Constraint Satisfaction Problems: Definitions and Naive Search

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Constraint Satisfaction ProblemsDefinitions and
Naive Search

• G51IAI Introduction to AI
• Andrew Parkes
• http//www.cs.nott.ac.uk/ajp/

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Map Colouring

• Given a map of countries
• Assign a colour to each country
• such that
• IF two countries share a border THEN they
  are given different colours
• We can only use a limited number of colours
• Alternatively, we must use the minimal possible
  number of colours

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Map Colouring Example

• Consider some square countries

• How many colours are needed?

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Example 3 colours?
Take the 3 colours to be red, green, blue
How would you be sure that you have not missed
out some possible 3 colouring?
Need some complete search method!
NO COLOUR LEFT!
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Example 4 colours?
WITH JUST ONE MORE COLOUR IT IS POSSIBLE
NO COLOUR LEFT!
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Motivations

• Map Colouring is a specific problem
• so why care?
• Map Colouring is a typical Constraint
  Satisfaction Problem (CSP)
• CSPs have many uses
• scheduling
• timetabling
• window (task pane) manager in a GUI
• and many other common optimization problems with
  industrial applications

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Scope

• Russell-Norvig (AIMA) Chapter 5
• We will just cover parts of sections
• 5.1 Constraint Satisfaction Problems
• 5.2 Backtracking Search for CSP, etc
• The intention of this part of the module is only
  to give the a brief summary of the basic ideas of
  CSPs

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Potential Follow-up

• Module G5BAIP Artificial Intelligence
  Programming
• http//www.cs.nott.ac.uk/rxq/teaching.htmg5baip
• CSPs and advanced search methods for CSPs
• Constraint Programming
• A different programming paradigm
• you tell it what to solve, the system decides
  how to solve
• a program to solve Sudoku can be only 20 lines of
  code!
• e.g. constraints on each row that all numbers in
  a row are different is just forall( j in
  1..9 ) alldifferent( all(i in 1..9) posi,j )
• Very different from the usual paradigms
• Procedural (Fortran, Pascal, C)
• Object-Oriented (C, Java, C)
• Functional (Lisp, ML, Haskell)
• Commercialised by ILOG, www.ilog.com, and others

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From Maps to CSPs

• Will convert the map colouring into general idea
  of a CSP
• Assign values
• such that
• the assigned values satisfy some constraints

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Map Colouring

• Firstly add some labels

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Map Colouring Constraints

• Graphs are more common than maps so convert to
  a graph
• Edge means share a border

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Graph Colouring

• Can discard the map, and just use the graph
• Edge means must have a different colour than

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Graph 3-Colouring

• Node, or variable, must be given a value from
  the set of colours r, g , b
• E.g. B b
• Edge between two nodes ? only allowed pairs of
  values from the set (r,g), (r,b), (g, r),
  (g, b), (b, r), (b, g)

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From 3-colouring to CSP

• Generalise from the form of the 3-colouring
  problem to a
• Constraint Satisfaction Problem (CSP)
• Each node, or variable, must be assigned a
  value from a given fixed set of allowed values
• usually called the domain, D
• Edge between two nodes ? only allowed pairs of
  values from some given fixed set of allowed pairs
  of values
• the edge is called a constraint
• and is synonymous with a set of allowed pairs

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Constraints Explicit vs. Implicit

• Edge between two nodes is a constraint
• A constraint is an explicit set of pairs of
  allowed pairs of values
• Usually, represented implicitly by means of a
  mathematical constraint such as
• x ! y
• x y 2
• x 2 lt y

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Solving a CSP

• We can take the domains to be finite
• Suppose, have n variables and, each variable is
  given a value from a set D
• Suppose size of D is d
• Number of possible assignments is d
  d d ... d dn
• Exponential in problem size
• Combinatorial Explosion Again!

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Generate-and-Test

• Generate all possible assignments
• Test each assignment to see if it satisfies all
  the constraints
• This is very inefficient
• We can do a lot better

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Partial Assignments

• Total assignment
• every variable is assigned a value
• Partial Assignment
• some, none, or all are assigned values

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Search Methods

• If want a complete search method then it is
  standard to use depth-first search with partial
  assignments
• Work with Partial Assignments
• Start with the empty assignment
• Generate children by adding a value for one more
  variable
• Analogy path-finding we started with the
  empty path, and then added one more segment at
  each time
• We already did this with the map colouring at the
  start of the lecture!

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Backtracking Search

• In the DFS we generated all children
• This has the disadvantage of needing extra space
  to store them
• e.g. suppose that have blocks world with 1000
  blocks
• Backtracking search delay the creation of the
  children until they are needed
• instead just remember that we also need to create
  them

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Backtracking Search Animation

• Nodes are not created until they are ready to be
  used to reduce memory usage

A
2nd child of A is not created immediately,just
remember that it is there
B
E
D
C
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Can we 2-colour a Triangle?

• Can we assign values from r , g with the
  following variables and constraints?

• Obviously not!
• But how can we see this using search?

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Can we 2-colour a Triangle?

• Can we assign values from r , g to nodes A,
  B, and C
• Generate-and-Test Colour nodes in the order,
  A, B, C

A
Ag
Ar
All the attempts to generate a satisfying
assignment fail
B
Br
Bg
Etc, etc
C
C
Cr
Cr
Cg
Cg
Fail
Fail
Fail
Fail
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Early Failure

• Suppose a partial assignment fails i.e. violates
  a constraint
• Whatever values we eventually give to the so-far
  un-assigned variables the constraint will stay
  violated, and the solution will fail
• In the backtracking search
• as soon as a constraint is violated by the
  current partial assignment then we can prune the
  search

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Can we 2-colour a Triangle?

• Naive Backtracking Search
• Backtracking with pruning of bad partial
  assignments

A
Ag
Ar
B
Br
Bg
C
Fail
Cr
Cg
Did not have to try values for C
Fail
Fail
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Summary

• Constraint Satisfaction Problems
• Motivations
• Example Map Colouring
• Informal Definition
• Combinatorial Explosion
• Partial Assignments
• DFS and naive backtracking search

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Questions?