Dicyclic group

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.

The two recursion theorems can be applied to construct fixed points of certain operations on computable functions, to generate quines, and to construct functions defined via recursive definitions. The application to construction of a fixed point of any computable function is known as Rogers' theorem and is due to Hartley Rogers, Jr. (Rogers, 1967).

Notation

The statement of the theorems refers to an admissible numbering $\varphi$ of the partial recursive functions, such that the function corresponding to index $e$ is $\varphi _{e}$. In programming terms, $e$ is the program and $\varphi _{e}$ its semantic denotation.

Rogers' fixed-point theorem

Given a function $F$, a fixed point of $F$ is, in this context, an index $e$ such that $\varphi _{e}\simeq \varphi _{F(e)}$; in programming terms, $e$ is semantically equivalent to $F(e)$.

Rogers' fixed-point theorem. If $F$ is (total) computable, it has a fixed point.

This theorem is Theorem I in (Rogers, 1967: §11.2) where it is described as "a simpler version" of Kleene's (second) recursion theorem.

Proof of the fixed-point theorem

The proof uses a particular total computable function $h$, defined as follows. Given a natural number $x$, the function $h$ outputs the index of the partial computable function that performs the following computation:

Given an input $y$, first attempt to compute $\varphi _{x}(x)$. If that computation returns an output $e$, then compute $\varphi _{e}(y)$ and return its value, if any.

Thus, for all $x$, if $\varphi _{x}(x)$ halts, then $\varphi _{h(x)}=\varphi _{\varphi _{x}(x)}$, and if $\varphi _{x}(x)$ does not halt then $\varphi _{h(x)}\,$ does not halt; this is denoted $\varphi _{h(x)}\simeq \varphi _{\varphi _{x}(x)}$. The function $h$ can be constructed from the partial computable function $g(x,y)=\varphi _{\varphi _{x}(x)}(y)$ and the s-m-n theorem: for each $x$, $h(x)$ is the index of a program which computes the function $y\mapsto g(x,y)$.

To complete the proof, let $F$ be any total computable function, and construct $h$ as above. Let $e$ be an index of the composition $F\circ h$, which is a total computable function. Then $\varphi _{h(e)}\simeq \varphi _{\varphi _{e}(e)}$ by the definition of $h$. But, because $e$ is an index of $F\circ h$, $\varphi _{e}(e)=(F\circ h)(e)$, and thus $\varphi _{\varphi _{e}(e)}\simeq \varphi _{F(h(e))}$. By the transitivity of $\simeq$, this means $\varphi _{h(e)}\simeq \varphi _{F(h(e))}$. Hence $\varphi _{n}\simeq \varphi _{F(n)}$ for $n=h(e)$.

Fixed-point free functions

A function $F$ such that $\varphi _{e}\not \simeq \varphi _{F(e)}$ for all $e$ is called fixed point free. The fixed-point theorem shows that no computable function is fixed point free, but there are many non-computable fixed-point free functions. Arslanov's completeness criterion"Potter or Ceramic Artist Truman Bedell from Rexton, has interests which include ceramics, best property developers in singapore developers in singapore and scrabble. Was especially enthused after visiting Alejandro de Humboldt National Park. states that the only recursively enumerable Turing degree that computes a fixed point free function is 0´, the degree of the halting problem.

Kleene's second recursion theorem

An informal interpretation of the second recursion theorem is that self-referential programs are acceptable.

The second recursion theorem. For any partial recursive function $Q(x,y)$ there is an index $p$ such that $\varphi _{p}\simeq \lambda y.Q(p,y)$.

This can be used as follows. Suppose that we have a self-referential program, namely one that evaluates a computable function $Q$ of two arguments where the first is supposed to be the index of that very program, and the second represents input. By the theorem, we have a program $p$ that does exactly that. Note that $p$ only has $y$ as input; it does not have to be supplied with its own index but satisfies the "self referential" equation by construction.

The theorem can be proved from Rogers' theorem by letting $F(p)$ be a function such that $\varphi _{F(p)}(y)=Q(p,y)$ (a construction described by the S-m-n theorem ). One can then verify that a fixed-point of this $F$ is an index $p$ as required.

The theorem is constructive in the sense that a fixed computable function maps an index for Q into the index p.

Comparison to Rogers' theorem

Kleene's second recursion theorem and Rogers' theorem can both be proved, rather simply, from each other (Jones, 1997: p. 229-230). However, a direct proof of Kleene's theorem (Kleene 1952: p. 352-353) does not make use of a universal program, which means that the theorem holds for certain subrecursive programming systems that do not have a universal program.

Application to elimination of recursion

Suppose that $g$ and $h$ are total computable functions that are used in a recursive definition for a function $f$:

$f(0,y)\simeq g(y),\,$

$f(x+1,y)\simeq h(f(x,y),x,y),\,$

The second recursion theorem can be used to show that such equations define a computable function, where the notion of computability does not have to allow, a priori, for recursive definitions (for example, it may be defined by μ-recursion, or by Turing machines). This recursive definition can be converted into a computable function $\varphi _{F}(e,x,y)$ that assumes $e$ is an index to itself, to simulate recursion:

$\varphi _{F}(e,0,y)\simeq g(y),\,$

$\varphi _{F}(e,x+1,y)\simeq h(\varphi _{e}(x,y),x,y).\,$

The recursion theorem establishes the existence of a computable function $\varphi _{f}$ such that $\varphi _{f}(x,y)\simeq \varphi _{F}(f,x,y)$. Thus $f$ satisfies the given recursive definition.

Application to quines

A classic example using the second recursion theorem is the function $Q(x,y)=x$. The corresponding index $p$ in this case yields a computable function that outputs its own index when applied to any value (Cutland 1980: p. 204). When expressed as computer programs, such indices are known as quines.

The following example in Lisp illustrates how the $p$ in the corollary can be effectively produced from the function $Q$. The function s11 in the code is the function of that name produced by the S-m-n theorem.

Q can be changed to any two-argument function.

(setq Q '(lambda (x y) x))
(setq s11 '(lambda (f x) (list 'lambda '(y) (list f x 'y))))
(setq n (list 'lambda '(x y) (list Q (list s11 'x 'x) 'y)))
(setq p (eval (list s11 n n)))

The results of the following expressions should be the same. $\varphi$ p(nil)

(eval (list p nil))

Q(p, nil)

(eval (list Q p nil))

Reflexive programming

Reflexive, or reflective, programming refers to the usage of self-reference in programs. Jones (1997) presents a view of the second recursion theorem based on a reflexive language. It is shown that the reflexive language defined is not stronger than a language without reflection (because an interpreter for the reflexive language can be implemented without using reflection); then, it is shown that the recursion theorem is almost trivial in the reflexive language.

The first recursion theorem

The first recursion theorem is related to fixed points determined by enumeration operators, which are a computable analogue of inductive definitions. An enumeration operator is a set of pairs (A,n) where A is a (code for a) finite set of numbers and n is a single natural number. Often, n will be viewed as a code for an ordered pair of natural numbers, particularly when functions are defined via enumeration operators. Enumeration operators are of central importance in the study of enumeration reducibility.

Each enumeration operator Φ determines a function from sets of naturals to sets of naturals given by

$\Phi (X)=\{n\mid \exists A\subseteq X[(A,n)\in \Phi ]\}.$

A recursive operator is an enumeration operator that, when given the graph of a partial recursive function, always returns the graph of a partial recursive function.

A fixed point of an enumeration operator Φ is a set F such that Φ(F) = F. The first enumeration theorem shows that fixed points can be effectively obtained if the enumeration operator itself is computable.

First recursion theorem. The following statements hold.

1. For any computable enumeration operator Φ there is a recursively enumerable set F such that Φ(F) = F and F is the smallest set with this property.
2. For any recursive operator Ψ there is a partial computable function φ such that Ψ(φ) = φ and φ is the smallest partial computable function with this property.

Example

Like the second recursion theorem, the first recursion theorem can be used to obtain functions satisfying systems of recursion equations. To apply the first recursion theorem, the recursion equations must first be recast as a recursive operator.

Consider the recursion equations for the factorial function f:

f(0) = 1,

f(n+1) = (n + 1) · f(n).

The corresponding recursive operator Φ will have information that tells how to get to the next value of f from the previous value. However, the recursive operator will actually define the graph of f. First, Φ will contain the pair $(\varnothing ,(0,1))$. This indicates that f(0) is unequivocally 1, and thus the pair (0,1) is in the graph of f.

Next, for each n and m, Φ will contain the pair $(\{(n,m)\},(n+1,(n+1)\cdot m))$. This indicates that, if f(n) is m, then f(n + 1) is (n + 1)m, so that the pair (n + 1, (n + 1)m) is in the graph of f. Unlike the base case f(0) = 1, the recursive operator requires some information about f(n) before it defines a value of f(n + 1).

The first recursion theorem (in particular, part 1) states that there is a set F such that Φ(F) = F. The set F will consist entirely of ordered pairs of natural numbers, and will be the graph of the factorial function f, as desired.

The restriction to recursion equations that can be recast as recursive operators ensures that the recursion equations actually define a least fixed point. For example, consider the set of recursion equations:

g(0) = 1,

g(n + 1) = 1,

g(2n) = 0.

There is no function g satisfying these equations, because they imply g(2) = 1 and also imply g(2) = 0. Thus there is no fixed point g satisfying these recursion equations. It is possible to make an enumeration operator corresponding to these equations, but it will not be a recursive operator.

Proof sketch for the first recursion theorem

The proof of part 1 of the first recursion theorem is obtained by iterating the enumeration operator Φ beginning with the empty set. First, a sequence F_k is constructed, for $k=0,1,\ldots$. Let F₀ be the empty set. Proceeding inductively, for each k, let F_{k + 1} be $F_{k}\cup \Phi (F_{k})$. Finally, F is taken to be $\bigcup F_{k}$. The remainder of the proof consists of a verification that F is recursively enumerable and is the least fixed point of Φ. The sequence F_k used in this proof corresponds to the Kleene chain in the proof of the Kleene fixed-point theorem.

The second part of the first recursion theorem follows from the first part. The assumption that Φ is a recursive operator is used to show that the fixed point of Φ is the graph of a partial function. The key point is that if the fixed point F is not the graph of a function, then there is some k such that F_k is not the graph of a function.

Comparison to the second recursion theorem

Compared to the second recursion theorem, the first recursion theorem produces a stronger conclusion but only when narrower hypotheses are satisfied. Rogers [1967] uses the term weak recursion theorem for the first recursion theorem and strong recursion theorem for the second recursion theorem.

One difference between the first and second recursion theorems is that the fixed points obtained by the first recursion theorem are guaranteed to be least fixed points, while those obtained from the second recursion theorem may not be least fixed points.

A second difference is that the first recursion theorem only applies to systems of equations that can be recast as recursive operators. This restriction is similar to the restriction to continuous operators in the Kleene fixed-point theorem of order theory. The second recursion theorem can be applied to any total recursive function.

Generalized theorem by A.I. Maltsev

Anatoly Maltsev proved a generalized version of the recursion theorem for any set with a precomplete numberingPotter or Ceramic Artist Truman Bedell from Rexton, has interests which include ceramics, best property developers in singapore developers in singapore and scrabble. Was especially enthused after visiting Alejandro de Humboldt National Park.. A Gödel numbering is a precomplete numbering on the set of computable functions so the generalized theorem yields the Kleene recursion theorem as a special case.

Given a precomplete numbering $\nu$ then for any partial computable function $f$ with two parameters there exists a total computable function $t$ with one parameter such that

$\forall n\in \mathbb {N} :\nu \circ f(n,t(n))=\nu \circ t(n).$

See also

• 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.

References

• Cutland, N.J., Computability: An introduction to recursive function theory, Cambridge University Press, 1980. ISBN 0-521-29465-7
• One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting
  
  In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang
  
  Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
  
  Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.
  
  A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
  
  The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
  
  There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
• Kleene, Stephen, Introduction to Metamathematics, North-Holland, 1952. ISBN 0-7204-2103-9
• Rogers, H. Theory of Recursive Functions and Effective Computability, MIT Press, 1967. ISBN 0-262-68052-1; ISBN 0-07-053522-1
• Jones, N.D.J., Computability and Complexity from a programming perspective, MIT Press, 1997. ISBN 0-262-10064-9

External links

• Template:SEP.