Suppose that instead of the usual linear work tape the input is given in a binary tree structure with n leaves and log n depth, the initial position being the root. At every node, we can step to its parent, left child or right child.
I am interested in what "tree-languages" can be accepted by what machines. More specifically, take the language where in every non-leaf we have the gate of a boolean circuit (ie. and or not), in the leaves either a constant or a variable (different for each leaf) and such a "tree-word" is in the language if and only if it has a satisfying assignment. Can this language be decided by a turing machine that has an o(log n) space worktape?
Note that it is accepted by a non-deterministic finite state machine and also by a deterministic turing machine with O(log n) space. Similarly, if all the variables are constant, then it can be decided by a deterministic finite state machine. (If there are no constants, then every formula is satisfiable.)