NBC News Scripts
WBAP-TV (Television station : Fort Worth, Tex.)
1954-12-31
Search results
73 records were found.
We prove that the determinacy of Gale-Stewart games whose winning sets are accepted by real-time 1-counter Büchi automata is equivalent to the determinacy of (effective) analytic Gale-Stewart games which is known to be a large cardinal assumption. We show also that the determinacy of Wadge games between two players in charge of omega-languages accepted by 1-counter Büchi automata is equivalent to the (effective) analytic Wadge determinacy. Using some results of set theory we prove that one can effectively construct a 1-counter Büchi automaton A and a Büchi automaton B such that: (1) There exists a model of ZFC in which Player 2 has a winning strategy in the Wadge game W(L(A), L(B)); (2) There exists a model of ZFC in which the Wadge game W(L(A), L(B)) is not determined. Moreover these are the only two possibilities, i.e. there are no m...
We prove that the determinacy of Gale-Stewart games whose winning sets are accepted by real-time 1-counter Büchi automata is equivalent to the determinacy of (effective) analytic Gale-Stewart games which is known to be a large cardinal assumption. We show also that the determinacy of Wadge games between two players in charge of omega-languages accepted by 1-counter Büchi automata is equivalent to the (effective) analytic Wadge determinacy. Using some results of set theory we prove that one can effectively construct a 1-counter Büchi automaton A and a Büchi automaton B such that: (1) There exists a model of ZFC in which Player 2 has a winning strategy in the Wadge game W(L(A), L(B)); (2) There exists a model of ZFC in which the Wadge game W(L(A), L(B)) is not determined. Moreover these are the only two possibilities, i.e. there are no m...
We prove that the determinacy of Gale-Stewart games whose winning sets are accepted by real-time 1-counter Büchi automata is equivalent to the determinacy of (effective) analytic Gale-Stewart games which is known to be a large cardinal assumption. We show also that the determinacy of Wadge games between two players in charge of omega-languages accepted by 1-counter Büchi automata is equivalent to the (effective) analytic Wadge determinacy. Using some results of set theory we prove that one can effectively construct a 1-counter Büchi automaton A and a Büchi automaton B such that: (1) There exists a model of ZFC in which Player 2 has a winning strategy in the Wadge game W(L(A), L(B)); (2) There exists a model of ZFC in which the Wadge game W(L(A), L(B)) is not determined. Moreover these are the only two possibilities, i.e. there are no m...
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 Büchi 1-counter automaton $A$. We prove the following surprising result: there exists a 1-counter Büchi 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 ...
It was noticed by Harel in [Har86] that ''one can define $\Sigma_1^1$-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 $\omega$-language is $\Sigma_1^1$-complete, hence located beyond the arithmetical hierarchy and highly undecidable. We infer from this result that it is $\Pi_1^1$-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 $\omega$-rational functions realized by finite state Büchi transducers. Indeed Prieur proved in [Pri01, Pri02] that it is decidable whether a given $\omega$-rational function is continuous, while we show here that it is $\Sigma_1^1$-complete to determine whether a given ...
Altenbernd, Thomas and Wöhrle have considered in [ATW02] acceptance of languages of infinite two-dimensional words (infinite pictures) by finite tiling systems, with the usual acceptance conditions, such as the Büchi and Muller ones, firstly used for infinite words. Many classical decision problems are studied in formal language theory and in automata theory and arise now naturally about recognizable languages of infinite pictures. We first review in this paper some recent results of [Fin09b] where we gave the exact degree of numerous undecidable problems for Büchi-recognizable languages of infinite pictures, which are actually located at the first or at the second level of the analytical hierarchy, and ''highly undecidable". Then we prove here some more (high) undecidability results. We first show that it is $\Pi_2^1$-complete to dete...
We give in this paper additional answers to questions of Lescow and Thomas [Logical Specifications of Infinite Computations, In:"A Decade of Concurrency", Springer LNCS 803 (1994), 583-621], proving new topological properties of omega context free languages : there exist some omega-CFL which are non Borel sets. And one cannot decide whether an omega-CFL is a Borel set. We give also an answer to questions of Niwinski and Simonnet about omega powers of finitary languages, giving an example of a finitary context free language L such that L^omega is not a Borel set. Then we prove some recursive analogues to preceding properties: in particular one cannot decide whether an omega-CFL is an arithmetical set.
This paper is a continuation of the study of topological properties of omega context free languages (omega-CFL). We proved before that the class of omega-CFL exhausts the hierarchy of Borel sets of finite rank, and that there exist some omega-CFL which are analytic but non Borel sets. We prove here that there exist some omega context free languages which are Borel sets of infinite (but not finite) rank, giving additional answer to questions of Lescow and Thomas [Logical Specifications of Infinite Computations, In:"A Decade of Concurrency", Springer LNCS 803 (1994), 583-621].
We prove in this paper that there exists some infinitary rational relations which are Sigma^0_3-complete Borel sets and some others which are Pi^0_3-complete. This implies that there exists some infinitary rational relations which are Delta^0_4-sets but not (Sigma^0_3U Pi^0_3)-sets. These results give additional answers to questions of Simonnet and of Lescow and Thomas.
A dictionary is a set of finite words over some finite alphabet X. The omega-power of a dictionary V is the set of infinite words obtained by infinite concatenation of words in V. Lecomte studied in [Omega-powers and descriptive set theory, JSL 2005] the complexity of the set of dictionaries whose associated omega-powers have a given complexity. In particular, he considered the sets $W({\bf\Si}^0_{k})$ (respectively, $W({\bf\Pi}^0_{k})$, $W({\bf\Delta}_1^1)$) of dictionaries $V \subseteq 2^\star$ whose omega-powers are ${\bf\Si}^0_{k}$-sets (respectively, ${\bf\Pi}^0_{k}$-sets, Borel sets). In this paper we first establish a new relation between the sets $W({\bf\Sigma}^0_{2})$ and $W({\bf\Delta}_1^1)$, showing that the set $W({\bf\Delta}_1^1)$ is ``more complex" than the set $W({\bf\Sigma}^0_{2})$. As an application we improve the lowe...
