8. Suppose that S = {A, B, ..., F} is a set of states and I = O = {0, 1} are the input and output alphabets for the Mealy machine described by the transition table below. ? A | C | E | 1 | 0 B | D | F | 0 | 1 C | B | D | 1 | 1 D | A | B | 0 | 0 E | C | A | 0 | 1 F | E | F | 1 | 1 (a) Construct a state diagram for this Mealy machine. (Layout the states so that there is no need to have transition arrows crossing each other.) (b) Find the output string corresponding to the input string '0110010100101', when starting in state A. In which state does the machine finish? (c) Design a Moore machine which produces the same output as the Mealy machine in the previous question. In removing any inaccessible states, discuss which state or states are valid as the initial state; choosing that which keeps the total number of states to a minimum, if this is possible.
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Let M = ({q0, q1, q2, q3, q4, q5}, {a, b, n}, q0, fs, {q2, q3, q4}) be the Deterministic Finite Automaton (DFA) with state transition function, fs defined as follows: f(q0, a) = q1 f(q0, b) = q0 f(q0, n) = q2 f(q1, a) = q2 f(q1, b) = q1 f(q1, n) = q1 f(q2, a) = q3 f(q2, b) = q2 f(q2, n) = q4 f(q3, a) = q1 f(q3, b) = q3 f(q3, n) = q4 f(q4, a) = q3 f(q4, b) = q5 f(q4, n) = q5 f(q5, a) = q4 f(q5, b) = q3 f(q5, n) = q5
Sri K.
In Exercises $58-62$ we introduce a technique for constructing a deterministic finite-state machine equivalent to a given deterministic finite-state machine with the least number of states possible. Suppose that $M=\left(S, I, f, s_{0}, F\right)$ is a finite-state automaton and that $k$ is a non negative integer. Let $R_{k}$ be the relation on the set $S$ of states of $M$ such that $s R_{k} t$ if and only if for every input string $x$ with $l(x) \leq k$ [where $l(x)$ is the length of $x,$ as usual $], f(s, x)$ and $f(t, x)$ are both final states or both not final states. Furthermore, let $R_{*}$ be the relation on the set of states of $M$ such that $s R_{*} t$ if and only if for every input string $x,$ regardless of length, $f(s, x)$ and $f(t, x)$ are both final states or both not final states. a) Show that if $M$ is a finite-state automaton, then the quotient automaton $\overline{M}$ recognizes the same language as $M .$ b) Show that if $M$ is a finite-state automaton with the property that for every state $s$ of $M$ there is a string $x \in I^{*}$ such that $f\left(s_{0}, x\right)=s,$ then the quotient automaton $\overline{M}$ has the minimum number of states of any finite-state automaton equivalent to $M .$
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