Journal of Formalized Mathematics
Volume 4, 1992
University of Bialystok
Copyright (c) 1992 Association of Mizar Users

On a Mathematical Model of Programs


Yatsuka Nakamura
Shinshu University, Nagano
Andrzej Trybulec
Warsaw University, Bialystok

Summary.

We continue the work on mathematical modeling of hardware and software started in [11]. The main objective of this paper is the definition of a program. We start with the concept of partial product, i.e. the set of all partial functions $f$ from $I$ to $\bigcup_{i\in I} A_i$, fulfilling the condition $f.i \in A_i$ for $i \in dom f$. The computation and the result of a computation are defined in usual way. A finite partial state is called autonomic if the result of a computation starting with it does not depend on the remaining memory and an AMI is called programmable if it has a non empty autonomic partial finite state. We prove the consistency of the following set of properties of an AMI: data-oriented, halting, steady-programmed, realistic and programmable. For this purpose we define a trivial AMI. It has only the instruction counter and one instruction location. The only instruction of it is the halt instruction. A preprogram is a finite partial state that halts. We conclude with the definition of a program of a partial function $F$ mapping the set of the finite partial states into itself. It is a finite partial state $s$ such that for every finite partial state $s' \in dom F$ the result of any computation starting with $s+s'$ includes $F.s'$.

MML Identifier: AMI_2

The terminology and notation used in this paper have been introduced in the following articles [15] [14] [7] [20] [2] [17] [3] [21] [5] [6] [12] [13] [18] [1] [8] [16] [9] [10] [4] [19]

Contents (PDF format)

Bibliography

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Received December 29, 1992


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