# Theory of Automata, Abstraction and Applications

## Abstract

We introduce computational models, such as sequential machines and automata, using the category theory. In particular, we introduce a generalized theorem which states the existence of the most efficient finite state automaton, called the minimal realization. First, we introduce set theoretical elementary models using sets and functions. We then consider a category of sequential machines which is an abstract model of finite automata. In the category theory, we consider several properties of compositions of morphisms. When we look at the category of sets and functions, we describe properties using equations of compositions of functions. Since the theory of category is a general theory, we can have many concrete properties from a general theorem by assigning it to specific categories such as sets and functions, linear space and linear transformations, etc.

### Keywords

Sequential machine Automaton Category theory### References

- 1.M.A. Arbib, E.G. Manes, Machines in a category, an expository introduction. SIAM Rev.
**16**, 163–192 (1974)MathSciNetCrossRefMATHGoogle Scholar - 2.M.A. Arbib, E.G. Manes,
*Arrows, Structures, and Functors: The Categorical Imperative*(Academic Press, New York, 1975)MATHGoogle Scholar - 3.C.E. Shannon, J. McCarthy (eds.),
*Automata Studies*(Princeton University Press, Princeton, 1956)MATHGoogle Scholar - 4.T.L. Booth,
*Sequential Machines and Automata Theory*(John Wiley & Sons, New York, 1967)MATHGoogle Scholar - 5.E.F. Moore (ed.), Sequential Machines: Selected Papers (Addison-Wesley, Reading, 1964).Google Scholar
- 6.M.O. Rabin, D. Scott, Finite automata and their decision problems. IBM J.
**3**, 114–125 (1959)MathSciNetCrossRefGoogle Scholar - 7.A-H. Dediu, C. Martin-Vide (ed.), in Language and Automata Theory and Applications, 6th International Conference, LATA2012, Lecture Notes in Computer Science, vol. 7183 (2012).Google Scholar
- 8.E. Formenti (ed.), in Proceedings of 18th International Workshop on Cellular Automata and Discrete Complex Systems (Automata2012) and 3rd International Symposium Journées Automates Cellulaires (JAC2012), Electronic Proceedings in Theoretical Computer Science, vol. 90 (2012).Google Scholar
- 9.G.C. Sirakoulis, S. Bandini (ed.), in Cellular Automata, 10th International Conference on Cellular Automata for Research and Industry, ACRI2012, Lecture Notes in Computer Science, vol. 7495 (2012).Google Scholar
- 10.S. Mac Lane, Categories for the Working Mathematicians. (Springer, New York, 1972).Google Scholar