Span Programs

Abstract

In 1993 Karchmer and Wigderson introduced an interesting linear algebraic model for computing boolean functions — the span program. A span program for a function f(x ₁, ..., x _n ) is presented as a matrix over some field, with rows labeled by variables x _i or their negations x̄ _i (one variable can label many rows) . The span program accepts an input assignment if and only if the all-1 vector can be obtained as a linear combination of the rows whose labels are satisfied by the input. The size of the span program is the number of rows in the matrix. A span program is monotone if only positive literals are used as labels of the rows, i.e. negated variables are not allowed.