Kleene's recursion theorem
WebIn computing terms, Kleene’s s-m-n theorem says that programs can be specialized with respect to partially known arguments, ... and in the case of the recursion theorem, the programs constructed in the standard proofs are extremely inefficient. These results were thus of no computational interest until new methods were recently developed [12 ... WebWe can use the recursion Theorem to prove that f is recursive. Consider the following definition by cases: g(n,0,y)=y +1, g(n,x+1,0) = ϕ univ(n,x,1), g(n,x+1,y+1)=ϕ univ(n,x,ϕ …
Kleene's recursion theorem
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WebApr 23, 2024 · Section 3 provides an overview of computability theory, inclusive of the so-called Recursion Theorem (Section 3.4)—a result which highlights the centrality of recursion to computation in general as well as its relationship to self-reference.
WebKleene’s Recursion Theorem formalises the notion of program self-reference: It says that given a... The present paper explores the interaction between two recursion-theoretic … 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 operators, which are a computable analogue of inductive definitions. An … 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 • "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 • 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
WebThere are some very strong counterexamples to this, which illustrate the importance of so-called acceptable enumerations of partial recursive functions. Joel gave a nice one with total recursive functions, here is another striking counterexample due to Friedberg [Three theorems on recursive enumeration, JSL 23 (1953), 309-316, MR0109125]. WebThe Kleene Fixed Point Theorem (Recursion Theorem) asserts that for every Turing computable total function f(x) there is a xed point nsuch that ’ f(n) = ’ n. This gives the following recursive call as described in [93, pp. 36{38]. Using the Kleene s-m-n-theorem we can de ne a computable function f(x) by specifying ’
WebOct 19, 2015 · In a lecture note by Weber, following statement gives as a corollary of Kleene's recursion theorem: For total computable function f there is infinitely many n s.t. …
WebMar 2, 2024 · Below are two versions of Kleene's recursion theorem. How are they related? Are they equivalent? If not, does one of them (which one?) imply the other? Note that both … sewage engineering companyWebThe present paper explores the interaction between two recursion-theoretic notions: program self-reference and learning partial recursive functions in the limit. Kleene’s Recursion Theorem formalises the notion of program self-reference: It says that given a partial-recursive function ψ p there is an index e such that the e-th function ψ e ... sewage emptyingWebLemma 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 … the trees in my forestWebThe Kleene Fixed Point Theorem (Recursion Theorem) asserts that for every Turing computable total function f(x) there is a xed point nsuch that ’ f(n) = ’ n. This gives the … the trees in frenchWeb2.2 Kleene’s second recursion theorem Kleene’s second recursion theorem (SRT for short) is an early and very general consequence of the Rogers axioms for computability. It clearly has a flavor of self-application, as it in effect asserts the existence of programs that can refer to their own texts. The statement and proof are short, though the the tree shop ukWebChapter 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 … the trees in my forest bernd heinrichWebThe Recursion Theorem De nitions: A \partial function" is a function f∶N →N∪{⊥} (think of ⊥as \unde ned"). A partial function f is called a \partial recursive" function if it is computed … the tree simply reaped from what it sowed