1 * _3,
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_5 =:= _6 - 5,
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_6 =:= _4 * _2,
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_7 =:= _1 * _2,
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_7 =:= -(_8) + 10,
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_8 =:= _3 * _4
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*/
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example zmul(1#2,B,(-5)#10).
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%Success, test = 0.0183334
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%YES B = 3.0 # 4.0
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% pythagorean numbers
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% loops, see fib for explaination
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% changed order of subgoals
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nat(X):- X=:=1.
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nat(X):- X>1,nat(Y),X=:=Y+1.
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% loops immediately
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pyth(X,Y,Z):-
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X*X+Y*Y=:=Z*Z,nat(X),nat(Y),nat(Z).
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/*
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%From lim@scorpio Fri Jan 31 17:09:44 1992
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minimize x1 - 2x2
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subject to
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x1 + x2 >= 2,
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-x1 + x2 >= 1,
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x2 <= 3.
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chr: X+Y>=2,Y-X>=1,Y=<3,X-2*Y=:=M,minimize(M),X=< -1.
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Y = 3
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M = -7
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X = -1
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% does not work
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chr: X+Y>=2,Y-X>=1,Y=<3,X-2*Y=:=M,minimize(M).
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X = X_m176
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Y = Y_m196
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M = M_m2048
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yes if
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eq0([slack(X_g7001_m242) * 1, slack(X_g7001_m470) * 1, slack(X_g7602_m1106) * 2], -3)
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eq0([X_m176 * 1, Y_m196 * -2, M_m2048 * -1], 0)
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eq0([Y_m196 * 1, M_m2048 * 1, slack(X_g7001_m470) * 1], 1)
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eq0([M_m2048 * -1, slack(X_g7001_m470) * -1, slack(X_g7602_m1106) * 1], -4)
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minimize(M_m2048)
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Delayed goals:
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X_g7001_m242 >= 0
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X_g7001_m470 >= 0
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X_g7602_m1106 >= 0
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minimize -3x1 + 4x2
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subject to
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x1 + x2 <= 4,
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2x1 + 3x2 >= 18.
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chr: X+Y =<4,2*X+3*Y>=18,M=:=4*Y-3*X,minimize(M).
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Y = 10
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X = -6
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M = 58
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chr: X+Y =<4,2*X+3*Y>=18,M=:=4*Y-3*X,M<58.
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no (more) solution.
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minimize -x1 + 2x2 -3x3
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subject to
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x1 + x2 + x3 = 6,
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-x1 + x2 + 2x3 = 4,
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2x2 + 3x3 = 10,
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x3 <= 2.
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chr: X+Y+Z=:=6,Y+2*Z-X=:=4,2*Y+3*Z=:=10,Z=<2,M=:=2*Y-3*Z-X,minimize(M).
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Y = 2
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Z = 2
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X = 2
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M = -4
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yes.
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chr: X+Y+Z=:=6,Y+2*Z-X=:=4,2*Y+3*Z=:=10,Z=<2,M=:=2*Y-3*Z-X,M< -4.
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no (more) solution.
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*/
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% from CACM May 91 34(5) p. 59
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% given solution is wrong because last inequation is wrong - replace 26 by 19
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%I1=3,I2=2,I3=3,I4=0,I5=4,I6=2,I7=5,I8=0,I9=3,I10=5,I11=(-2),I12=3,I13=4,I14=3,
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example I8+I7+I6+I5+I4+I3+I2+6=:=22, I9+I8+I7+I6+I5+I4+I3+I2+6=:=25,
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I1=:=3, I2>=2, I3>=3, I4+I3+I2+1>=4, I5+I4+1>=5,
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I6+I5+1>=7, I6>=2, I7>=5, I10+I9+1>=2, I11+I10+1>=4,
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I12+I11+2=<3, I12+1=<4, I12+I11+1>=2, I12>=3, I13>=4,
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I14>=3, I14+I13+I12+I11+4=<22, I14+I13+I12+I11+3=<25,
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I14+I13+I12+I11+I10+I9+7>=23, I14+I13+I12+I11+I10+6>=26.
|
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/*
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YES I1 = 3 , I9 = 3 , I12 = 3 , I11 = -2,
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I7 =:= slack(_1) + 5,
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I10 =:= slack(_14) + slack(_13) - 9,
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I13 =:= -(slack(_10)) - slack(_9) + 14,
|
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I14 =:= slack(_9) + 3,
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I2 =:= -(slack(_7)) + slack(_6) - slack(_5) + slack(_4) - slack(_3),
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I3 =:= slack(_3) + 3,
|
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I4 =:= slack(_7) - slack(_6) + slack(_5),
|
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I5 =:= -(slack(_7)) + slack(_6) + 4,
|
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I6 =:= slack(_7) + 2,
|
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I8 =:= -(slack(_1)) - slack(_6) - slack(_4) + 2,
|
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slack(_10) =:= slack(_13) - 4,
|
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slack(_11) =:= slack(_14) + slack(_13) - 7,
|
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slack(_12) =:= slack(_14) + slack(_13) - 14,
|
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slack(_14) =:= slack(_15) + 7,
|
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slack(_2) =:= -(slack(_7)) + slack(_6) - slack(_5) + slack(_4) - slack(_3) - 2,
|
|
slack(_8) =:= -(slack(_10)) - slack(_9) + 10
|
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*/
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/*
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|
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% Examples that take longer --------------------------------------------------
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example sendmory(L). % too slow maybe
|
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%YES L = [9, 5, 6, 7, 1, 0, 8, 2]
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example stocks1(69,S). % takes long!
|
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/*
|
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YES S = 10,
|
|
_1 =:= slack(_2),
|
|
_3 =:= slack(_4),
|
|
_5 =:= slack(_6),
|
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_7 =:= slack(_8)
|
|
;
|
|
YES S = 188,
|
|
_A =:= slack(_B),
|
|
_C =:= slack(_D),
|
|
_E =:= slack(_F),
|
|
_G =:= slack(_H)
|
|
*/
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%------------------------------------------------------------------------------
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|
|
chr: X+Y>=2,Y-X>=1,3>=Y.
|
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|
|
X = X_m208
|
|
Y = Y_m228
|
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|
|
yes if
|
|
eq0([X_m208 * 1, Y_m228 * -1, slack(X_g7001_m4754) * 1], 1)
|
|
eq0([Y_m228 * 1, slack(X_g7001_m13874) * 1], -3)
|
|
eq0([slack(X_g7001_m802) * 1, slack(X_g7001_m4754) * 1, slack(X_g7001_m13874) * 2], -3)
|
|
|
|
|
|
Delayed goals:
|
|
X_g7001_m802 >= 0
|
|
X_g7001_m4754 >= 0
|
|
X_g7001_m13874 >= 0
|
|
yes.
|
|
chr: X+2*Y=<3,-X-Y=<1.
|
|
|
|
X = X_m206
|
|
Y = Y_m226
|
|
|
|
yes if
|
|
eq0([X_m206 * -1, Y_m226 * -1, slack(X_g7602_m5512) * 1], -1)
|
|
eq0([Y_m226 * 1, slack(X_g7602_m808) * 1, slack(X_g7602_m5512) * 1], -4)
|
|
|
|
|
|
Delayed goals:
|
|
X_g7602_m808 >= 0
|
|
X_g7602_m5512 >= 0
|
|
yes.
|
|
chr: X+Y-Z=:=0,-Y+3*Z=:=0.
|
|
|
|
X = X_m222
|
|
Y = Y_m242
|
|
Z = Z_m262
|
|
|
|
yes if
|
|
eq0([X_m222 * 1, Y_m242 * 1, Z_m262 * -1], 0)
|
|
eq0([Y_m242 * -1, Z_m262 * 3], 0)
|
|
|
|
yes.
|
|
chr: 2*X-3*Y+4*Z=:=5,X+2*Y-Z=:=6,-3*X+Y+3*Z=:=1.
|
|
|
|
X = 2.5714285714285712
|
|
Y = 2.714285714285714
|
|
Z = 2
|
|
*/
|
|
|
|
% end of file examples-thom1.math ============================================
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