Turing machine where the next step is determined by the state and the symbols up to the read/write head
Suppose we are given a language $L$ of words of finite length. We will construct machine $M_L$ that runs as the following.



$M_L$ starts at state $q_0$. On input $x$ of finite length that is put to the right of its head, $M_L$ will move right all the way to the end of input, keeping the input and state intact. (This might mean nothing if the input is empty.) When $M_L$ read the first blank symbol, what is written on the tape up to the current point exclusive is exactly the input $x$. (If current point should be included, let $M_L$ go back a cell.) It halts, changing state to 


$q_1$, an accepting state if $x\in L$,
or $q_2$, a rejecting state otherwise.


It is clear that $M_L$ decides $L$.



Well, it is reasonable to say that that $M_L$ does not compute $L$ at all. $M_L$ just knows effectively each and every word in $L$ without any computation. When $L$ is infinite or even undecidable, we are not able to build or truly simulate such a machine within any bounded time by any known physical means and intelligence. Hence, we would not believe those imaginary machines contradict the Church–Turing thesis.



Exercise.  Given a modified type of turning machine where

$$\delta: Q\times \Gamma^* \to Q\times \Gamma \times \{L,R\}$$

where the next step of the machine is determined by the current state and the content of last $\lceil\log_2n\rceil$ cells that has been read by it, where $n$ is the number of steps the machine has run. Can some unrecognized languages be recognized with these types of machine?