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.
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Notes
- 1.
\(e:X \rightarrow Z\) is a surjection if for any element \(z \in Z\), there exists an element \(x \in X\) such that \(e(x)=z\). \(m:Z \rightarrow Y\) is an injection if \(m(z_1)\not =m(z_2)\) for any elements \(z_1, z_2 \in Z\) and \(z_1\not =z_2\).
- 2.
We follow the definition of the Moore type sequential machine. The Mealy type sequential machine uses an output function \(\lambda :Q \times X \rightarrow Y\) instead of \(\beta \). These two models are equivalent. If we omit the output for an initial state, they are mutually transformable. We note that there is no output of a Mealy-type sequential machine for an initial state. We can define a sequential machine as a pentad without an initial state.
- 3.
Note that \(f_m(q):X^*\rightarrow Y\) (\(q \in Q\)).
- 4.
Note that the total response map is a dynamorphism. \(\tau _M:(\varSigma ^*, \mu _0 1) \rightarrow (Y^{\varSigma ^*},LY)\).
References
M.A. Arbib, E.G. Manes, Machines in a category, an expository introduction. SIAM Rev. 16, 163–192 (1974)
M.A. Arbib, E.G. Manes, Arrows, Structures, and Functors: The Categorical Imperative (Academic Press, New York, 1975)
C.E. Shannon, J. McCarthy (eds.), Automata Studies (Princeton University Press, Princeton, 1956)
T.L. Booth, Sequential Machines and Automata Theory (John Wiley & Sons, New York, 1967)
E.F. Moore (ed.), Sequential Machines: Selected Papers (Addison-Wesley, Reading, 1964).
M.O. Rabin, D. Scott, Finite automata and their decision problems. IBM J. 3, 114–125 (1959)
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).
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).
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).
S. Mac Lane, Categories for the Working Mathematicians. (Springer, New York, 1972).
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Mizoguchi, Y. (2014). Theory of Automata, Abstraction and Applications. In: Nishii, R., et al. A Mathematical Approach to Research Problems of Science and Technology. Mathematics for Industry, vol 5. Springer, Tokyo. https://doi.org/10.1007/978-4-431-55060-0_25
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DOI: https://doi.org/10.1007/978-4-431-55060-0_25
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