Download Artificial Intelligence

Artificial Intelligence
Constraint
Programming 1
Ian Gent
[email protected]
Artificial Intelligence
Part I :
Constraint
Programming 1
Constraint Satisfaction Problems
Constraint Satisfaction Problems
 CSP = Constraint Satisfaction Problems
 AI exams, CSP =/= Communicating Sequential Processes
 A CSP consists of:
 a set of variables, X
 for each variable xi in X, a domain Di
 Di is a finite set of possible values
 a set of constraints restricting tuples of values
 if only pairs of values, it’s a binary CSP
 A solution is an assignment of a value in Di to each
variable xi such that every constraint satisfied
3
Who Cares?
 Many problems can be represented as CSP’s
 e.g. scheduling, timetabling, graph coloring, puzzles,
supply chain management, parcel routing, party arranging
…
 Many Constraint Programming toolkits available




CHIP
ILOG Solver [expensive commercially]
Eclipse [free academically]
Mozart [available as rpm for Linux]
4
Colouring as CSP
 Can we colour all 4 nodes
with 3 colours so that no
two connected nodes the
same colour?
 Variable for each node
 All Di = { red, green, blue}
 Constraint for each edge
 all constraints of the form
 xi  xj
 Solution gives a colouring
 It’s a binary CSP
5
SAT as a CSP
 Variable in CSP for each variable/letter in SAT
 Each domain Di = {true, false}
 Constraint corresponds to each clause
 disallows unique tuple which falsifies clause
 e.g. (not A) or (B) or (not C)
  not < A = true, B = false, C = true >
 Not binary CSP unless all clauses 2-clauses
6
N-Queens as a CSP
 Chessboard puzzle
 e.g. when n = 8…
 place 8 queens on a 8x8 chessboard so that no two attack
each other
 Variable xi for each row i of the board
 Domain = {1, 2, 3 … , n} for position in row
 Constraints are:
 xi  xj
 xi - xj  i-j
 xj - xi  i - j
queens not in same column
queens not in same SE diagonal
queens not in SW diagonal
7
Constraint Satisfaction Problems
 CSP = Constraint Satisfaction Problems
 AI exams, CSP =/= Communicating Sequential Processes
 A CSP consists of:
 a set of variables, X
 for each variable xi in X, a domain Di
 Di is a finite set of possible values
 a set of constraints restricting tuples of values
 if only pairs of values, it’s a binary CSP
 A solution is an assignment of a value in Di to each
variable xi such that every constraint satisfied
8
Formal Definition of Constraints
 A constraint Cijk… involving variables xi, xj, xk …
 is any subset of combinations of values from Di, Dj, Dk …
 I.e. Cijk...  Di x Dj x Dk …
 indicating the allowed set of values
 Most constraint programming languages/toolkits
allow a number of ways to write constraints:
 e.g. if D1 = D2 = {1,2,3} …
 { (1,2), (1,3), (2,1), (2,3), (3,1), (3,2) }
 x1  x2
 CtNeq(x1,x2)
 I’ll use whatever notation seems right at the time
9
More Complex Constraints
 Constraints don’t need to be simple

 +
 =
D
G
R
O
E
O
N
R
B
A
A
E
L
L
R
D
D
T
 Cryptarithmetic: all letters different and sum correct
 Variables are D, O, N, A, L, G, E, R, B, T
 Domains:
 {0,1,2,3, … , 9} for O, N, A L,E,R,B,T
 {1,2,3, … , 9} for D, G
 How do we express it?
10
Donald + Gerald = Robert
 We can write one long constraint for the sum:
 100000*D + 10000*O + 1000*N + 100*A+ 10*L + D
+ 100000*G + 10000*E + 1000*R + 100*A+ 10*L + D
= 100000*R + 10000*O + 1000*B + 100*E+ 10*R + T
 But what about the difference between variables?
 Could write D =/= O, D=/=N, … B =/= T
 Or express it as a single constraint on all variables
 AllDifferent(D,O,N,A,L,G,E,R,B,T)
 These two constraints
 express the problem precisely
 both involve all the 10 variables in the problem
11
Search in Constraints
 The basics of search are (guess what) the same as
usual
 Depth first search is the most commonly used
 What toolkits (Solver, Mozart … ) add is
 propagation during search done automatically
 In particular they offer efficient implementations of
propagation algorithms like
 Forward checking
 Maintaining Arc consistency
12
Forward Checking
 The simplest good propagation is Forward Checking
 The idea is very simple
 If you set a variable and it is inconsistent with some other
variable, backtrack!
 To do this we need to keep up to date the current
state of each variable
 Add a data structure to do this, the current domain CD
 Initially, CDi = Di
 When we set variable xj = v
 remove xi = u from CDi if some constraint is not
consistent with both xj = v, xi= u
13
Forward Checking
 For implementation, we have to cope with undoing
the effects of forward checking after backtracking
 One way is to store CDi on the stack at each depth in
search, so it can be restored
 expensive on space
 very easy in languages which make copies
automatically, e.g. Lisp, Prolog
 Another is to store only the changes to CDi
 then undo destructive changes to data structures on
backtracking
 usually faster but can be more fiddly to implement
14
Forward Checking, example





Variables x, y
Dx = Dy = {1,2,3,4,5}
Constraint x < y - 1
Initially CDx = CDy = {1,2,3,4,5}
If we set x = 2, then:
 the only possible values are y = 4, y = 5
 so set CDy = {4,5}
 If we set x = 4, then
 no possible values of y remain, I.e. CDy = { }
 retract choice of x = 4 and backtrack
15
Heuristics
 As usual we need efficient variable ordering
heuristics
 One is very important, like unit propagation in SAT:
 If any CD is of size 1
 we must set that variable to the remaining value
 More generally, a common heuristic is “minimum
remaining value”
 I.e. choose variable with smallest size CD
 motivated by most constrained first, or also some
theoretical considerations.
16
Next time




Arc Consistency
Special kinds of constraints, like all different
Formulation of constraint problems
How to organise a party
17