Locally computable coding for unary operations

Coding plays an important role in the design of parallel algorithms. When each digit of a result of a desired operation depends only on a part of digits of operands under a coding scheme, we say the operation is locally computable under the coding scheme. There is a closed relation between local computability and redundancy of codes. Several excellent algorithms utilizing local computability by redundant coding schemes are developed and used practically. The problem to examine the relation among coding schemes, local computability and algebraic structures of target operations is a basic problem on the design of parallel algorithms. In this paper, we discuss a relation between redundancy of coding schemes and local computability of unary operations defined on finite sets. We show that it is generally impossible to realize local computability by nonredundant coding schemes. If we introduce redundancy into coding, we can construct a coding under which every digit of a result depends only on 2 digits of an operand for any unary operations. These results are closely related with a state assignment problem of finite state machines. The above 2-locally computable coding derives a state assignment of a pipelined sequential circuit for an arbitrary sequential machine.