Constraints

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Constraints

• COMP4418 Lecture 2

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Constraint Models

• In constraint programming, we express the problem
  using
• Variables, representing unknowns and parameters
• Constraints, describing the relationship between
  the variables

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Constraint

• A constraint consists of
• a list of n variables
• a relation with n columns

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Constraint

• A constraint consists of
• a list of n variables
• a relation with n columns
• Relation can be defined
• implicitly
• X Y 5

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Constraint

• A constraint consists of
• a list of n variables
• a relation with n columns
• Relation can be defined (X, Y)
• implicitly or explicitly
• X Y 5

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Constraint Models

• A constraint model is
• a conjunctive set of constraints

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Constraint Models

• A constraint model is
• a conjunctive set of constraints
• a program that generates conjunctions of
  constraints

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Constraint Models

• A constraint model is
• a conjunctive set of constraints
• a program that generates conjunctions of
  constraints
• (not even limited to conjunctions)

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Solving Constraint Models

• Finding a solution
• Algorithmically
• Uniform class of constraints
• Constraint solving
• Search propagation
• General purpose, less efficient
• Constraint satisfaction
• Both

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Constraint Solving

• Given a signature ?
• naming relevant function and predicate symbols
  and their arities
• A constraint domain consists of
• a language
• describes what formulas are constraints
• a structure
• defines the set of values
• defines the meaning of function and predicate
  symbols as functions and relations

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Constraint Solving

• Linear arithmetic constraints over the real
  numbers
• ? , -, , lt, ?, 0, 1, 2, , -1, -2,
• Terms are constructed by , -, and numerals
• Primitive constraints
• term term, term lt term, term ? term
• Constraints are conjunctions of primitive
  constraints

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Constraint Solving

• Linear arithmetic constraints over the real
  numbers
• The structure consists of
• The set of values R
• Mapping
• numerals to the corresponding number
• to addition
• - to subtraction
• , lt, ? to equality, less-than, less-than-or-equal

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Constraint Solving

• Linear equations over the real/rational numbers
• Gauss, Gauss-Jordan algorithms
• Equations over Herbrand terms
• Unification algorithm
• Linear equations and inequalities over the
  real/rational numbers
• Simplex algorithm, interior point algorithms
• Polynomial equations and inequalities over the
  real numbers
• Collins method

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Constraint Solving

• Transcendental equations and inequalities over
  the real numbers
• undecidable
• Linear equations and inequalities over the
  integers
• NP-complete
• Polynomial equations and inequalities over the
  integers
• undecidable

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Constraint Solving

• Constraints that are easy to solve
• Bounds on real/rational numbers
• Bounds on integers
• Bounds on sets
• Bounds on multisets/bags
• Membership of finite sets

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Constraint Satisfaction Problems

• CSPs are used to formalize a class of conjunctive
  constraint problems
• A CSP is a tuple ltV, D, Cgt where
• V is a set of variables
• D maps each variable to its domain the set of
  its possible values, usually finite
• C is a set of constraints

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Rock Star Dressing Problem

• A rock star has
• red, green, orange and blue shirts
• pink and yellow ties
• black, khaki, purple and blue pants
• Fashion sense
• A yellow tie only goes with red, green or blue
  shirts
• A pink tie only goes with blue shirts
• etc
• Find a satisfactory outfit

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Rock Star Dressing Problem

• Variables shirt, tie pants
• Domains
• D(shirt) red, green, orange, blue
• D(tie) pink, yellow
• D(pants) black, khaki, purple, blue
• There are three constraints, specifying the
  colours of pairs of items of clothing that are
  compatible

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Rock Star Dressing Problem

• Constraints
• C1 C2
  C3
• C1(P,S), C2(S,T), C3(T,P)

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Binary CSPs

• A binary CSP is one where all relations involve 2
  variables (or fewer)
• Binary CSPs can be represented as graphs

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Rock Star Dressing Problem
blue black purple khaki
red green orange blue
C1
C2
C3
yellow pink
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Rock Star Dressing Problem
blue black purple khaki
red green orange blue
C
C
C
yellow pink
Equivalent formulation using only one relation
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Binary CSPs

• Binary CSPs are universal
• Every CSP can be encoded as a binary CSP
• Variables range over tuples in constraints of
  original CSP
• Constraints permit only tuples that are
  compatible in original CSP

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CSP Satisfiability

• The satisfiability problems for CSPs is
  NP-complete
• In NP because a solution is a PTIME checkable
  certificate
• NP-hard by reduction from 3SAT
• 8 primitive constraints, corresponding to 8
  3-clauses
• All domains are 0, 1

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CSP Satisfiability
Rpnn

• X ? ?Y ? ?Z
• Bool var 0/1 var
• clause constraint
• SAT problem CSP

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Binary CSPs

• Binary CSPs are universal
• Every CSP can be encoded as a binary CSP
• Variables range over tuples in constraints of
  original CSP
• Constraints permit only tuples that are
  compatible in original CSP

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Binary CSPs

• A binary CSP is arc-consistent if
• for every variable x in V,
• and every constraint c(x, y) involving x,
• for every value d in D(x)
• there is a value d' in D(y)
• such that c(d, d') is true ?
• (equivalently, the tuple (d, d') is in c)
• d is called a support for d

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Binary CSPs

• If a CSP is arc-consistent, then
• all the information about possible variable
  values (i.e. domains)
• that can be extracted from the constraints
  (looked at one-at-a-time)
• is represented in the domains
• Sometimes called domain consistency
• By making a CSP arc-consistent, we simplify the
  problem, without losing solutions

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Binary CSPs

• We can make a CSP arc consistent by deleting all
  domain values that do not have a support
• these are values that cannot participate in a
  solution
• By making a CSP arc-consistent, we simplify the
  problem, without losing solutions
• For table constraints, arc-consistency can be
  enforced in O(nD2)
• n constraints
• maximum domain size D

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Rock Star Dressing Problem
blue black purple khaki
red green orange blue
C1
C2
C3
yellow pink
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Rock Star Dressing Problem
blue black purple khaki
red green orange blue
C1
C2
C3
yellow pink
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Rock Star Dressing Problem
blue black purple khaki
red green orange blue
C1
C2
C3
yellow pink
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Generalized Arc Consistency

• A CSP is generalized arc-consistent if
• for every variable x in V,
• and every constraint c(x, y1, , yn),
• for every value d in D(x)
• there are values d1dn in D(y1), , D(yn)
• such that c(d, d1dn) is true ?
• (equivalently, the tuple (d, d1dn) is in c)
• Complexity O(nDk) for constraints of arity k

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Rock Star Dressing Problem
blue black purple khaki
red green orange blue
C1
C2
C3
yellow pink