In the Collatz Conjecture, the question is wether the sequence of iterates will always reach $1$ for all $n>0$ where $n\in\mathbb{Z^+}$. The formal Collatz function consists of the operations: $n/2$ for all even-parities of $n$, or $3n+1$ for all odd-parities.
There are three well known forms of the Collatz function; the formal, the shortcut and the reduced form. A conclusive proof of one form is analogous to the outcome of another form. I.e. the proof that one form is true implies that the other form is true and vice versa.
Starting with $n=12$, the following sequence is produced: $\{12,6,3,10,5,16,8,4,2,1\}$, and has stopping time $=9$.
Thats roughly a definition of the Collatz conjecture.
My question pivots around the idea of using a Markov model, preferably the markov chain to emulate Collatz sequences.
"..a Markov model is a stochastic model used to model randomly changing systems. It is assumed that future states depend only on the current state, not on the events that occurred before it (that is, it assumes the Markov property).
"..the term Markov property refers to the memoryless property of a stochastic process." -Wikipedia
Mathworld also mentions that the problem is connected to ergodic theory and markov chains.
Let $n_{i+1}$ be the future state, $n_{i}$ be the current state, and $n_{i-1}$ be the previous state.
$n_{i-1} \rightarrow n_{i} \rightarrow n_{i+1}$.
It is easy to show that it is impossible to know the previous state, by starting with two previous states $n_{i-1}=7$ and $n_{i-1}=44$.
For $n_{i-1}=7$, we get $n_{i} = 3n_{i-1}+1 = 3\cdot7+1 = 22$.
For $n_{i-1}=44$, we get $n_{i} = n_{i-1}/2 = 44/2 = 22$.
$7 \rightarrow 22 \rightarrow n_{i+1}$.
$44 \rightarrow 22 \rightarrow n_{i+1}$.
So when $n_{i}=22$, we do not know wether $n_{i-1} = 7$ or $n_{i-1} = 44$, refering to the memoryless property in the definition. So, for randomly chosen $n_{i}$ it is impossible to know what $n_{i-1}$ were.
If it is a hidden markov model, in that, "the state is not directly visible, but the output (in the form of data or "token" in the following), dependent on the state, is visible." what can be said about the model?
"A Markov chain is a stochastic model describing a sequence of possible events in which the probability of each event depends only on the state attained in the previous event"
I want to contract the process into the events where $n_{i} \rightarrow n_{i+1}$ or $n_{i-1} \rightarrow n_{i}$, using the binary expansion of $n$.
What kinds of things can be proven using a Markov model, in this case a markov chain on the bit-states of $n_{i}$ and $n_{i+1}$ in the Collatz conjecture ?
If we look at simple discrete dynamical systems in general, where the systems are based on binary states what kinds of things can be proven about it, with markov models?
Markov chains in the main apply to processes wherein at any given state $x_n$, the successor state $x_{n+1}$ takes a range of values, each with probability less than $1$. In the case of the Collatz conjecture, any given state $x_n$ has a single fixed successor $x_{n+1}$ with probability $1$. Therefore there is no probability involved.
In order to introduce probability and the idea of a Markov chain, one has to abstract away the actual numbers in question as Terrence Tao does in his 2011 blog post which I believe we have discussed before, in which he shows that on average one would expect any sequence to fall.
He does this by recognising that, in the absence of any knowledge of the specific sequence in question, one can deduce that $x_n$ will be odd half of the time and even the other half. Then multiplying each odd number by $\approx3/2$ and dividing each even by $2$ we get that on average the sequence will fall. This is the example of what can be done with the idea of a Markov chain.
Unfortunately, this says little about what any individual sequence will do - since (as he alludes to in that post) $\Bbb Z_2$ comes with its own Haar measure and $\Bbb Z$ is a measure zero subset of $\Bbb Z_2$ - and therefore any subset of $\Bbb Z$ could diverge irrespective of that the overall probabilistic statement might be.
Another way of understanding how abstracting away the specific numbers in question renders us unable to draw comprehensive conclusions, is that by abstracting them away, one must first introduce the assumption that the behaviour of $x_{n+1}$ is independent of $x_{n}$. In actual fact, if we know $x_{n}$ then the behaviour of $x_{n+1}$ is NOT a random variable and $x_{n+1}$ isn't a random number even or odd with probability half. Rather it is a determined number which is either even or odd with probability one.
It is also important (and I think you are in danger of not doing this) to recognise that $x_n$ being independent of $x_{n+1}$ is different to $x_{n+1}$ being independent of $x_n$.