1 Variable Assignments

Transcription

1 1 Variable Assignments Definition of a Variable Assignment An Assignment over Variables Given a set V of variables, an assignment over V is a function a : V x V dom(x), with the property that a(x) dom(x), for every variable x V. We say that a(x) is the value assigned to x by a. var(a) refers to variable set V. The Set of all Assignments A(V ): the set of all possible assignments over variable set V Example of a Variable Assignment V = {x 1, x 2, x 3, x 4 }. Each variable has domain {1, 2, 3}. v V a(v) x 1 3 x 2 2 x 3 3 x 1 2 Writing a as a Tuple No assumed ordering of the variables (unordered tuple notation): a = (x 3 = 3, x 1 = 3, x 4 = 2, x 2 = 2) Assuming the ordering as shown above ((ordered tuple notation): a = (3, 2, 3, 2). Omitting commas when there is no ambiguity: a = (3232) Partial Variable Assignments a is called a partial assignment over V iff it is an assignment over a subset of U of V. Writing a as a Tuple Within the Larger Context of V Example: U = {x 1, x 3 }. Then using ordered tuple notation, a = (3,, 3, ) or a = (3 3 ) 1

2 Projecting an Assignment π U (a) Given an assignment a over V, the projection of a onto U V, denoted π U (a) is the assignment â over U for which â(x) = a(x) for every x U. Projection Examples V = {x 1, x 2, x 3, x 4 }, and a = (1, 3, 2, 1) U = {x 1, x 4 } π U (a) = (1, 1) U = {x 2, x 3, x 4 } π U (a) = (3, 2, 1) Extending an Assignment if a is an assignment over U, and â is an assignment over V with U V, then â is called an extension of a provided π U (â) = V. Extension Examples U = {x 1, x 4 }, and a = (2, 3) V = {x 1, x 2, x 3, x 4 } â = (2, 1, 4, 3) is an extension of a. U = {x 1, x 4, x 7 } â = (2, 3, 5) is an extension of a. 2 Relations and Constraints Definition of a Relation A relation R over a set of variables V is a subset of A(V ). V is called the scope of R, written scope(r), or the variables of R, written var(r) A synonym for relation is constraint, since a relation restricts (i.e. constrains) the set of possible assignments over V. Relation Example 2

3 V = {x 1, x 2, x 3 }, dom(x 1 ) = dom(x 2 ) = {0, 1}, dom(x 3 ) = {1, 2, 3} R A(V ) = {(0, 1, 2), (0, 1, 3), (1, 0, 1), (1, 0, 2), (1, 1, 3)}. Assignments that Satisfy Constraints Satisfaction of Constraint c If a c, then a is said to satisfy c. Note: π c (a) is short for πvar(c)(a). In general, if a is an assignment over V, and var(c) V, then a is said to satisfy c iff π c (a) c. Satisfaction Example V = {x 1, x 2, x 3 }, dom(x 1 ) = dom(x 2 ) = {0, 1}, dom(x 3 ) = {1, 2, 3} c = {(0, 1, 2), (0, 1, 3), (1, 0, 1), (1, 0, 2), (1, 1, 3)} a = (0, 1, 3) satisfies c since a c. a = (0, 1, 2, 0, 2) also satisfies c if we assume a is an assignment over {x 0, x 1, x 4, x 2, x 3 }. This is true since π c (a) = (1, 0, 2) c. Assignments that are Consistent with Constraints Two ways that a can be Consistent with c 1. Case 1: var(c) var(a). Then a satisfies c. 2. Case 2: var(c) var(a). Then a can be extended to an assignment that satisfies c. Case 2 Example V = {x 1, x 2, x 3 }, dom(x 1 ) = dom(x 2 ) = {0, 1}, dom(x 3 ) = {1, 2, 3} c = {(0, 1, 2), (0, 1, 3), (1, 0, 1), (1, 0, 2), (1, 1, 3)} a = (6, 1, 3, 4) is an assignment over {x 0, x 2, x 3, x 4 }. a can be extended to the assignment â = (6, 0, 1, 3, 4) over {x 0, x 1, x 2, x 3, x 4 } which satisfies c since π c (â) = (0, 1, 3) c. 3

4 Representing a Constraint Extensional Representation An extensional representation of a constraint is an enumeration of all the assignments that satisfy the constraint. Intensional Representation An intensional representation of a constraint occurs when the constraint is represented using a formula or some other functional mechanism that does not rely on explicit knowledge of all the satisfying tuples. Intensional Representation Example Defining the constraint c is defined over Boolean variables {x, y, z} and by formula (x y) z Truth Table x y z (x y) z Extensional Representation c = {(1, 1, 0), (0, 0, 1), (0, 1, 1), (1, 0, 1), (1, 1, 1)} Extensional versus Intensional Pros Extensional representations can make use of data structures, such as arrays, trees, and hash tables. Thus, satisfaction and consistency queries can be performed efficiently. Extensional representations allow for ease of constraint propagation. Intensional representations are normally the representation of choice for the constraint programmer. 4

5 Extensional versus Intensional Cons Extensional representations are often computationally infeasible, due to the (large) number of satisfying assignments, and/or to the computational work needed to translate an intensional representation to an extensional one. Intensional representations often require more computational work to answer queries about satisfaction and consistency. Intensional representations often to not allow for ease in constraint propagation. 3 Definition of a Constraint Satisfaction Problem Joining two Relations If R 1 is a relation over V 1 and R 2 a relation over V 2, then the join of R 1 and R 2, writtten R 1 R 2, is the set of all assignments over V 1 V 2 that satisfy both R 1 and R 2. Join Example R 1 = {(1, 2, 1), (1, 3, 2), (2, 1, 4), (1, 4, 2)} is a relation over {x, y, z}. R 2 = {(1, 2, 3), (2, 4, 1), (1, 2, 2), (1, 4, 2)} is a relation over {x, z, w}. R 1 R 2 = {(1, 3, 2, 3), (1, 3, 2,, 2), (2, 1, 4, 1), (1, 4, 2, 3), (1, 4, 2, 2) is a relation over {x, y, z, w}. Constraint Satisfaction Problem: Formal Definition A Constraint Satisfaction Problem (CSP) is a triple P = (V, D, C), where V = {x 1,..., x n } is a set of variables. D = {D 1,..., D n } is a set of domains, where dom(x i ) = D i. C = {c 1,..., c m } is a set of constraints. Solution to a Constraint Satisfaction Problem A solution to P = (V, D, C) is any assignment over V that satisfies every c C. sol(p ) = c C c is the set of all solutions to P. 5

6 Exercises 1. Let R 1 = {(a, b), (a, e), (c, d), (d, e)} be a relation over {x, y}, and R 2 = {(b, c), (e, a), (b, d)} be a relation over {y, z}. Compute the following: (a) π y (R 1 ); (b) R 1 R 2 ; (c) the extensions of (, e) that satisfy R Given constraints c 1 = x (y z), and c 2 = y (z w), compute the following: (a) extensional representations for both c 1 and c 2 ; (b) c 1 c 2 ; (c) π yz (c 2 ); (d) an assignment a over {x, y} that is not consistent with c 1 ; (e) an extension of a (from part d) that satisfies c The Graph Coloring Problem. Let G = (V, E) be a simple graph where V = {a, b, c, d, e, f, g} and E = {(a, b), (a, d), (b, c), (b, d), (b, g), (c, g), (d, e), (d, f), (d, g), (f, g)}. The problem is to find a coloring of V using colors red,blue, and yellow, so that no two adjacent vertices are assigned the same color. Define a CSP for this problem. Clearly define the variables, domains, and constraints. Find at least one solution to the CSP. 6