Abstract
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.
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Notes
- 1.
The proof of the general case can be found in [7], and, hopefully, will also appear elsewhere.
- 2.
It has been shown in [8] that alternating non-deterministic and deterministic one-way wPA have the same expressive power.
- 3.
We omit the subscript \({\varvec{w}}\) of \(\vdash \), if it is clear from the context.
- 4.
That is, pebble \(i + 1\) is placed at the position of pebble i, whereas in the strong PA model this pebble is placed at the beginning of the input word, i.e., at the leftmost position.
- 5.
By definition, \(P_1 = \emptyset \) and, therefore, is redundant.
- 6.
Note that in [7] the pebbles are placed in the reversed order, i.e., the computation start with pebble k and pebble i is placed after pebble \(i + 1\), \(i = 1,\ldots , k - 1\).
- 7.
Recall that we identify \(\theta \) with the tuple of its values and, by the observation in the previous section, we omit the P-component of transitions.
- 8.
As usual, \(\equiv _{\ell _2}\) is the congruence modulo \(l_2\).
- 9.
The automaton enters configurations \([1,t,(p_1^\prime )]\) and \([1,t,(p_1^{\prime \prime })]\) after pebble 2 falls down from the right end of the input entering state t.
- 10.
Thus, both \({\varvec{u}}^\prime {\varvec{u}}^{\prime \prime } {\varvec{v}}\ \mathrm{and}\ {\varvec{x}}\) are in \(L_{{{\texttt {diff}}}}\) and \([ {\varvec{u}}^\prime {\varvec{u}}^{\prime \prime } {\varvec{v}}] = [ {\varvec{x}}].\)
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Genkin, D., Kaminski, M., Peterfreund, L. (2018). Closure Under Reversal of Languages over Infinite Alphabets. In: Fomin, F., Podolskii, V. (eds) Computer Science – Theory and Applications. CSR 2018. Lecture Notes in Computer Science(), vol 10846. Springer, Cham. https://doi.org/10.1007/978-3-319-90530-3_13
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