Canonical forms of functions in p-valued logics

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Abstract
The functions considered are p-valued functions of n p-valued arguments; they may conveniently be represented by functions over the field Jp of integers modulo some prime p. It is noted that if every function can be uniquely written as a mod-p linear combination (equation 1) then (1) may be thought of equivalently as a canonical form or as a vector-space representation, with the bi forming a basis. This latter interpretation suggests the use of matrix multiplication to transform functions from one canonical form to another. The present paper is devoted to two main topics: 1. A consideration of various canonical forms and their analogies to the Taylor and Maclaurin expansions and the Lagrange interpolation formula of real-variable function theory. 2. A derivation of the matrices relating these forms and of expedient matrix-inversion techniques. The inversion of a pn times pn matrix is reduced, in general, to the inversion of n p times p matrices and in some cases simply to transposition or rotation of the matrix. These simplifications greatly facilitate the evaluation of 'power' series expansions for all inputs and the generation of power series from function tables.
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