Explorations in Artificial Intelligence Prof Carla P Gomes

Explorations in Artificial Intelligence Prof. Carla P. Gomes gomes@cs. cornell. edu Module 4. 2 Binary CSP

Binary CSP Restrict form of CSP with only Unary constraints Binary contraints Constraint graph Note: unary constraints can be satisfied by reducing the domain of the constrained variable (node consistency) Any CSP with n-ary constraints can be converted to another equivalent binary CSP.

Example Original constraint and variables: X+Y=Z X: : [1, 2, 3]; Y: : [2, 3, 4]; Z: : [5, 6] X < Y; How to represent this problem using only binary constraints?

From N-Ary CSP to Binary Encapsulated variable • For each constraint: A new variable that encapsulates the set of constrained variables; • Domain – set of values that satisfy the constraint (cartesian product of domains – invalid tuples) (assumed finite domains)

Example: original constraint and variables: X+Y=Z X: : [1, 2, 3]; Y: : [2, 3, 4]; Z: : [5, 6] (ternary constraint) X<Y (binary constraint) encapsulated variable and reduced U={(X, Y, Z)| X+Y=Z} domain: U: : [(1, 4, 5), (2, 3, 5), (2, 4, 6), (3, 2, 5)(3, 3, 6)] In fact, this transformation solves individual constraints. How to combine solutions from different constraints?

Hidden variable encoding (keeping original variables) Dual encoding

With original variables (hidden variable encoding) • A constraint binds the original variable to the corresponding position of the encapsulated variable; i. e. : X original variable; U encapsulated variable; X=ith_argument_of(U) – i is the "position of X within U". original (non-binary) CSP: X+Y=Z, X<Y X: : [1, 2, 3]; Y: : [2, 3, 4]; Z: : [5, 6] equivalent binary CSP: [(1, 4, 5), (2, 3, 5), (2, 4, 6), (3, 2, 5)(3, 3, 6)] [1, 2, 3] [2, 3, 4]

Without original variables (dual encoding) Constraints bind the encapsulated variables via common components in a following way i_th_argument_of(U)=j_th_argument_of(V), where U and V are encapsulated variables and i and j resply are the "positions of common component". original (non-binary) CSP: ective. X+Y=Z, X<Y X: : [1, 2]; Y: : [3, 4]; Z: : [5, 6] • Each constraint from the original CSP is represented by an encapsulated variable (even binary constraints). • Constraint network smaller than when using hidden variable • The valuation of original variables, has to be extracted from the valuation of encapsulated variables.