If we use the notation where when we say:
$$M = M(G)$$
We mean to say that $M$ is a automata with states and alphabet elements of $G$.
From here, I am posed this question (Abstract Algebra by Pinter):
Describe $M(Z_4)$, give the table of its next-state function as well as its state diagram.
I cannot understand how I am to do this, since
- I'm not given an operation (although I suspect it is assumed to be multiplication)
- If I do multiplication over $Z_4$, it gives me a infinite state diagram, which I do not quite understand.
Any help?
I’ll write $\nu(s,i)=t$ to mean that if the automaton is in state $s$, and the input is $i$, the next state is $t$. I’ll use $0,1,2$, and $3$ for the elements of the group. If the intended operation is multiplication mod $4$, you want $\nu(s,i)$ to be $s\cdot i$; if, as I suspect, it’s addition mod $4$, you want $\nu(s,i)$ to be $s+i$. Thus, for example, if it’s multiplication, you’ll have $\nu(s,0)=0$ and $\nu(s,1)=s$ for every $s$, while if it’s addition, you’ll have $\nu(s,0)=s$ for every $s$. Can you take it from there?