Topic
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.
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TL;DR: It is shown that every recursively enumerable chain code picture language is described by a picture-unambiguous context-sensitive language.
6 citations
TL;DR: A characterization of the recursively enumerable languages in terms of the iterated deletion operation is given, which solves two open problems posed by Ito and Silva on the closure properties and decidability of iterated delete.
6 citations
TL;DR: It is shown that there is a countably infinite antichain of prime ideals of recursively enumerable degrees that solves a generalized form of Post's problem.
6 citations
TL;DR: Godel's second incompleteness theorem is generalized by showing that if the set of axioms of a theory T ⊇ PA is not+1-definable and T isn-sound, then T dose not prove the sentencen-Sound(T) that expresses then-soundness of T.
Abstract: Godel's second incompleteness theorem is generalized by showing that if the set of axioms of a theory T ⊇ PA isn+1-definable and T isn-sound, then T dose not prove the sentencen-Sound(T) that expresses then-soundness of T. The optimal- ity of the generalization is shown by presenting an+1-definable (indeed a complete �n+1-definable) andn−1-sound theory T such that PA ⊆ T andn−1-Sound(T) is provable in T. It is also proved that no recursively enumerable and �1-sound theory of arithmetic, even very weak theories which do not contain Robinson's Arithmetic, can prove its own �1-soundness.
6 citations
TL;DR: It is shown that intersection types are capable of precisely characterizing both notions of termination inside a single system of types: the probability of convergence of any λ-term can be underapproximated by its type, while the underlying derivation's weight gives a lower bound to the term’s expected number of steps to normal form.
Abstract: Randomized higher-order computation can be seen as being captured by a lambda calculus endowed with a single algebraic operation, namely a construct for binary probabilistic choice. What matters about such computations is the probability of obtaining any given result, rather than the possibility or the necessity of obtaining it, like in (non)deterministic computation. Termination, arguably the simplest kind of reachability problem, can be spelled out in at least two ways, depending on whether it talks about the probability of convergence or about the expected evaluation time, the second one providing a stronger guarantee. In this paper, we show that intersection types are capable of precisely characterizing both notions of termination inside a single system of types: the probability of convergence of any lambda-term can be underapproximated by its type, while the underlying derivation's weight gives a lower bound to the term's expected number of steps to normal form. Noticeably, both approximations are tight -- not only soundness but also completeness holds. The crucial ingredient is non-idempotency, without which it would be impossible to reason on the expected number of reduction steps which are necessary to completely evaluate any term. Besides, the kind of approximation we obtain is proved to be optimal recursion theoretically: no recursively enumerable formal system can do better than that.
6 citations