Journal of Formalized Mathematics
Volume 6, 1994
University of Bialystok
Copyright (c) 1994 Association of Mizar Users

## Minimization of Finite State Machines

Miroslava Kaloper
University of Alberta, Department of Computing Science
Piotr Rudnicki
University of Alberta, Department of Computing Science

### Summary.

We have formalized deterministic finite state machines closely following the textbook [10], pp. 88-119 up to the minimization theorem. In places, we have changed the approach presented in the book as it turned out to be too specific and inconvenient. Our work also revealed several minor mistakes in the book. After defining a structure for an outputless finite state machine, we have derived the structures for the transition assigned output machine (Mealy) and state assigned output machine (Moore). The machines are then proved similar, in the sense that for any Mealy (Moore) machine there exists a Moore (Mealy) machine producing essentially the same response for the same input. The rest of work is then done for Mealy machines. Next, we define equivalence of machines, equivalence and $k$-equivalence of states, and characterize a process of constructing for a given Mealy machine, the machine equivalent to it in which no two states are equivalent. The final, minimization theorem states: \begin{quotation} \noindent {\bf Theorem 4.5:} Let {\bf M}$_1$ and {\bf M}$_2$ be reduced, connected finite-state machines. Then the state graphs of {\bf M}$_1$ and {\bf M}$_2$ are isomorphic if and only if {\bf M}$_1$ and {\bf M}$_2$ are equivalent. \end{quotation} and it is the last theorem in this article.

This work was partially supported by NSERC Grant OGP9207.

#### MML Identifier: FSM_1

The terminology and notation used in this paper have been introduced in the following articles [15] [7] [19] [2] [17] [12] [9] [18] [16] [14] [20] [4] [6] [5] [8] [3] [1] [13] [11]

#### Contents (PDF format)

1. Preliminaries
2. Definitions and Terminology
3. Mealy and Moore Machines
4. Equivalence of States and Machines
5. The Reduction of a Mealy Machine
6. Machine Isomorphism
7. Reduced and Connected Machines
8. Machine Union
9. The Minimization Theorem

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