By Fabrizio Luccio
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Additional resources for An Introduction to the Theory of Automata: Course Held at the Department for Automation and Information July 1971
Namely: Proposition 4. Necessary and sufficient condition for automaton A to be observable, is that A is in minimal form. The validity of Proposition 4 relies upon the very general definition of observability (Definition 10), where no constraint was put on the experiment to be used for determining q~. Bounds on the length and size of such experiment have been implicitly derived by Ginsburg  • Some of his results are summarized in the following proposition: Proposition 5. An automaton any state q i.
An experiment on automaton A is a collection of pairs x~, y~, 1 '~' m such that: * i. ~ ii. II is an input sequence; is the output sequence generated by the application 111 of Xd • although iii. , such a state is unknown. The length of the Iongest sequence • is x~ the length of the experiment. mis the size of the experiment. An experiment of size 1 is a simple experiment. Related to previous definitions of equivalence (Defs. 5 and 7), the notion of experiment has a particular relevance. In fact, we can state that: 1.
Minimization of A is still centered on a merging process for its states. However, a new relation specified in the following definition substitutes the equivalence, for the determination of states that can be merged into a single one. Definition 15. States qi. and qi of automaton A are compatible 6. II and YJ* are 1'den- tical, wherever both are specified. In symbols, compatibility is indicated as Cfi. ~ CfJ Between states of a single automaton, compatibility is a less restrictive relation than inclusion.
An Introduction to the Theory of Automata: Course Held at the Department for Automation and Information July 1971 by Fabrizio Luccio