Closure Under Reversal of Languages over Infinite Alphabets

It is shown that languages definable by weak pebble automata are not closed under reversal. For the proof, we establish a kind of periodicity of an automaton's computation over a specific set of words. The periodicity is partly due to the finiteness of the automaton description and partly due to the word's structure. Using such a periodicity we can find a word such that during the automaton's run on it there are two different, yet indistinguishable, configurations. This enables us to remove a part of that word without affecting acceptance. Choosing an appropriate language leads us to the desired result.