This time: constraint satisfaction - Constraint Satisfaction Problems (CSP) - Backtracking search for CSPs - Local search for CSPs CS 561, Session 8 1 Constraint satisfaction problems Standard search problem: state is a black box any data structure that supports successor function, heuristic function, and goal test

CSP: state is defined by variables Xi with values from domains Di goal test is a set of constraints specifying allowable combinations of values for subsets of variables Simple example of a formal representation language

Allows useful general-purpose algorithms with more power than standard CS 561, Session 8 2 search algorithms Example: map coloring problem Variables: WA, NT, Q, NSW, V, SA, T Domains: Di = {red, green, blue}

Constraints: Ci = where scope is a tuple of variables and rel is the relation over the values of these variables E.g., here, adjacent regions must have different colors e.g., WA NT, or (WA,NT) in {(red,green), (red,blue), (green,red), (green,blue), (blue,red), (blue,green)} (one for each variable) CS 561, Session 8 3

Example: map coloring problem Assignment: values are given to some or all variables Consistent (legal) assignment: assigned values do not violate any constraint Complete assignment: every variable is assigned Solution to a CSP: a consistent and complete assignment CS 561, Session 8

4 Example: map coloring problem Solutions are complete and consistent assignments, e.g., WA = red, NT = green, Q = red, NSW = green, V = red, SA = blue, T = green CS 561, Session 8 5 Example: map coloring problem

See map coloring applet CS 561, Session 8 6 Constraint graph Binary CSP: each constraint relates two variables

Constraint graph: nodes are variables, arcs are constraints CS 561, Session 8 7 Varieties of CSPs Discrete variables finite domains:

infinite domains: n variables, domain size d O(dn) complete assignments e.g., Boolean CSPs, incl. Boolean satisfiability (NP-complete) integers, strings, etc. e.g., job scheduling, variables are start/end days for each job need a constraint language, e.g., StartJob1 + 5 StartJob3

Continuous variables e.g., start/end times for Hubble Space Telescope observations linear constraints solvable in polynomial time by linear programming CS 561, Session 8 8 Varieties of constraints

Unary constraints involve a single variable, Binary constraints involve pairs of variables, e.g., SA green e.g., SA WA

Higher-order (sometimes called global) constraints involve 3 or more variables, e.g., cryptarithmetic column constraints CS 561, Session 8 9 Example: cryptarithmetic

Variables: F T U W R O X1 X2 X3 Domains: {0,1,2,3,4,5,6,7,8,9} Constraints: Alldiff (F,T,U,W,R,O) O + O = R + 10 X1 X1 + W + W = U + 10 X2 X2 + T + T = O + 10 X3

X3 = F, T 0, F 0 Constraint hypergraph Circles: nodes for variable Squares: hypernodes for n-ary constraints CS 561, Session 8 10 Real-world CSPs

Assignment problems e.g., who teaches what class Timetabling problems e.g., which class is offered when and where? Transportation scheduling

Factory scheduling Notice that many real-world problems involve real-valued variables CS 561, Session 8 11 Example: sudoku ?

ariables: each square (81 variables) omains: [1 .. 9] onstraints: each column, each row, and each of the nine 33 sub-grids that compose the g ontain all of the digits from 1 to 9 CS 561, Session 8 12 Example: sudoku ariables: each square (81 variables) omains: [1 .. 9] onstraints: each column, each row, and each of the nine 33 sub-grids that compose the g ontain all of the digits from 1 to 9 CS 561, Session 8

13 Formulation as a search problem Let's start with the straightforward approach, then fix it States are defined by the values assigned so far Initial state: the empty assignment { } Successor function: assign a value to an unassigned variable that does not conflict with current assignment fail if no legal assignments

Goal test: the current assignment is complete 1. 2. 3. 4. This is the same for all CSPs Every solution appears at depth n with n variables use depth-first search Path is irrelevant, so can be discarded

b = (n - l )d at depth l, hence n! dn leaves CS 561, Session 8 14 Backtracking search Variable assignments are commutative, i.e., [ WA = red then NT = green ] same as [ NT = green then WA = red ] Only need to consider assignments to a single variable at each

node b = d and there are dn leaves Depth-first search for CSPs with single-variable assignments is called backtracking search Backtracking search is the basic uninformed algorithm for CSPs

Can solve n-queens for n 25 CS 561, Session 8 15 Backtracking search CS 561, Session 8 16 Backtracking example 17

Backtracking example 18 Backtracking example 19 Backtracking example 20 Improving backtracking efficiency General-purpose methods can give huge gains in speed (like using

heuristics in informed search): Which variable should be assigned next? In what order should its values be tried? Can we detect inevitable failure early? 21 Most constrained variable Most constrained variable: choose the variable with the fewest legal values a.k.a. minimum remaining values (MRV) heuristic

22 Most constraining variable Tie-breaker among most constrained variables Most constraining variable: choose the variable with the most constraints on remaining variables 23 Least constraining value Given a variable, choose the least constraining value: the one that rules out the fewest values in the remaining variables

Combining these heuristics makes 1000 queens feasible 24 Forward checking Idea: Keep track of remaining legal values for unassigned variables (inference step) Terminate search when any variable has no legal values 25 Forward checking Idea: Keep track of remaining legal values for unassigned variables (inference step)

Terminate search when any variable has no legal values 26 Forward checking Idea: Keep track of remaining legal values for unassigned variables Terminate search when any variable has no legal values 27 Forward checking Idea: Keep track of remaining legal values for unassigned variables

Terminate search when any variable has no legal values 28 Constraint propagation Forward checking propagates information from assigned to unassigned variables, but doesn't provide early detection for all failures: NT and SA cannot both be blue! Constraint propagation repeatedly enforces constraints locally 29 Node and Arc consistency A single variable is node-consistent if all the values

in its domain satisfy the variables unary constraints A variable is arc-consistent if every value in its domain satisfies the binary constraints i.e., Xi arc-consistent with Xj if for every value in Di there exists a value in Dj that satisfies the binary constraints on arc (Xi, Xj) A network is arc-consistent if every variable is arcconsistent with every other variable. Arc-consistency algorithms: reduce domains of 30 Arc consistency Simplest form of propagation makes each arc consistent

X Y is consistent iff for every value x of X there is some allowed y 31 Arc consistency Simplest form of propagation makes each arc consistent X Y is consistent iff for every value x of X there is some allowed y 32 Arc consistency Simplest form of propagation makes each arc consistent

X Y is consistent iff for every value x of X there is some allowed y If X loses a value, neighbors of X need to be rechecked 4 Feb 2004 CS 3243 - Constraint Satisfaction 33 Arc consistency Simplest form of propagation makes each arc consistent X Y is consistent iff for every value x of X there is some allowed y

If X loses a value, neighbors of X need to be rechecked Arc consistency detects failure earlier than forward checking After running AC-3, either every arc is arc-consistent or some variable has empty domain, indicating the CSP cannot be solved. Can be run as a preprocessor or after each assignment 34 Arc consistency algorithm AC-3

Start with a queue that contains all arcs Pop one arc (Xi, Xj) and make Xi arc-consistent with respect to Xj If Di was not changed, continue to next arc, Otherwise, Di was revised (domain was reduced), so need to check all arcs connected to Xi again: add all connected arcs (Xk, Xi) to the queue. (this is because the reduction in Di may yield further reductions in Dk) If Di is revised to empty, then the CSP problem has no solution. 35 Arc consistency algorithm AC-3 Time complexity: ? (n variables, d values)

36 Arc consistency algorithm AC-3 Time complexity: O(n2d3) (n variables, d values) (each arc can be queued only d times, n 2 arcs (at most), checking one arc is O(d2)) 37 Local search for CSPs Hill-climbing, simulated annealing typically work with "complete" states, i.e., all variables assigned

To apply to CSPs: allow states with unsatisfied constraints operators reassign variable values Variable selection: randomly select any conflicted variable

Value selection by min-conflicts heuristic: choose value that violates the fewest constraints i.e., hill-climb with h(n) = total number of violated constraints 38 Example: 4-Queens

States: 4 queens in 4 columns (44 = 256 states) Actions: move queen in column Goal test: no attacks Evaluation: h(n) = number of attacks

Given random initial state, can solve n-queens in almost constant time for arbitrary n with high probability (e.g., n = 10,000,000) 39 Example: 4-Queens 40 Example: 4-Queens 41 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 one variable assigned per node Variable ordering and value selection heuristics help significantly

Forward checking prevents assignments that guarantee later failure Constraint propagation (e.g., arc consistency) does additional work to constrain values and detect inconsistencies 42