What is the complexity of the set of closed boolean tautologies assuming well formed formulae?
The answer to all three questions is negative. Consider e.g. the prefix version (3), and assume for contradiction that it is decidable by a (wlog deterministic) automaton $A$ with $m$ states. Consider the strings

$$\begin{align}
u_k&amp;={+}^{m-k}{*}{+}^k,\\
v_l&amp;=1^{l+1}0^{m-l+1}.
\end{align}$$

Notice that $u_kv_l$ is a well-formed Boolean sentence for every $0\le k\le m$, $0\le l\le m+1$, and it is easy to see that $u_kv_l$ evaluates to $1$ iff $k&lt;l$. By the pigeonhole principle, there are $0\le k&lt;l\le m$ such that $u_k$ and $u_l$ land in the same state of $A$, thus $u_kv_l$ and $u_lv_l$ are both accepted or both rejected. However, the former evaluates to $1$, and the latter to $0$.

For infix formulas (questions 1,2), one can use in a similar way formulas of the form

$$(0+((0+\cdots((0+((1+((0+\cdots((0+0)*1))*1))\dots*1))*0))*1))\dots*1)).$$

The argument in fact shows that any deterministic automaton evaluating well-formed Boolean sentences of length $n$ must have size $\Omega(n)$, or in other words, any online algorithm for evaluation of such sentences needs space at least $\Omega(\log n)$. This bound is tight: as shown by Buss, it is possible to evaluate Boolean sentences in infix notation in online space $O(\log d)$, where $d$ is the depth of the formula (i.e., maximum nesting of brackets). (The paper advertises it as an online $O(\log\log n)$-space algorithm for evaluation of balanced formulas, using the fact that these have depth $d=\log n$.)

On the other hand, as also shown by Buss, Boolean sentence evaluation (in either prefix or infix notation) can be done in ALOGTIME, aka uniform $\mathrm{NC}^1$, and it is in fact complete for this class under DLOGTIME many-one reductions. Since regular languages with unsolvable syntactic monoid (for example, $\bigl((a|b)^3(ab^*a|b)\bigr)^*$) are also $\mathrm{NC}^1$-complete, it follows that one can reduce Boolean sentence evaluation in DLOGTIME to a regular language.