15,431 research outputs found

### The Complexity of Infinite Computations In Models of Set Theory

We prove the following surprising result: there exist a 1-counter B\"uchi
automaton and a 2-tape B\"uchi automaton such that the \omega-language of the
first and the infinitary rational relation of the second in one model of ZFC
are \pi_2^0-sets, while in a different model of ZFC both are analytic but non
Borel sets.
This shows that the topological complexity of an \omega-language accepted by
a 1-counter B\"uchi automaton or of an infinitary rational relation accepted by
a 2-tape B\"uchi automaton is not determined by the axiomatic system ZFC.
We show that a similar result holds for the class of languages of infinite
pictures which are recognized by B\"uchi tiling systems.
We infer from the proof of the above results an improvement of the lower
bound of some decision problems recently studied by the author

### Highly Undecidable Problems For Infinite Computations

We show that many classical decision problems about 1-counter
omega-languages, context free omega-languages, or infinitary rational
relations, are $\Pi_2^1$-complete, hence located at the second level of the
analytical hierarchy, and "highly undecidable". In particular, the universality
problem, the inclusion problem, the equivalence problem, the determinizability
problem, the complementability problem, and the unambiguity problem are all
$\Pi_2^1$-complete for context-free omega-languages or for infinitary rational
relations. Topological and arithmetical properties of 1-counter
omega-languages, context free omega-languages, or infinitary rational
relations, are also highly undecidable. These very surprising results provide
the first examples of highly undecidable problems about the behaviour of very
simple finite machines like 1-counter automata or 2-tape automata.Comment: to appear in RAIRO-Theoretical Informatics and Application

### Wadge Degrees of $\omega$-Languages of Petri Nets

We prove that $\omega$-languages of (non-deterministic) Petri nets and
$\omega$-languages of (non-deterministic) Turing machines have the same
topological complexity: the Borel and Wadge hierarchies of the class of
$\omega$-languages of (non-deterministic) Petri nets are equal to the Borel and
Wadge hierarchies of the class of $\omega$-languages of (non-deterministic)
Turing machines which also form the class of effective analytic sets. In
particular, for each non-null recursive ordinal $\alpha < \omega\_1^{{\rm CK}}$ there exist some ${\bf \Sigma}^0\_\alpha$-complete and some ${\bf
\Pi}^0\_\alpha$-complete $\omega$-languages of Petri nets, and the supremum of
the set of Borel ranks of $\omega$-languages of Petri nets is the ordinal
$\gamma\_2^1$, which is strictly greater than the first non-recursive ordinal
$\omega\_1^{{\rm CK}}$. We also prove that there are some ${\bf
\Sigma}\_1^1$-complete, hence non-Borel, $\omega$-languages of Petri nets, and
that it is consistent with ZFC that there exist some $\omega$-languages of
Petri nets which are neither Borel nor ${\bf \Sigma}\_1^1$-complete. This
answers the question of the topological complexity of $\omega$-languages of
(non-deterministic) Petri nets which was left open in [DFR14,FS14].Comment: arXiv admin note: text overlap with arXiv:0712.1359, arXiv:0804.326

### Borel Ranks and Wadge Degrees of Context Free Omega Languages

We show that, from a topological point of view, considering the Borel and the
Wadge hierarchies, 1-counter B\"uchi automata have the same accepting power
than Turing machines equipped with a B\"uchi acceptance condition. In
particular, for every non null recursive ordinal alpha, there exist some
Sigma^0_alpha-complete and some Pi^0_alpha-complete omega context free
languages accepted by 1-counter B\"uchi automata, and the supremum of the set
of Borel ranks of context free omega languages is the ordinal gamma^1_2 which
is strictly greater than the first non recursive ordinal. This very surprising
result gives answers to questions of H. Lescow and W. Thomas [Logical
Specifications of Infinite Computations, In:"A Decade of Concurrency", LNCS
803, Springer, 1994, p. 583-621]

- …