CIS 471/571(Fall 2020): Introduction to Artificial Intelligence
Lecture 4: Constraint Satisfaction Problems (Part 1)
Thanh H. Nguyen
Source: http://ai.berkeley.edu/home.html
Reminder
§ Homework 1: Search
§ Deadline: Oct 10th, 2020
§ Project 1: Search
§ Deadline: Oct 13th, 2020
Thanh H. Nguyen
10/11/20
2
Today
§Constraint Satisfaction Problems §Backtracking Search
§Filtering §Ordering
Constraint Satisfaction Problems
Constraint Satisfaction Problems
§ Standard search problems:
§ State is a “black box”: arbitrary data structure § Goal test can be any function over states
§ Successor function can also be anything
§ Constraint satisfaction problems (CSPs):
§ A special subset of search problems
§ State is defined by variables Xi 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
§ Allows useful general-purpose algorithms with more power than standard search algorithms
CSP Examples
Example: Map Coloring
§ Variables:
§ Domains:
§Constraints: adjacent regions must have different colors
Implicit: Explicit:
§Solutions are assignments satisfying all constraints, e.g.:
Example: N-Queens
§Formulation 1: § Variables:
§ Domains:
§ Constraints
Example: N-Queens
§Formulation 2: § Variables:
§ Domains:
§ Constraints:
Implicit: Explicit:
Constraint Graphs
Constraint Graphs
§Binary CSP: each constraint relates (at most) two variables
§Binary 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!
Example: Sudoku
§ Variables:
§ Each (open) square
§ Domains:
§ {1,2,…,9} § Constraints:
9-way alldiff for each column 9-way alldiff for each row
9-way alldiff for each region
(or can have a bunch of pairwise inequality constraints)
Varieties of CSPs and Constraints
Varieties of CSPs
§Discrete Variables § Finite domains
§ E.g., Boolean CSPs, including Boolean satisfiability (NP-complete) § Infinite domains (integers, strings, etc.)
§ E.g., job scheduling, variables are start/end days for each job
§Continuous variables
§ E.g., start/end times for Hubble Telescope observations
Varieties of Constraints
§Varieties of Constraints
§Unary constraints involve a single variable
(equivalent to reducing domains), e.g.:
§Binary constraints involve pairs of variables, e.g.: §Higher-order constraints involve 3 or more variables
§Preferences (soft constraints):
§E.g., red is better than green
§ Often representable by a cost for each variable assignment
§Gives constrained optimization problems
Real-World CSPs
§ Scheduling problems: e.g., when can we all meet?
§ Timetabling problems: e.g., which class is offered when and where? §Assignment problems: e.g., who teaches what class
§Hardware configuration
§Transportation scheduling
§Factory scheduling
§Circuit layout
§Fault diagnosis
§… lots more!
§Many real-world problems involve real-valued variables…
Solving CSPs
Standard Search Formulation
§Standard search formulation of CSPs
§States defined by the values assigned so far (partial assignments)
§ Initial state: the empty assignment, {}
§ Successor function: assign a value to an
§ Goal test: the current assignment is complete and satisfies all constraints
§We’ll start with the straightforward, naïve approach, then improve it
unassigned variable
Search Methods
§What would BFS do?
§What would DFS do?
§What problems does naïve search have?
[Demo: coloring — dfs]
Backtracking Search
Backtracking Search
§ Backtracking search is the basic uninformed algorithm for solving CSPs
§ Idea 1: One variable at a time
§ Variable assignments are commutative, so fix ordering
§ 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
§ Idea 2: Check constraints as you go
§ I.e. consider only values which do not conflict with previous assignments § Might have to do some computation to check the constraints
§ “Incremental goal test”
§ Depth-first search with these two improvements is called backtracking search (not the best name)
§ Can solve n-queens for n » 25
Backtracking Example
Backtracking Search
§ Backtracking = DFS + variable-ordering + fail-on-violation § What are the choice points?
Improving Backtracking
§General-purpose ideas give huge gains in speed
§ Ordering:
§Which variable should be assigned next? § In what order should its values be tried?
§Filtering: Can we detect inevitable failure early? §Structure: Can we exploit the problem structure?
Filtering
Filtering: Forward Checking
§ Filtering: Keep track of domains for unassigned variables and cross off bad options
§Forward checking: Cross off values that violate a constraint when added to the existing assignment
WA NT Q
SA NSW V
Filtering: Constraint Propagation
§ Forward checking propagates information from assigned to unassigned variables, but doesn’t provide early detection for all failures:
NT SA
Q
WA
NSW
V
§ NT and SA cannot both be blue!
§Why didn’t we detect this yet?
§Constraint propagation: reason from constraint to constraint
Consistency of A Single Arc
§An arc X ® Y is consistent iff for every x in the tail there is some y in the head which could be assigned without violating a constraint
Delete from the tail!
NT SA
Q
WA
NSW
V
§ Forward checking: Enforcing consistency of arcs pointing to each new assignment
Arc Consistency of an Entire CSP
§ A simple form of propagation makes sure all arcs are consistent:
NT SA
Q
WA
NSW
V
§ Important: If X loses a value, neighbors of X need to be rechecked! § Arc consistency detects failure earlier than forward checking
§ Can be run as a preprocessor or after each assignment
§ What’s the downside of enforcing arc consistency?
Remember: Delete from the tail!
Enforcing Arc Consistency in a CSP
§ Runtime: O(n2d3)
Limitations of Arc Consistency
§After enforcing arc consistency:
§Can have one solution left
§Can have multiple solutions left
§Can have no solutions left (and not know it)
§Arc consistency still runs inside a backtracking search!
What went wrong here?
K-Consistency
K-Consistency
§Increasing degrees of consistency
§ 1-Consistency (Node Consistency): Each single node’s domain
has a value which meets that node’s unary constraints
§ 2-Consistency (Arc Consistency): For each pair of nodes, any consistent assignment to one can be extended to the other
§ K-Consistency: For each k nodes, any consistent assignment to k-1 can be extended to the kth node.
§Higher k more expensive to compute
§ (You need to know the k=2 case: arc consistency)
Strong K-Consistency
§ Strong k-consistency: also k-1, k-2, … 1 consistent
§Claim: strong n-consistency means we can solve without backtracking!
§ Why?
§ Choose any assignment to any variable
§ Choose a new variable
§ By 2-consistency, there is a choice consistent with the first
§ Choose a new variable
§ By 3-consistency, there is a choice consistent with the first 2 §…
§ Lots of middle ground between arc consistency and n-consistency! (e.g. k=3, called path consistency)