MathDB

Suppose we have some proteins that each protein is a sequence of 7 "AMINO-ACIDS"

A,\ B,\ C,\ H,\ F,\ N

. For example

AFHNNNHAFFC

is a protein. There are some steps that in each step an amino-acid will change to another one. For example with the step

NA\rightarrow N

the protein

BANANA

will cahnge to

BANNA

("in Persian means workman"). We have a set of allowed steps that each protein can change with these steps. For example with the set of steps: \\ 1)\ AA\longrightarrow A\\ 2)\ AB\longrightarrow BA\\ 3)\ A\longrightarrow \mbox{null} Protein

ABBAABA

will change like this:

\\ ABB\underline{AA}BA\\ \underline{AB}BABA\\ B\underline{AB}ABA\\ BB\underline{AA}BA\\ BB\underline{AB}A\\ BBB\underline{AA}\\ BBB\underline{A}\\ BBB

You see after finite steps this protein will finish it steps. Set of allowed steps that for them there exist a protein that may have infinitely many steps is dangerous. Which of the following allowed sets are dangerous? a)

NO\longrightarrow OONN

\left\{\begin{array}{c}HHCC\longrightarrow HCCH\\ CC\longrightarrow CH\end{array}\right.

c) Design a set of allowed steps that change

\underbrace{AA\dots A}_{n}\longrightarrow\underbrace{BB\dots B}_{2^{n}}

d) Design a set of allowed steps that change

\underbrace{A\dots A}_{n}\underbrace{B\dots B}_{m}\longrightarrow\underbrace{CC\dots C}_{mn}

You see from

c

and

d

that we acn calculate the functions

F(n)=2^{n}

and

G(M,N)=mn

with these steps. Find some other calculatable functions with these steps. (It has some extra mark.)

Problem Statement