# Constraint Satisfaction Problems

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1 Constraint Satisfaction Problems Introduction to Artificial Intelligence CSE 150 Lecture 4 April 12, 2007

2 Last lecture A* search ptimality of A* IDA* Recursive Best First Search (briefly!) Beam Search (really briefly!)

3 This lecture General search applied to CSPs Backtracking Forward checking Heuristics for CSPs Reading Chapter 5 (Meaning, you should have read 1-4 by now!)

4 Constraint Satisfaction Problems

5 Constraint Satisfaction Problems (CSPs( CSPs) A state-space search problem where The state is defined by n variables V i (i=1,,n) The possible values for each variable are from a domain D i There are a set of constraints between the variable values The goal test checks that all variables have been assigned and no constraints are violated. We will restrict ourselves to examples with discrete, finite sets of values for each variable.

6 Sudoku Toy Examples N-Queens: Place N queens on an N N board such that none can take any other Cryptarithmetic problems: SEND + MRE = MNEY Map coloring (no two neighboring countries the same color) Propositional satisfiability: assign T or F to a set of Boolean variables so that a sentence is True (e.g., 3-SAT)

7 Example: 4-Queens as a CSP Place 4 queens on a 4 by 4 chess board such that no queen can take another Assume one queen in each column. Which row does each one go in? Variables: Q i (i is the column) Domains: D i = {1, 2, 3, 4} (the row) Constraints Q i Q j (cannot be in same row) Q i - Q j i - j (or same diagonal) Translate each constraint into set of allowable values for its variables E.g., values for (Q 1, Q 2 ) are (1,3) (1,4) (2,4) (3,1) (4,1) (4,2)

8 Constraint graph Binary CSP: each constraint relates at most two variables Constraint graph: nodes are variables, arcs show constraints Q 1 Q 2 Q 3 Q 4

9 Example: Cryptarithmetic Variables D, E, M, N,, R, S, Y Domains {0, 1, 2, 3,4, 5, 6, 7, 8, 9 } S E N D + M R E M N E Y Constraints M 0, S 0 (unary constraints) Y = D + E R Y = D + E 10. D E, D M, D N, etc. (ALL-DIFFERENT)

10 Example: Map coloring Color a map so that no adjacant countries have the same color Variables: Countries Domains: C i {Red, Blue, Green} Constraints: C 1 C 2, C 1 C 5, etc. C 3 C C 2 1 C5 C 6 C 4 Constraint graph:

11 Real-world CSPs Assignment problems e.g., who teaches what class Timetabling problems e.g., which class is offered when and where? Hardware configuration Spreadsheets Transportation scheduling Factory scheduling Planning bservations for the Hubble Space Telescope Note that many real-world problems involve real-valued variables as well as binary and discrete variables.

12 Applying standard search Let's start with the straightforward, dumb approach, and then fix it Initial state: all variables unassigned perators: assign one of value in domain to one of the unassigned variables. Goal test: all variables assigned, no constraints violated Notice that this is the same for all CSPs

13 Implementation CSP state keeps track of which variables have values so far Each variable has a domain and a current value Datatype CSP-STATE components UNASSIGNED: a list of variables not yet assigned ASSIGNED: a list of variables that have values Datatype CSP-VARIABLE components NAME, for I/ purposes DMAIN: a list of possible values VALUE, current value (if any) Constraints can be represented explicitly as sets of allowable values, or implicitly by a function that tests for satisfaction of the constraint

14 Depth-first search applied to 4-Queens Start: No variables assigned Level 1: Q1 =1 Q1 = 2 Q4=3 Q4=4 Unassigned: Q2-Q4 Un: Q2-Q4 Un: Q1-Q3 Un: Q1-Q3 Level 2: Q1=1, Q2=1 Q1=1, Q2=2 Un: Q3,Q4 Un: Q3,Q4 This algorithm is called generate-and-test, because each state is fully instantiated before it is tested.

15 Complexity of the Dumb Approach Max. depth of space = Depth of solution state: d = n (all vars assigned) Search algorithm to use: Depth-first Branching factor b = Σ D i (at top of tree) Time cost (b n ) How can we do better?

16 How to not be so Dumb! Note Branching factor b = Σ D i (at top of tree) But the assignment of row 1 to Q 1 and row 3 to Q 2 is equivalent first assigning row 3 to Q 2 and then row 1 to Q 1 - the order of assignment doesn t matter. Hence the idea of backtracking search Second: We generated partial assignments that violated the constraints! Should check as we generate!

17 Backtracking search Use depth-first search, but 1. Fix the order of assignment Branching factor, b = D i 2. Check for constraint violations after each assignment 3. Backtrack if constraints have been violated The constraint violation check can be implemented in two ways: 1. Modify Successors function to assign only values that are allowed, given the values already assigned 2. Check constraints are satisfied before expanding a state Backtracking search is the basic uninformed algorithm for CSPs Can solve n-queens for n 15

18 Heuristics for CSPs More intelligent decisions on: 1. which variable to assign next 2. which value to choose for each variable C 3 C C 2 1 C5 C 6 C 4 Given C 1 = RED, C 2 = GREEN, which variable to chose next? C 5 : it s the one with the minimum remaining values (MRV) Given C 1 = RED, C 2 = GREEN, what s a good color for C 3? C 3 = Green : least constraining value, I.e. Green doesn t constrain the choices for C5, but blue would Can solve n-queens for n 1,000.

19 What more can we do? Checking for constraint violations as we generate is good - but we can do more: Constraint propagation is the idea of propagating the constraints to variables we haven t checked yet. The simplest form is forward checking: 1. Whenever a variable is assigned, check all variables Y connected to by a constraint and delete from Y s domain any value that is inconsistent with the value chosen for. 2. As soon as the domain of any variable becomes empty, backtrack.

20 Forward checking Another way of saying it: Delete illegal values for unassigned variables - but only the ones connected to the most recent assignment Backtrack when any variable has no legal values Simplified map-coloring example C 1 C 2 C 3 C 4 C 5 RED BLUE GREEN C 2 C 5 C 3 C 1 C 4 Can solve n-queens for n 30.

21 Forward checking Another way of saying it: Delete illegal values for unassigned variables - but only the ones connected to the most recent assignment Backtrack when any variable has no legal values Simplified map-coloring example C 1 C 2 C 3 C 4 C 5 RED BLUE GREEN C 2 C 5 C 3 C 1 C 4 Can solve n-queens for n 30.

22 Forward checking Another way of saying it: Delete illegal values for unassigned variables - but only the ones connected to the most recent assignment Backtrack when any variable has no legal values Simplified map-coloring example C 1 C 2 C 3 C 4 C 5 RED BLUE GREEN C 2 C 5 C 3 C 1 C 4 Can solve n-queens for n 30.

23 Forward checking Another way of saying it: Delete illegal values for unassigned variables - but only the ones connected to the most recent assignment Backtrack when any variable has no legal values Simplified map-coloring example C 1 C 2 C 3 C 4 C 5 RED BLUE GREEN C 2 C 5 C 3 C 1 C 4 Can solve n-queens for n 30.

24 Note: Forward checking + MRV The obvious partner for Forward checking is the Most Restricted Value (aka Most Constrained Variable) heuristic Note that Forward checking is actually helping compute what MRV needs to do its job! C 5 RED BLUE GREEN C 1 C 2 C 3 C 4 Can solve n-queens for n 30. C 2 C 5 C 3 C 1 C 4

25 Constraint Propagation Forward checking on steroids: Start with forward checking, but continue checking any variables connected to variables you have deleted values for - until no more values have been deleted. The algorithm for this is called arc-consistency (see book!) C 1 C 2 C 3 C 4 C 5 RED BLUE GREEN C 2 C 5 C 3 C 1 C 4 Note: Last three steps were only constraint propagation!

26 Hill-climbing (or gradient ascent/descent) Define quality function on state space h(n) Make local changes of state to increase value of function. ``Like climbing Everest in thick fog with amnesia''

27 Hill Climbing Continued Depending upon initial conditions, can get stuck in local maxima Plateaus lead to random walk around state space Simulated annealing

28 Iterative algorithms for CSPs Hill-climbing, simulated annealing typically work with complete states, i.e., all variables assigned To apply to CSPs: allow states with violated constraints operators reassign variable values Reducing constraints Variable selection: randomly select any conflicted variable min-conflicts heuristic: choose value that violates the fewest constraints i.e., hill-descend with h(n) = total number of violated constraints

29 Example: 4-Queens States: 4 queens in 4 columns (4 4 = 256 states) perators: Move queen in column Goal test: No attacks Evaluation: h(n) = number of attacks Can solve n-queens with high probability (n=10,000,000)

30 Summary CSPs are a special kind of problem: States defined by values of a fixed set of variables Goal test defined by constraints on variable values Backtracking = depth-first search with 1) fixed variable order 2) only legal successors Forward checking prevents assignments that guarantee later failure Variable ordering and value selection heuristics help significantly Iterative min-conflicts is usually effective in practice

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