Recursively enumerable language

About: Recursively enumerable language is a research topic. Over the lifetime, 1508 publications have been published within this topic receiving 32382 citations.

The DPRM Theorem in Isabelle (Short Paper).

Abstract: Hilbert's 10th problem asks for an algorithm to tell whether or not a given diophantine equation has a solution over the integers. The non-existence of such an algorithm was shown in 1970 by Yuri Matiyasevich. The key step is known as the DPRM theorem: every recursively enumerable set of natural numbers is Diophantine. We present the formalization of Matiyasevich's proof of the DPRM theorem in Isabelle. To represent recursively enumerable sets in equations, we implement and arithmetize register machines. Using several number-theoretic lemmas, we prove that exponentiation has a diophantine representation. Further, we contribute a small library of number-theoretic implementations of binary digit-wise relations. Finally, we discuss and contribute an is_diophantine predicate. We expect the complete formalization of the DPRM theorem in the near future; at present it is complete except for a minor gap in the arithmetization proofs of register machines and extending the is_diophantine predicate by two binary digit-wise relations.

Axiomatizing provable $n$-provability

Abstract: A formula $\phi$ is called \emph{$n$-provable} in a formal arithmetical theory $S$ if $\phi$ is provable in $S$ together with all true arithmetical $\Pi_{n}$-sentences taken as additional axioms. While in general the set of all $n$-provable formulas, for a fixed $n>0$, is not recursively enumerable, the set of formulas $\phi$ whose $n$-provability is provable in a given r.e.\ metatheory $T$ is r.e. This set is deductively closed and will be, in general, an extension of $S$. We prove that these theories can be naturally axiomatized in terms of progressions of iterated local reflection principles. In particular, the set of provably 1-provable sentences of Peano arithmetic PA can be axiomatized by $\varepsilon_0$ times iterated local reflection schema over PA. Our characterizations yield additional information on the proof-theoretic strength of these theories (w.r.t. various measures of it) and on their axiomatizability. We also study the question of speed-up of proofs and show that in some cases a proof of $n$-provability of a sentence can be much shorter than its proof from iterated reflection principles.

The Role of True Finiteness in the Admissible Recursively Enumerable Degrees

Abstract: Introduction Coding into the R. E. degrees Coding effective successor models A negative result concerning effective successor models A nonembedding result Embedding the 1-3-1 lattice Appendix A. Basics Appendix B. The jump Appendix C. The projectum Appendix D. The admissible collapse Appendix E. Prompt permission Appendix. Bibliography.

Using Pseudo-Stochastic Rational Languages in Probabilistic Grammatical Inference

Abstract: In probabilistic grammatical inference, a usual goal is to infer a good approximation of an unknown distribution P called a stochastic language. The estimate of P stands in some class of probabilistic models such as probabilistic automata (PA). In this paper, we focus on probabilistic models based on multiplicity automata (MA). The stochastic languages generated by MA are called rational stochastic languages; they strictly include stochastic languages generated by PA; they also admit a very concise canonical representation. Despite the fact that this class is not recursively enumerable, it is efficiently identifiable in the limit by using the algorithm DEES, introduced by the authors in a previous paper. However, the identification is not proper and before the convergence of the algorithm, DEES can produce MA that do not define stochastic languages. Nevertheless, it is possible to use these MA to define stochastic languages. We show that they belong to a broader class of rational series, that we call pseudo-stochastic rational languages. The aim of this paper is twofold. First we provide a theoretical study of pseudo-stochastic rational languages, the languages output by DEES, showing for example that this class is decidable within polynomial time. Second, we have carried out a lot of experiments in order to compare DEES to classical inference algorithms such as ALERGIA and MDI. They show that DEES outperforms them in most cases.