Infinitary Rational Relations Research Articles

Infinite games specified by 2-tape automata

We prove that the determinacy of Gale–Stewart games whose winning sets are infinitary rational relations accepted by 2-tape Büchi automata is equivalent to the determinacy of (effective) analytic Gale–Stewart games which is known to be a large cardinal assumption. Then we prove that winning strategies, when they exist, can be very complex, i.e. highly non-effective, in these games. We prove the same results for Gale–Stewart games with winning sets accepted by real-time 1-counter Büchi automata, then extending previous results obtained about these games.1.There exists a 2-tape Büchi automaton (respectively, a real-time 1-counter Büchi automaton) A such that: (a) there is a model of ZFC in which Player 1 has a winning strategy σ in the game G(L(A)) but σ cannot be recursive and not even in the class (Σ21∪Π21); (b) there is a model of ZFC in which the game G(L(A)) is not determined.2.There exists a 2-tape Büchi automaton (respectively, a real-time 1-counter Büchi automaton) A such that L(A) is an arithmetical Δ30-set and Player 2 has a winning strategy in the game G(L(A)) but has no hyperarithmetical winning strategies in this game.3.There exists a recursive sequence of 2-tape Büchi automata (respectively, of real-time 1-counter Büchi automata) An, n≥1, such that all games G(L(An)) are determined, but for which it is Π21-complete hence highly undecidable to determine whether Player 1 has a winning strategy in the game G(L(An)). Then we consider the strengths of determinacy for these games, and we prove the following results.1.There exists a 2-tape Büchi automaton (respectively, a real-time 1-counter Büchi automaton) A♯ such that the game G(A♯) is determined iff the effective analytic determinacy holds.2.There is a transfinite sequence of 2-tape Büchi automata (respectively, of real-time 1-counter Büchi automata) (Aα)α<ω1CK, indexed by recursive ordinals, such that the games G(L(Aα)) have strictly increasing strengths of determinacy. We also show that the determinacy of Wadge games between two players in charge of infinitary rational relations accepted by 2-tape Büchi automata is equivalent to the (effective) analytic Wadge determinacy and thus also equivalent to the (effective) analytic determinacy.

THREE APPLICATIONS TO RATIONAL RELATIONS OF THE HIGH UNDECIDABILITY OF THE INFINITE POST CORRESPONDENCE PROBLEM IN A REGULAR ω-LANGUAGE

It was noticed by Harel in [Har86] that "one can define [Formula: see text]-complete versions of the well-known Post Correspondence Problem". We first give a complete proof of this result, showing that the infinite Post Correspondence Problem in a regular ω-language is [Formula: see text]-complete, hence located beyond the arithmetical hierarchy and highly undecidable. We infer from this result that it is [Formula: see text]-complete to determine whether two given infinitary rational relations are disjoint. Then we prove that there is an amazing gap between two decision problems about ω-rational functions realized by finite state Büchi transducers. Indeed Prieur proved in [Pri01, Pri02] that it is decidable whether a given ω-rational function is continuous, while we show here that it is [Formula: see text]-complete to determine whether a given ω-rational function has at least one point of continuity. Next we prove that it is [Formula: see text]-complete to determine whether the continuity set of a given ω-rational function is ω-regular. This gives the exact complexity of two problems which were shown to be undecidable in [CFS08].

Some problems in automata theory which depend on the models of set theory

We prove that some fairly basic questions on automata reading infinite words depend on the models of the axiomatic system ZFC. It is known that there are only three possibilities for the cardinality of the complement of an omega-language $L(A)$ accepted by a B\"uchi 1-counter automaton $A$. We prove the following surprising result: there exists a 1-counter B\"uchi automaton $A$ such that the cardinality of the complement $L(A)^-$ of the omega-language $L(A)$ is not determined by ZFC: (1). There is a model $V_1$ of ZFC in which $L(A)^-$ is countable. (2). There is a model $V_2$ of ZFC in which $L(A)^-$ has cardinal $2^{\aleph_0}$. (3). There is a model $V_3$ of ZFC in which $L(A)^-$ has cardinal $\aleph_1$ with $\aleph_0<\aleph_1<2^{\aleph_0}$. We prove a very similar result for the complement of an infinitary rational relation accepted by a 2-tape B\"uchi automaton $B$. As a corollary, this proves that the Continuum Hypothesis may be not satisfied for complements of 1-counter omega-languages and for complements of infinitary rational relations accepted by 2-tape B\"uchi automata. We infer from the proof of the above results that basic decision problems about 1-counter omega-languages or infinitary rational relations are actually located at the third level of the analytical hierarchy. In particular, the problem to determine whether the complement of a 1-counter omega-language (respectively, infinitary rational relation) is countable is in $\Sigma_3^1 \setminus (\Pi_2^1 \cup \Sigma_2^1)$. This is rather surprising if compared to the fact that it is decidable whether an infinitary rational relation is countable (respectively, uncountable).

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 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.

Wadge Degrees of Infinitary Rational Relations

We show that, from the topological point of view, 2-tape Buchi automata have the same accepting power as Turing machines equipped with a Buchi acceptance condition. The Borel and the Wadge hierarchies of the class RAT_omega of infinitary rational relations accepted by 2-tape Buchi automata are equal to the Borel and the Wadge hierarchies of omega-languages accepted by real-time Buchi 1-counter automata or by Buchi Turing machines. In particular, for every non-null recursive ordinal $\alpha$, there exist some $\Sigma^0_\alpha$-complete and some $\Pi^0_\alpha$-complete infinitary rational relations. And the supremum of the set of Borel ranks of infinitary rational relations is an ordinal $\gamma^1_2$ which is strictly greater than the first non-recursive ordinal $\omega_1^{CK}$. This very surprising result gives answers to questions of Simonnet (1992) and of Lescow and Thomas (1988,1994).

On infinite real trace rational languages of maximum topological complexity

We consider the set ℝω(Γ, D) of infinite real traces, over a dependence alphabet (Γ,D) with no isolated letter, equipped with the topology induced by the prefix metric. We prove that all rational languages of infinite real traces are analytic sets. We also reprove that there exist some rational languages of infinite real traces that are analytic but non-Borel sets; in fact, these sets are even Σ 11 -complete, hence have maximum possible topological complexity. For this purpose, we give an example of a Σ 11 -complete language that is fundamentally different from the known example of a Σ 11 -complete infinitary rational relation given by Finkel (2003). Bibliography: 35 titles.

Topology and ambiguity in ω-context free languages

We study the links between the topological complexity of an omega context free language and its degree of ambiguity. In particular, using known facts from classical descriptive set theory, we prove that non Borel omega context free languages which are recognized by Buchi pushdown automata have a maximum degree of ambiguity. This result implies that degrees of ambiguity are really not preserved by the operation of taking the omega power of a finitary context free language. We prove also that taking the adherence or the delta-limit of a finitary language preserves neither unambiguity nor inherent ambiguity. On the other side we show that methods used in the study of omega context free languages can also be applied to study the notion of ambiguity in infinitary rational relations accepted by Buchi 2-tape automata and we get first results in that direction.

Undecidability of Topological and Arithmetical Properties of Infinitary Rational Relations

We prove that for every countable ordinal α one cannot decide whether a given infinitary rational relation is in the Borel class (respectively ). Furthermore one cannot decide whether a given infinitary rational relation is a Borel set or a -complete set. We prove some recursive analogues to these properties. In particular one cannot decide whether an infinitary rational relation is an arithmetical set. We then deduce from the proof of these results some other ones, like: one cannot decide whether the complement of an infinitary rational relation is also an infinitary rational relation.

On the Topological Complexity of Infinitary Rational Relations

We prove in this paper that there exists some infinitary rational relations which are analytic but non Borel sets, giving an answer to a question of Simonnet [20].

Relations rationnelles infinitaires

In this paper, we build a theory of infinitary rational relations, which is an extension of the theory of finitary rational relations, i. e. sets ofK-vectors of finite words which are recognized by finite automata withK tapes, and at the same time an extension of the theory of infinitary rational languages, i.e., sets of finite and infinite words which are recognized by finite automata (the condition of recognizability of an infinite word is that its reading by the automaton must go through a state, wich belongs to a designated subset, infinitly time).