Kleene's recursion theorem
In computability theory, Kleene's recursion theorems are a pair of fundamental results about the application of computable functions to their own descriptions. The theorems were first proved by Stephen Kleene in 1938 and appear in his 1952 book Introduction to Metamathematics. A related theorem, which … See more Given a function $${\displaystyle F}$$, a fixed point of $${\displaystyle F}$$ is an index $${\displaystyle e}$$ such that $${\displaystyle \varphi _{e}\simeq \varphi _{F(e)}}$$. Rogers describes the following result as "a simpler … See more While the second recursion theorem is about fixed points of computable functions, the first recursion theorem is related to fixed points determined by enumeration … See more • Denotational semantics, where another least fixed point theorem is used for the same purpose as the first recursion theorem. • Fixed-point combinators, which are used in lambda calculus for the same purpose as the first recursion theorem. See more • "Recursive Functions" entry by Piergiorgio Odifreddi in the Stanford Encyclopedia of Philosophy, 2012. See more The second recursion theorem is a generalization of Rogers's theorem with a second input in the function. One informal interpretation of the second recursion theorem is that it is possible to construct self-referential programs; see "Application to quines" below. See more In the context of his theory of numberings, Ershov showed that Kleene's recursion theorem holds for any precomplete numbering. A Gödel numbering is a precomplete … See more • Jockusch, C. G.; Lerman, M.; Soare, R.I.; Solovay, R.M. (1989). "Recursively enumerable sets modulo iterated jumps and extensions of Arslanov's completeness criterion". The Journal of Symbolic Logic. 54 (4): 1288–1323. doi: See more WebKleene uses the theorem in the very next page to prove that there is a largest initial segment of the countable ordinals which can be given “constructive nota-tions”, in the first …
Kleene's recursion theorem
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WebThe second half-century of recursive function theory is marked by the introduction of such a characterization, in a number of equivalent versions. At the beginning of the 1930's, no overview was possible on the most fundamental problems of the foundations of mathematics without this step. WebChapter 7: Kleene’s Theorem Transition Graph Regular Expression Algorithm (and proof) 1. Add (if necessary) a unique start state without incoming edges and a unique final state …
WebMar 24, 2024 · Kleene's Recursion Theorem Let denote the recursive function of variables with Gödel number , where (1) is normally omitted. Then if is a partial recursive function, … WebRecursion Theory In recursion theory one of basic notions is the notion of a recursively enumerable set – a set whose elements can be arranged in a computable sequence. From: Studies in Logic and the Foundations of Mathematics, 1999 View all Topics Add to Mendeley About this page Handbook of Computability Theory
WebLemma 2.3. Let r be a regular expression. Then r √ if and only if ε ∈ L(r). Lemma 2.4. Let r ∈ R (Σ)be a regular expression over Σ, a ∈ Σ, and x ∈ Σ∗.Then ax ∈ L(r)if Both lemmas may be proved using strong induction on the size of regular expression r. WebKleene Theorem • A language L over Σis regular iff there exists an FA that accepts L. 1. If L is regular there exists an FA M such that L = L(M) 2. For any FA, M, L(M) is regular L(M), the language accepted by the FA can be expressed as a regular expression. Proving Kleene Theorem • Approach – Define 2 variants of the Finite Automata
WebIn the mathematical areas of order and lattice theory, the Kleene fixed-point theorem, named after American mathematician Stephen Cole Kleene, states the following: Kleene Fixed …
WebMar 24, 2024 · Kleene's s-m-n Theorem A theorem, also called the iteration theorem, that makes use of the lambda notation introduced by Church. Let denote the recursive … ffpyplayer输出设备http://www.people.cs.uchicago.edu/~soare/History/handbook.pdf dennis whiteWebSep 1, 1999 · From this it follows that if intuitionistic logic is consistent, then (P ∨ ¬P) is not a theorem if P is prime. Kleene [1945, 1952] proved that intuitionistic first-order ... , including Beth's tableaus, Rasiowa and Sikorski's topological models, formulas-as-types, Kleene's recursive realizabilities, and the Kleene and Aczel slashes. ... dennis white fisherman