“…The leaves of the run may be labeled by states of a strictly lower stratum or negations thereof, or by states of the same stratum whose transition function is tautological, i.e., by some (q , w) such that ∆(q , λ(w)) has ∅, ∅ as a satisfying assignment. Intuitively, if we disallow negation in transitions, our automata amount to the alternating two-way automata used by [22], with the simplification that they do not need parity acceptance conditions (because we only work with finite trees), and that they are isotropic: the run for each positive child state of an internal node may start indifferently on any neighbor of w in the tree (its parent, a child, or w itself), no matter the direction. (Note, however, that the run for negated child states must start on w itself.…”