Constraint Satisfaction Problems Problems Artificial Intelligence - PDF document

Constraint Satisfaction Problems Problems Artificial Intelligence AIMA 2 nd edition, chapter 5 Section 1 – 3 Hadi Moradi 1 Outline � Constraint Satisfaction Problems (CSP) � Backtracking search for CSPs B kt ki h f CSP � Local search for CSPs 2 1

Constraint satisfaction problems (CSPs) � CSP: � � State � � goal test � 3 Example: Map-Coloring Variables � Domains � Constraints: � 4 2

Example: Map-Coloring � Solutions are complete and consistent assignments, 5 Constraint graph � Binary CSP: each constraint relates two variables � � Constraint graph: nodes are variables, arcs are constraints � 6 3

Varieties of CSPs � Discrete variables � finite domains: � infinite domains: � Continuous variables � 7 Varieties of constraints � Unary constraints involve a single variable, � Binary constraints involve pairs of variables, � Higher-order constraints involve 3 or more variables, 8 4

Real-world CSPs � Assignment problems � Timetabling problems � Transportation scheduling p g � Factory scheduling 9 Standard search formulation (incremental) Let's start with the straightforward approach, then fix it Initial state: Initial state: � Successor function � � � Goal test: � Branching factor : B anching facto 1. 10 5

Backtracking search Variable assignments are commutative}, i.e., � [ WA = red then NT = green ] same as [ NT = green then WA = red ] Depth-first search for CSPs with single-variable assignments is called � backtracking search Can solve n -queens for n ≈ 25 � 11 Backtracking search 12 6

Backtracking example 13 Backtracking example 14 7

Backtracking example 15 Backtracking example 16 8

Improving backtracking efficiency � General-purpose methods can give huge gains in speed: gains in speed: � Which variable should be assigned next? � In what order should its values be tried? � Can we detect inevitable failure early? 17 Minimum Remaining Variable 18 9

Degree Heuristic � Choose the Most constraining variable: 19 Least constraining value � Given a variable, choose the least constraining value: constraining value: � � 20 10

Forward checking � Idea: � Keep track of remaining legal values for unassigned variables p g g g � Terminate search when any variable has no legal values 21 Forward checking 22 11

Forward checking 23 Forward checking 24 12

Constraint propagation � Forward checking doesn't provide early detection for all failures: failures: � 25 Arc consistency � Simplest form of propagation makes each arc consistent � X � Y is consistent iff X � Y i i t t iff � for every value x of X there is some allowed y 26 13

Arc consistency � Simplest form of propagation makes each arc consistent � X � Y is consistent iff X � Y i i t t iff � for every value x of X there is some allowed y 27 Arc consistency � If X loses a value, neighbors of X need to be rechecked � 28 14

Arc consistency � � 29 Arc consistency algorithm AC-3 30 15

Local search for CSPs � Hill-climbing, simulated annealing typically work with "complete" states, i.e., all variables assigned � To apply to CSPs: 31 Example: 4-Queens � States: 4 queens in 4 columns (4 4 = 256 states) � � Actions: move queen in column � � Goal test: no attacks � � Evaluation: h(n) = number of attacks � 32 16

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 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 constrain values and detect inconsistencies Iterative min-conflicts is usually effective in practice � 33 17