The original problem is involved in the first-order Markov process, I transform it into a matrix problem. There are two $M \times M $ matrix ` $$ T_A = \left( \begin{array}{lllll} a_0& 1-a_0 & 0 & 0&...\\ 0 &a_1& 1-a_1&0&...\\ 0&0&a_2&1-a_2&...\\ 0&0&0&a_3&...\\ \cdot ..&..&..&..&1 \end{array} \right ). $$
$$ T_B = \left( \begin{array}{lllll} 1 & 0 & 0 & 0&...\\ b_1 &1-b_1& 0&0&...\\ 0&b_1&1-b_2&0&...\\ 0&0&b_3&1-b_3&...\\ \cdot ..&..&..&..&1-b_M \end{array} \right ). $$ where $a_i,b_i \in (0,1)$
Also Two $M \times 1$ vectors P and D
where $P=[0,0,..0,1,0...0]'$ that is only one of the value is $1$, the rest is $0$.
$D=[1,1,1..1,0,...0,0,0]'$ that is there is some cutoff, before is $1$, after is $0$.
$T_A,T_B,D,P$ are all unknown to us.
Given a sequence consist of A and B, for example [AABBAB], we define a function $$ F([s_1,s_2...s_t]) =P' T_{s_1} T_{s_2}..T_{s_t} D $$
where $s_i \in \{A,B\}$ and $t$ can be any nature number. For example $$F([AABBAB])=P' T_A T_AT_BT_BT_AT_B D $$ $$F(empty)=P'D$$
The outcome of F is in $\mathbb{R}$.
Now my question is suppose we only know F, that is to say for any given sequence, we know the outcome of F,( give AABBAB, we know the value of F(AABBAB) ), can we pin down a unique solution of $T_A,T_B,D,P$ ? The size of the matrix is also unknown.
Since there are so many polynomial equations we can construct, I expect we have a unique solution, but I don't know how to prove it at all. Please help Thanks.