CSci 533: Artificial Intelligence
Fall 2007
Lecture 6: CSP
10/04/2007
Derek Harter – Texas A&M University - Commerce
Many slides over the course adapted from Srini Narayanan,
Dan Klein, Stuart Russell or Andrew Moore
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Artificial Intelligence Lecture 6: Constraint Satisfaction Problems (CSP) - Prof. Derek Ha, Assignments of Computer Science
A lecture note from csci 533: artificial intelligence course at texas a&m university - commerce, fall 2007. The lecture covers constraint satisfaction problems (csps), their formal representation, varieties, and search methods such as backtracking search and constraint propagation. The document also discusses the concept of constraint graphs and their role in solving csps.
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CSci 533: Artificial Intelligence
Fall 2007
Lecture 6: CSP
Derek Harter – Texas A&M University - Commerce Many slides over the course adapted from Srini Narayanan, Dan Klein, Stuart Russell or Andrew Moore
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Assignment 2 is up
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CSP
Formulation
Propagation
Applications
Constraint Satisfaction Problems
Standard search problems:
(^) State is a “black box”: any old data structure (^) Goal test: any function over states (^) Successors: any map from states to sets of states
Constraint satisfaction problems (CSPs):
State is defined by variables X
i with values from a
domain D (sometimes D depends on i )
(^) 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 search algorithms
Example: Map-Coloring
Solutions are complete and consistent
assignments, e.g., WA = red, NT = green,Q =
red,NSW = green,V = red,SA = blue,T = green
Constraint Graphs
Binary CSP: each constraint
relates (at most) two variables
Constraint graph: nodes are
variables, arcs show constraints
General-purpose CSP
algorithms use the graph
structure to speed up search.
E.g., Tasmania is an
independent subproblem!
Varieties of CSPs
Discrete Variables
Finite domains
Size d means O( dn ) complete assignments
(^) E.g., Boolean CSPs, including Boolean satisfiability (NP-complete)
Infinite domains (integers, strings, etc.)
(^) E.g., job scheduling, variables are start/end times for each job (^) Need a constraint language , e.g., StartJob 1 + 5 < StartJob 3 (^) Linear constraints solvable, nonlinear undecidable
Continuous variables
E.g., start/end times for Hubble Telescope observations
Linear constraints solvable in polynomial time by LP methods
(see cs170 for a bit of this theory)
Varieties of Constraints
Varieties of Constraints
(^) Unary constraints involve a single variable (equiv. to shrinking domains): (^) Binary constraints involve pairs of variables: (^) Higher-order constraints involve 3 or more variables: e.g., cryptarithmetic column constraints
Preferences (soft constraints):
(^) E.g., red is better than green (^) Often representable by a cost for each variable assignment (^) Gives constrained optimization problems (^) (We’ll ignore these until we get to Bayes’ nets)
Standard Search Formulation
Standard search formulation of CSPs
(incremental)
Let's start with the straightforward, dumb
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
fail if no legal assignment
Goal test: the current assignment is complete and
satisfies all constraints
Search Methods
What does DFS do?
What’s the obvious problem here?
What’s the slightly-less-obvious problem?
Backtracking Search
Idea 1: Only consider a single variable at each point:
(^) 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 step (^) How many leaves are there?
Idea 2: Only allow legal assignments at each point
(^) I.e. consider only values which do not conflict previous assignments (^) Might have to do some computation to figure out whether a value is ok