Constraint Reasoning

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Constraint Reasoning
Florida Institute of Technology Computer Science
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Chapter 1) Introduction

• Constraint Satisfaction Problem(CSP) is an
  important sub-field of the artificial
  intelligence. CSP appears in many areas, for
  instance in computer vision, in resource
  allocation, in scheduling and in temporal
  reasoning.
• What is a constraint satisfaction problem
• A CSP is a problem composed of a finite set of
  variables, each of which is associated with a
  finite domain, and a set of constraints.
• The task is to assign a value to each variable
  satisfying all the constraints.
• For example the N-queens problem the problem
  is to place eight queens on an 8x8 chessboard
  satisfying the constraint that no two queens
  should be on the same row, column or diagonal.

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Example 2-car sequencing problem

• In a car production scenario, cars are placed on
  conveyor belts which move through different work
  areas.
• A production line is normally required to produce
  cars of different models. The number of cars
  required for each model is called the production
  requirement.
• Each work area is constrained by its resource
  constraint or Capacity constraint.
• variable - the car model of one position in the
  conveyor belt (I.e. if there are n cars to be
  scheduled, the problem consists of n variables).
• domain - the set of car models, for example from
  model A to D.
• The task - to assign a value (a car model) to
  each variable (a position in the conveyor belt),
  satisfying both the production requirements and
  capacity constraints.

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Definition of domain and labels

• Definition 1-1The domain of a variable is a set
  of all possible values that can be assigned to
  the variable. If x is a variable, then we use Dx
  to denote the domain of it.
• Definition 1-2A label is a variable-value pair
  that represents the assignment of the value to
  the variable. We use ltx,vgt to denote the label of
  assigning the value v to the variable x. ltx, vgt
  is only meaningful if v is in the domain of x
  (I.e. v ? Dx).
• Definition 1-3A compound label is the
  simultaneous assignment of values to a (possibly
  empty) set of variables. We use
  (ltx1,v1gtltx2,v2gtltxn,vngt)to denote the compound
  label of assigning v1, v2, , vn to x1, x2, , xn
  respectively.
• Definition 1-4A k-compound label is a compound
  label which assigns k values to k variables
  simultaneously.

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Definition of domain and labels

• Definition 1-5If m and n are integers such that
  mltn, then a projection of an n-compound label N
  to an m-compound label M, written as
  projection(N,M), (read as M is a projection of
  N) if the labels in M all appear in N.
• Definition 1-6The variables of a compound label
  is the set of all variables which appear in that
  compound label.

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Definition of constraints

• Definition 1-7A constraint on a set of
  variables is conceptually a set of compound
  labels for the subject variables. For
  convenience, we use Cs to denote the constraint
  on the set of variables S.
• Definition 1-8The variables of a constraint is
  the variables of the members of the constraint.
• Definition 1-9If m and n are integers such that
  mltn, then an m-constraint M is subsumed-by an
  n-constraint N(written as subsumed-by(M,N)) if
  for all elements c in M there exists an element d
  in N such that c is a projection of d.

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Definitions of satisfiability

• Definition 1-10aIf the variables of the
  compound label x are the same as those variables
  of the elements of the compound labels in
  constraint C, then x satisfies C if and only if X
  is an element of C.
• Definition 1-10bsatisfies(ltx,vgt,Cx) ? (ltx,vgt) ?
  Cx
• Definition 1-11Given a compound label L and a
  constraint C such that the variables of C are a
  subset of the variables of L, the compound label
  L satisfies constraint C if and only if the
  projection of L onto the variables of C is an
  element of C.

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Constraint satisfaction problem

• Definition 1-12A constraint satisfaction
  problem is a triple (Z, D, C)where, Za
  finite set of variables x1, x2, , xnDa
  function which maps every variable in Z to a set
  of objects of arbitrary typeD Z?finite set of
  objects (of any type)We shall take Dxi as the
  set of objects mapped from xi by D. We call these
  objects possible values of xi and the set Dxi the
  domain of xiCa finite (possibly empty) set of
  constraints on an arbitrary subset of variables
  in Z. In other words, C is a set of sets of
  compound lables. We use csp(P) to denote that P
  is a constraint satisfaction problem.

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Task in a CSP

• Definition 1-13A solution tuple of a CSP is a
  compound label for all those variables which
  satisfy all the constraints.
• Depending on the requirements of an application,
  CSPs can be classified into the following
  categories(1) CSPs in which one has to find any
  solution tuple(2) CSPs in which one has to find
  all solution tuples.(3) CSPs in which one has to
  find optimal solutions, where optimality is
  defined according to some domain knowledge.
  Optimal or near optimal solutions are often
  required in scheduling.

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Constraint
representation and binary CSPs

• Definition 1-14A binary CSP, or binary
  constraint problem, is a CSP with unary and
  binary constraints only. A CSP with constraints
  not limited to unary and binary will be referred
  to as a general CSP.
• Definition 1-15A graph is a tuple (V,U) where V
  is a set of nodes and U(?VxV) is a set of arcs. A
  node can be an object of any type and an arc is a
  pair of nodes. For convenience, we use graph(G)
  to denote that G is a graph.An undirected graph
  is a tuple (V,E) where V is a set of nodes and E
  is a set of edges, each of which being a
  collection of exactly two elements in V.
• Definition 1-16For all graphs (V,E), node x is
  adjacent to node y if and only if (x,y) is in E.

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Graph-related concepts

• Definition 1-17A hypergraph is a tuple (V,E)
  where V is a set of nodes and E is a set of
  hyperedges, each of which is a set of nodes. For
  convenience we use hypergraph((v,E)) to denote
  that (V,E) is a hypergraph, hyperedges(F,V) to
  denote that F is a set of hyperedges for the
  nodes V (I.e. F is a set of set of nodes in V),
  and nodes_of(e) to denote the nodes involved in
  the hyperedge e.
• Definition 1-18The constraint hypergraph of a
  CSP (Z, D, C) is a hypergraph in which each node
  represents a variable in Z, and each hyperedge
  represents a constraint in C. We denote the
  constraint hypergraph of a CSP P by H(P). If P is
  a binary CSP and we exclude hyperedges on single
  nodes, then H(P) is a graph. We denote the
  constraint graph of a CSP by G(P).

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Graph-related concepts

• Definition 1-20A path of length n is a path
  which goes through n1 (not necessarily distinct)
  nodes.
• Definition 1-21A node y is accessible from
  another node x if there exists a path from x to
  y.
• Definition 1-22A graph is connected if there
  exists a path between every pair of nodes.
• Definition 1-23A loop in a graph is an edge or
  an arc which goes from a node to itself, I.e. a
  loop is (x,x), where x is a node.
• Definition 1-24A network is a graph which is
  connected and without loops.
• Definition 1-25A cycle is a path on which
  end-points coincide.

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Graph-related concepts

• Definition 1-26An acyclic graph is a graph
  which has no cycles.
• Definition 1-27A tree is a connected acyclic
  graph
• Definition 1-28A binary relation(lt) on a set S
  is called an ordering of S when it is
  irreflexive, asymmetric and transitive.
• Definition 1-29A set S is totally ordered if
  every two elements in S are ordered. Such an
  ordering is called a total ordering of the
  elements in S.

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The N-queens problem

• This N-queen problem has very specific features
• firstly, it is a binary constraints,
• secondly, every variable is constrained by every
  other variable,
• More importantly, constraint get looser as N
  grows larger.
• Such features may not be shared by many other
  CSPs therefore, benchmarks on different
  algorithms produced using this problem must be
  interpreted with caution.
• Problem formalization
• The set of variables Z Q1, Q2, , Q8
• Domain DQ1 DQ2 DQ8 1,2,3,4,5,6,7,8
• Constraint (1) ?i,j if Qia and Qjb, then a?b
• Constraint (2) ?i,j, if Qia and Qjb, then i-j
  ? a-b, and i-j ? b-a.

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The N-queens problem

• Alternative formalization of the N-queens problem
• The set of variables Z Q1, Q2, , Q8
• Domain DQ1 DQ2 DQ8 0,1,2,3,4, ,63
• Constraints Ri is (Ni div 8)1, Ci is (Ni mod
  8)1Rj is (nj div 8)1, Cj is (Nj mod 8)1
• Row (number div 8)1
• Column (number mod 8)1
• Ri ? Rj
• Ci ? Cj
• Ri-Rj ? Ci-Cj
• Ri-Rj ? Cj-Ci
• Formalization can be a hidden constraint.
• The first one allows 88 combinations, the second
  one 648.
• A problem can be formulated as a CSP in different
  ways.

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The graph coloring problem

• As a map can be represented by a graph where each
  node represents an area in the map, and every
  pair of nodes which represent two adjacent areas
  in the map is connected by an edge Figure 1.5.
• Problem formalization
• The set of variables Z w, x, y, z
• Domain Dw Dx Dy Dz r, g, b
• Constraints C Cw,x, Cw,y, Cx,y, Cx,z, Cy,z
• Cx1,x2,,xn (Vx1?Vx2 ? ?Vxn)

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The scene labeling problem

• The task is to label all the edges in Figure 1.7,
  satisfying all the constraints.
• The set of variables Z A, B, C, D, E, F, G,
  h, IOne way is to use one variable to represent
  the value of a line in the scene.
• Domain D , -, ?, ?

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Temporal reasoning

• Events are all temporally related to each other.
  Depending on the structure of time one uses,
  different types of temporal relations apply.
• In planning and scheduling, one has to determine
  the temporal relationship between events.
• One approach is to pick up one temporal relation
  per pair of events in each step, and backtrack
  when the hypothetical situation has been proved
  to be unsatisiable.
• The other approach is to reason with all
  disjunctive temporal relations simultaneously.

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Resource allocation

• Resource allocation, especially when time and
  shared resources are involved, is basically a
  CSP. Each variable in the CSP represents one
  shared resource requirement.
• variable X may represent the machine requirement
  of a job (each variable in the CSP represents one
  shared resource requirement).
• The domain may be the set of machines available
  in the workshop which have the capacity to do the
  job.
• The allocation of resources is normally
  constrained in many ways.

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Graph matching

• Given two graphs G1 and G2, the problem is to
  check whether G2 has a subgraph which matches G1.
  Graph (V1, E1) contains graph (V2, E2) if(1)
  every node in V2 can be mapped to a distinct node
  in V1 and(2) for all x1, y1 in V1 and x2, y2
  in V2, if x2 and y2 are mapped to x1 and y1,
  respectively, then whenever (x2, y2) is an edge
  in E2, then (x1, y1) is an edge in E1.

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Other application

• The grammar of the language restricts the roles
  that a string of words can take simultaneously.
• Database query optimization is an important
  database research area in which CSP solving
  techniques can be applied.
• CSP techniques have also been applied to
  parameter setting for greenhouses in agricultural
  applications, and demonstrated to be successful.

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Constraint programming

• CLP one of the main objectives of developing
  CLP is to define a class of logic programming
  languages with well defined semantics under a
  particular equational theory.
• Prolog III It was developed with similar goals
  as CLP, but with refined manipulation on trees
  (including infinite trees), lists and Boolean
  variables.
• CHIP It is a logic programming language for
  handling symbolic, Boolean as well as numerical
  variables. The basic search strategy used in CHIP
  is called forward checking (FC). It is used
  together with a heuristic called the fail first
  principle (FFP). Its success has led to the of
  Charme and PECOS, which are mainly built to
  handle symbolic variables.
• Charme It uses the syntax of C, and one of its
  merits is that it can easily be integrated into
  the users of the C programs.
• PECOS It uses LISP syntax.