Introduction to
Constraint Programming
Kris Kuchcinski
kku@ida.liu.se
5/31/2016
1
Programming with Constraints
“Constraint programming represents one of the closest
approaches computer science has yet made to the Holy
Grail of programming: the user states the problem, the
computer solves it.”
Eugene C. Freuder
CONSTRAINTS, April 1997
5/31/2016
2
Programming with Constraints
“Were you to ask me which programming paradigm is likely
to gain most in commercial significance over the next 5
years I’d have to pick Constrained Logic Programming
(CLP), even though it’s perhaps currently one of the
least known and understood”
Dick Pountain
BYTE, February 1995
5/31/2016
3
Course organization
Date
16 Feb
23 Feb
2 Mar
Content
Organization, motivation, limitations, etc.
Introductory notions (Ch 1)
Basic operations on constraints (Ch 2)
(Satisfiability, entailment, projection, optimization)
9 Mar Finite domain constraints (Ch 3)
(Incompleteness, consistency techniques).
16 Mar Logic programming (*)
23 Mar Constraint logic programming (Ch 4, 5)
30 Mar Advanced logic programming (Ch 6, 7)
5/31/2016
Lect
KK
JM
JM
JM
UN
UN
UN
4
Course organization (cont’d)
Date Content
13 Apr Exercises, batch 1
20 Apr Modeling with finite domain constraints I
(Global constraints- diffn, cumulative,
precedence, element and search (Ch 8)
27 Apr Modeling with finite domain constraints II
(Global constraints and search (CHIP based)
4 May Reducing search
Redundant, disjunctive and global constraintscycle, among, sequence)
11 May Exercises, batch 2
18 May Advanced algorithms (consistency, SA, TS)
25 May Exercises, batch 3
5/31/2016
Lect
-KK
KK
UN
-??
-5
Course organization (cont’d)
 Three obligatory course assignments with the following
deadlines:
 25 March
 20 April
 18 May
 Course credit points 4
 Possibility for additional project and up to 6 credit points
5/31/2016
6
A simple definition
Constraint programming is the study of computational
systems based on constraints.
The idea of constraint programming is to solve problems
by exploring constraints which must be satisfied by the
solution.
5/31/2016
7
Different constraint systems
 Real/rational constraints- CLP(R), CLP(Q)
 CLP(R), Sicstus, CHIP
 Finite domains constraints- CLP(FD)
 Sicstus, CHIP
 Boolean constraints- CLP(B)
 Sicstus, CHIP
 Interval constraints- CLP(I)
 CLP(BNR), Numerica
5/31/2016
8
The programming paradigm
Logic
programming
Constraint
satisfaction/solving
CLP
Optimisation
5/31/2016
9
Logic programming
 Logic for problem description
 Declarative description style
 Problem description separated from its solving
 Unification
 Lists and recursion
 Standard backtracking
 Constraint programming does not need to use LP !!!
5/31/2016
10
Constraint satisfaction/solving
 Set of variables and constraints which limit the values
that can be assigned to the constraint variables
 Constraint propagation methods
 “Encapsulation” of specific knowledge from
mathematics, geometry, graph theory and operational
research
5/31/2016
11
Optimization
 Finding a solution which satisfies constraints and
minimizes/maximizes objective function
 Different types
 combinatorial optimization of discrete (finite domain) variables
 linear optimization for continuous variables
5/31/2016
12
Examples
 DC circuit using real constraints
 Digital circuits synthesis and verification using binary
constraints
 Allocation, scheduling and binding in high-level synthesis
using finite domain constraints
5/31/2016
13
Analysis of DC circuit
CLP(R)
I5
I2
I3
Is
I8
V
I1
I9
I4
R1
I6
I7
5/31/2016
Is - I1 - I2 - I8 = 0,
I1 + I7 - Is = 0,
I2 + I3 - I5 = 0,
I8 - I3 - I4 - I9 = 0,
I4 + I6 - I7 = 0,
I5 - I6 + I9 = 0,
R1*I1 = V,
2*I2 - 3*I3 - 8*I8 = 0,
3*I3 + 5*I5 -9*I9 = 0,
6*I6 - 4*I4 +9*I9 =0,
4*I4 - I1 + 7*I7 +8*I8 = 0
14
Analysis of DC circuit
CLP(R)
dc([Is, I1,
Is - I1 I1 + I7 I2 + I3 I8 - I3 I4 + I6 I5 - I6 +
(cont’d)
I2, I3, I4, I5, I6, I7, I8]) :I2 - I8 = 0,
Is = 0,
I5 = 0,
I4 - I9 = 0,
I7 = 0,
I9 = 0,
R1*I1 = V,
2*I2
3*I3
6*I6
4*I4
5/31/2016
+
-
3*I3
5*I5
4*I4
I1 +
- 8*I8
- 9*I9
+ 9*I9
7*I7 +
= 0,
= 0,
=0,
8*I8 = 0.
15
Analysis of DC circuit
CLP(R)
(cont’d)
V = 10, R1 = 1
V= 10, R1=3
V = 10, R1=?
I8
I7
I6
I5
I4
I3
I2
I1
IS
I8
I7
I6
I5
I4
I3
I2
I1
IS
I7
I6
I5
I4
I3
I2
I1
IS
10
=
=
=
=
=
=
=
=
=
0.259209
0.828299
0.296239
0.25726
0.53206
-0.31183
0.56909
10
10.8283
=
=
=
=
=
=
=
=
=
0.086403
0.2761
0.098746
0.085753
0.177353
-0.103943
0.189697
3.33333
3.60943
=
=
=
=
=
=
=
=
=
3.19549*I8
1.14286*I8
0.992481*I8
2.05263*I8
-1.20301*I8
2.19549*I8
38.5789*I8
41.7744*I8
I1 * R1
*** Maybe
5/31/2016
16
Analysis of DC circuit
CLP(I)
dc([Is,I1,I2,I3,I4,I5,I6,I7,I8])
Is>= -100, Is<=100, I1>= -100,
I2>= -100, I2<=100, I3>= -100,
I4>= -100, I4<=100, I5>= -100,
I6>= -100, I6<=100, I7>= -100,
I8>= -100, I8<=100, I9>= -100,
:I1<=100,
I3<=100,
I5<=100,
I7<=100,
I9<=100,
Is-I1-I2-I8=:=0,
I2+I3-I5=:=0,
I4+I6-I7=:=0,
I1+I7-Is=:=0,
I8-I3-I4-I9=:=0,
I5-I6+I9=:=0,
R>=1, R<=3,
R*I1=:=10,
2*I2-3*I3-8*I8=:=0,
6*I6-4*I4+9*I9=:=0,
3*I3+5*I5-9*I9=:=0,
4*I4-I1+7*I7+8*I8=:=0.
5/31/2016
17
Analysis of DC circuit
CLP(I)
(cont’d)
?- dc([Is,I1,I2,I3,I4,I5,I6,I7,I8]).
Is
I1
I2
I3
I4
I5
I6
I7
I8
in
in
in
in
in
in
in
in
in
5/31/2016
(3.6094328590907638,10.8282985772764136)
(3.3333333333333330,10.0000000000000000]
(0.1896966153445871,0.5690898460339618)
(-0.3118300526213467,-0.1039433508737487)
(0.1773533424283491,0.5320600272851572)
(0.0857532644708472,0.2572597934126064)
(0.0987461833300698,0.2962385499902706)
(0.2760995257584099,0.8282985772754369)
(0.0864029104138029,0.2592087312414950)
18
Digital Circuits- full adder
CLP(B)
A
0
0
0
0
1
1
1
1
B
0
0
1
1
0
0
1
1
Cin
0
1
0
1
0
1
0
1
Sum Cout
0
0
1
0
1
0
0
1
1
0
0
1
0
1
1
1
adder(_, A, B, C, SUM, CARRY) :sat(SUM =:= card([1,3], [A, B, C])),
sat(CARRY =:= card([2,3], [A, B, C])).
5/31/2016
19
Digital Circuits- full adder
CLP(B)
(cont’d)
| ?- adder(_, a, b, c, SUM, CARRY).
sat(SUM=:=a#b#c),
sat(CARRY=:=b*a#c*a#c*b).
5/31/2016
20
Digital Circuits- full adder
CLP(B)
(cont’d)
 N-transistor model
ntrans(_, B, X, Y) :sat((B * (X=:=Y)) + (B=:=0)).
 P-transistor model
ptrans(_, B, X, Y) :sat((~B * (X=:=Y)) + (B=:=1)).
X
X
B
B
Y
5/31/2016
Y
21
Digital Circuits- full adder
CLP(B)
(cont’d)
full_adder(_, A, B, C, SUM, CARRY) :carry_part(_, A, B, C, NCA, CARRY),
sum_part(_, A, B, C, NCA, SUM).
sum_part(_, A, B, C, NCA, SUM) :ptrans(_, NCA, T1, 1), ptrans(_, C, 1, T5), ptrans(_, B, T1, T5),
ptrans(_, A, T1, T2), ptrans(_, NCA, T5, T2), ptrans(_, T2, 1, SUM),
ntrans(_, A, T2, T3), ntrans(_, NCA, T2, T6), ntrans(_, T2, SUM, 0),
ntrans(_, B, T3, T6), ntrans(_, NCA, T3, 0), ntrans(_, C, T6, 0).
carry_part(_, A, B, C, NCA, CA) :ptrans(_, A, T1, 1), ptrans(_, B, T1, 1), ptrans(_, A, T2, 1),
ptrans(_, C, T1, NCA), ptrans(_, B, T2, NCA), ptrans(_, NCA, 1, CA),
ntrans(_, C, NCA, T3), ntrans(_, B, NCA, T4), ntrans(_, NCA, CA, 0),
ntrans(_, A, T3, 0), ntrans(_, B, T3, 0), ntrans(_, A, T4, 0).
5/31/2016
22
Digital Circuits- full adder
CLP(B)
(cont’d)
| ?- full_adder(_, a, b, c, SUM, CARRY).
sat(SUM=:=a#b#c),
sat(CARRY=:=b*a#c*a#c*b).
5/31/2016
23
High-level synthesis
CLP(FD)
3
x u dx 3
*
*
y u dx x
*
*
*
+
*
y
dx
dx
x u dx 3
+
c
a
x1
*
u
y1
dx
*
*
<
u
-
y u dx x
a
x1
+
3
dx
-
<
*
c
*
y *
-
+
u1
u1
data-flow graph
5/31/2016
y1
scheduled
data-flow graph
24
High-level synthesis
CLP(FD)
Op1
(cont’d)
Op2
T1 + D1 #=< T3,
T2 + D2 #=< T3,
T3 + D3 #=< T4,
Op4
Op3
T4 :: 0..50,
D4 :: 3..4,
R4 :: 0..1
5/31/2016
(T1 + D1 #=< T2 
T2 + D2 #=< T1 
R1 #\= R2).
25
High-level synthesis
CLP(FD)
(cont’d)
Scheduled design
5/31/2016
26
Methods for constraint solving
 CLP(R)
 Gauss-Jordan elimination
 simplex
 CLP(I)
 interval narrowing, box consistency,
 Gauss-Seidel elimination, interval Newton method,
 CLP(B)
 for example, operations on BDD’s
 CLP(FD)
 arc, node and path consistency methods
 constraint propagation (forward checking, look-ahead),
 branch-and-bound
5/31/2016
27
Constraint programming
limitations
 Many addressed problems in the area of FD constraints
are NP-complete (combinatorial problems, such as
scheduling, traveling salesman)
 search for (optimal) solution has exponential complexity in general case
 constraints and their propagation methods try to reduce the search
space
 possible heuristic search methods
 Boolean satisfaction problem is NP-complete
 Linear programming has polynomial time solving
algorithms (but in many practical cases it has too long
execution times)
5/31/2016
28
An Example
5/31/2016
29
An Example (cont’d)
5/31/2016
30
Web resources
 Constraints archive
http://www.cs.unh.edu/ccc/archive
 Guide to constraints programming
http://kti.ms.mff.cuni.cz/~bartak/constraints
 Sicstus manual
http://www.sics.se/isl/sicstus/sicstus_toc.html
 CHIP documentation
file:/sw/chip-5.1/chip.htm
 Prolog systems at IDA
 http://www.ida.liu.se/labs/logpro/lpsw/
5/31/2016
31