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Explorations in Artificial Intelligence. Prof. Carla P. Gomes gomes@cs.cornell.edu Module 6 Binary CSP. Binary CSP. Restrict form of CSP with only Unary constraints Binary contraints Constraint graph
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Explorations in Artificial Intelligence Prof. Carla P. Gomes gomes@cs.cornell.edu Module 6 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]; Y::[3,4]; Z::[5,6] • X < Z; • 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]; Y::[3,4]; Z::[5,6] • encapsulated variable and reduced • domain:U::[(1,4,5),(2,3,5),(2,4,6)] 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]; Y::[3,4]; Z::[5,6] equivalent binary CSP:
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 respectively are the "positions of common component". original (non-binary) CSP: 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.
