Volume 4, 1992

University of Bialystok

Copyright (c) 1992 Association of Mizar Users

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

- 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 $F.s'$.

Contents (PDF format)

- [1]
Grzegorz Bancerek.
The fundamental properties of natural numbers.
*Journal of Formalized Mathematics*, 1, 1989. - [2]
Grzegorz Bancerek.
The ordinal numbers.
*Journal of Formalized Mathematics*, 1, 1989. - [3]
Grzegorz Bancerek.
K\"onig's theorem.
*Journal of Formalized Mathematics*, 2, 1990. - [4]
Grzegorz Bancerek and Krzysztof Hryniewiecki.
Segments of natural numbers and finite sequences.
*Journal of Formalized Mathematics*, 1, 1989. - [5]
Czeslaw Bylinski.
Functions and their basic properties.
*Journal of Formalized Mathematics*, 1, 1989. - [6]
Czeslaw Bylinski.
Functions from a set to a set.
*Journal of Formalized Mathematics*, 1, 1989. - [7]
Czeslaw Bylinski.
Some basic properties of sets.
*Journal of Formalized Mathematics*, 1, 1989. - [8]
Czeslaw Bylinski.
A classical first order language.
*Journal of Formalized Mathematics*, 2, 1990. - [9]
Czeslaw Bylinski.
The modification of a function by a function and the iteration of the composition of a function.
*Journal of Formalized Mathematics*, 2, 1990. - [10]
Czeslaw Bylinski.
Subcategories and products of categories.
*Journal of Formalized Mathematics*, 2, 1990. - [11]
Yatsuka Nakamura and Andrzej Trybulec.
A mathematical model of CPU.
*Journal of Formalized Mathematics*, 4, 1992. - [12]
Dariusz Surowik.
Cyclic groups and some of their properties --- part I.
*Journal of Formalized Mathematics*, 3, 1991. - [13]
Andrzej Trybulec.
Domains and their Cartesian products.
*Journal of Formalized Mathematics*, 1, 1989. - [14]
Andrzej Trybulec.
Enumerated sets.
*Journal of Formalized Mathematics*, 1, 1989. - [15]
Andrzej Trybulec.
Tarski Grothendieck set theory.
*Journal of Formalized Mathematics*, Axiomatics, 1989. - [16]
Andrzej Trybulec.
Function domains and Fr\aenkel operator.
*Journal of Formalized Mathematics*, 2, 1990. - [17]
Andrzej Trybulec.
Subsets of real numbers.
*Journal of Formalized Mathematics*, Addenda, 2003. - [18]
Michal J. Trybulec.
Integers.
*Journal of Formalized Mathematics*, 2, 1990. - [19]
Wojciech A. Trybulec.
Pigeon hole principle.
*Journal of Formalized Mathematics*, 2, 1990. - [20]
Zinaida Trybulec.
Properties of subsets.
*Journal of Formalized Mathematics*, 1, 1989. - [21]
Edmund Woronowicz.
Relations and their basic properties.
*Journal of Formalized Mathematics*, 1, 1989.

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