  We construct recursively-presented finitely-generated torsion groups which
have bounded torsion and whose word problem is conjunctive equivalent (in
particular positive and Turing equivalent) to a given recursively enumerable
set. These groups can be interpreted as groups of finite state machines or as
subgroups of topological full groups, on effective subshifts over other torsion
groups. We define a recursion-theoretic property of a set of natural numbers,
called impredictability. It roughly states that a Turing machine can enumerate
numbers such that every Turing machine occasionally incorrectly guesses (by
either halting or not) whether they are in the set, even given an oracle for a
prefix of the set. We prove that impredictable recursively enumerable sets
exist. Combining these constructions and slightly adapting a result of [Salo
and T\"orm\"a, 2017], we obtain that four-headed group-walking finite-state
automata can define strictly more subshifts than three-headed automata on a
group containing a copy of the integers, confirming a conjecture of [Salo and
T\"orm\"a, 2017]. These are the first examples of groups where four heads are
better than three, and they show the maximal height of a finite head hierarchy
is indeed four.
