Pointless topology

In computability theory, primitive recursive functions are a class of functions that are defined using primitive recursion and composition as central operations and are a strict subset of the total µ-recursive functions (µ-recursive functions are also called partial recursive). Primitive recursive functions form an important building block on the way to a full formalization of computability. These functions are also important in proof theory. The term was coined by Rózsa Péter.

Most of the functions normally studied in number theory are primitive recursive. For example: addition, division, factorial, exponential and the nth prime are all primitive recursive. So are many approximations to real-valued functions. In fact, it is difficult to devise a computable function that is not primitive recursive, although some are known (see the section on Limitations below). The set of primitive recursive functions is known as PR in complexity theory.

Every primitive recursive function is a general recursive function.

Definition

The primitive recursive functions are among the number-theoretic functions, which are functions from the natural numbers (nonnegative integers) {0, 1, 2, ...} to the natural numbers. These functions take n arguments for some natural number n and are called n-ary.

The basic primitive recursive functions are given by these axioms:

1. Constant function: The 0-ary constant function 0 is primitive recursive.
2. Successor function: The 1-ary successor function S, which returns the successor of its argument (see Peano postulates), is primitive recursive. That is, S(k) = k + 1.
3. Projection function: For every n≥1 and each i with 1≤i≤n, the n-ary projection function P_iⁿ, which returns its i-th argument, is primitive recursive.

More complex primitive recursive functions can be obtained by applying the operations given by these axioms:

1. Composition: Given f, a k-ary primitive recursive function, and k m-ary primitive recursive functions g₁,...,g_k, the composition of f with g₁,...,g_k, i.e. the m-ary function $h(x_1,\dots ,x_m)=f(g_1(x_1,\dots ,x_m),\dots ,g_k(x_1,\dots ,x_m))$ is primitive recursive.
2. Primitive recursion: Given f, a k-ary primitive recursive function, and g, a (k+2)-ary primitive recursive function, the (k+1)-ary function h is defined as the primitive recursion of f and g, i.e. the function h is primitive recursive when

   $h(0,x_1,\dots ,x_k)=f(x_1,\dots ,x_k)$ and

   $h(S(y),x_1,\dots ,x_k)=g(y,h(y,x_1,\dots ,x_k),x_1,\dots ,x_k).$

The primitive recursive functions are the basic functions and those obtained from the basic functions by applying these operations a finite number of times.

Role of the projection functions

The projection functions can be used to avoid the apparent rigidity in terms of the arity of the functions above; by using compositions with various projection functions, it is possible to pass a subset of the arguments of one function to another function. For example, if g and h are 2-ary primitive recursive functions then

$f(a,b,c)=g(h(c,a),h(a,b))$

is also primitive recursive. One formal definition using projection functions is

$f(a,b,c)=g(h(P_3^3(a,b,c),P_1^3(a,b,c)),h(P_1^3(a,b,c),P_2^3(a,b,c))).$

Converting predicates to numeric functions

In some settings it is natural to consider primitive recursive functions that take as inputs tuples that mix numbers with truth values { t= true, f=false }, or that produce truth values as outputs (see Kleene [1952 pp. 226–227]). This can be accomplished by identifying the truth values with numbers in any fixed manner. For example, it is common to identify the truth value t with the number 1 and the truth value f with the number 0. Once this identification has been made, the characteristic function of a set A, which literally returns 1 or 0, can be viewed as a predicate that tells whether a number is in the set A. Such an identification of predicates with numeric functions will be assumed for the remainder of this article.

Computer language definition

An example of a primitive recursive programming language is one that contains basic arithmetic operators (e.g. + and -, or ADD and SUBTRACT), conditionals and comparison (IF-THEN, EQUALS, LESS-THAN), and bounded loops, such as the basic for loop, where there is a known or calculable upper bound to all loops (FOR i FROM 1 to n). No control structures of greater generality, such as while loops or IF-THEN plus GOTO, are admitted in a primitive recursive language. Douglas Hofstadter's Bloop in Gödel, Escher, Bach is one such. Adding unbounded loops (WHILE, GOTO) makes the language partially recursive, or Turing-complete; Floop is such, as are almost all real-world computer languages.

Arbitrary computer programs, or Turing machines, cannot in general be analyzed to see if they halt or not (the halting problem). However, all primitive recursive functions halt. This is not a contradiction; primitive recursive programs are a non-arbitrary subset of all possible programs, constructed specifically to be analyzable.

Examples

Most number-theoretic functions definable using recursion on a single variable are primitive recursive. Basic examples include the addition and "limited subtraction" functions.

Addition

Intuitively, addition can be recursively defined with the rules:

add(0,x)=x,

add(n+1,x)=add(n,x)+1.

To fit this into a strict primitive recursive definition, define:

add(0,x)=P₁¹(x) ,

add(S(n),x)=S(P₂³(n,add(n,x),x)).

Here P₂³ is the projection function that takes 3 arguments and returns the second one.

P₁¹ is simply the identity function; its inclusion is required by the definition of the primitive recursion operator above; it plays the role of f. The composition of S and P₂³, which is primitive recursive, plays the role of g. The term S(n) refers to "the successor of n".

Subtraction

Because primitive recursive functions use natural numbers rather than integers, and the natural numbers are not closed under subtraction, a limited subtraction function (also called "proper subtraction") is studied in this context. This limited subtraction function sub(a,b) [or b ∸ a] returns b - a if this is nonnegative and returns 0 otherwise.

The predecessor function acts as the opposite of the successor function and is recursively defined by the rules:

pred(0)=0,

pred(n+1)=n.

These rules can be converted into a more formal definition by primitive recursion:

pred(0)=0,

pred(S(n))=P₁²(n, pred(n)).

The limited subtraction function is definable from the predecessor function in a manner analogous to the way addition is defined from successor:

sub(0,x)=P₁¹(x),

sub(S(n),x)=pred(P₂³(n,sub(n,x),x)).

Here sub(a,b) corresponds to b∸a; for the sake of simplicity, the order of the arguments has been switched from the "standard" definition to fit the requirements of primitive recursion. This could easily be rectified using composition with suitable projections.

Other primitive recursive functions include exponentiation and primality testing. Given primitive recursive functions e, f, g, and h, a function that returns the value of g when e≤f and the value of h otherwise is primitive recursive.

Operations on integers and rational numbers

By using Gödel numberings, the primitive recursive functions can be extended to operate on other objects such as integers and rational numbers. If integers are encoded by Gödel numbers in a standard way, the arithmetic operations including addition, subtraction, and multiplication are all primitive recursive. Similarly, if the rationals are represented by Gödel numbers then the field operations are all primitive recursive.

Relationship to recursive functions

The broader class of partial recursive functions is defined by introducing an unbounded search operator. The use of this operator may result in a partial function, that is, a relation with at most one value for each argument, but does not necessarily have any value for any argument (see domain). An equivalent definition states that a partial recursive function is one that can be computed by a Turing machine. A total recursive function is a partial recursive function that is defined for every input.

Every primitive recursive function is total recursive, but not all total recursive functions are primitive recursive. The Ackermann function A(m,n) is a well-known example of a total recursive function that is not primitive recursive. There is a characterization of the primitive recursive functions as a subset of the total recursive functions using the Ackermann function. This characterization states that a function is primitive recursive if and only if there is a natural number m such that the function can be computed by a Turing machine that always halts within A(m,n) or fewer steps, where n is the sum of the arguments of the primitive recursive function.

An important property of the primitive recursive functions is that they are a recursively enumerable subset of the set of all total recursive functions (which is not itself recursively enumerable). This means that there is a single computable function f(e,n) such that:

• For every primitive recursive function g, there is an e such that g(n) = f(e,n) for all n, and
• For every e, the function h(n) = f(e,n) is primitive recursive.

However, the primitive recursive functions are not the largest recursively enumerable set of total computable functions.

Limitations

Primitive recursive functions tend to correspond very closely with our intuition of what a computable function must be. Certainly the initial functions are intuitively computable (in their very simplicity), and the two operations by which one can create new primitive recursive functions are also very straightforward. However the set of primitive recursive functions does not include every possible total computable function — this can be seen with a variant of Cantor's diagonal argument. This argument provides a total computable function that is not primitive recursive. A sketch of the proof is as follows:

The primitive recursive functions of one argument (i.e., unary functions) can be computably enumerated. This enumeration uses the definitions of the primitive recursive functions (which are essentially just expressions with the composition and primitive recursion operations as operators and the basic primitive recursive functions as atoms), and can be assumed to contain every definition once, even though a same function will occur many times on the list (since many definitions define the same function; indeed simply composing by the identity function generates infinitely many definitions of any one primitive recursive function). This means that the Template:Mvar-th definition of a primitive recursive function in this enumeration can be effectively determined from Template:Mvar. Indeed if one uses some Gödel numbering to encode definitions as numbers, then this Template:Mvar-th definition in the list is computed by a primitive recursive function of Template:Mvar. Let Buying, selling and renting HDB and personal residential properties in Singapore are simple and transparent transactions. Although you are not required to engage a real property salesperson (generally often known as a "public listed property developers In singapore agent") to complete these property transactions, chances are you'll think about partaking one if you are not accustomed to the processes concerned.

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Now define the "evaluator function" Template:Mvar with two arguments, by Buying, selling and renting HDB and personal residential properties in Singapore are simple and transparent transactions. Although you are not required to engage a real property salesperson (generally often known as a "public listed property developers In singapore agent") to complete these property transactions, chances are you'll think about partaking one if you are not accustomed to the processes concerned.

Professional agents are readily available once you need to discover an condominium for hire in singapore In some cases, landlords will take into account you more favourably in case your agent comes to them than for those who tried to method them by yourself. You need to be careful, nevertheless, as you resolve in your agent. Ensure that the agent you are contemplating working with is registered with the IEA – Institute of Estate Brokers. Whereas it might sound a hassle to you, will probably be worth it in the end. The IEA works by an ordinary algorithm and regulations, so you'll protect yourself in opposition to probably going with a rogue agent who prices you more than they should for his or her service in finding you an residence for lease in singapore.

There isn't any deal too small. Property agents who are keen to find time for any deal even if the commission is small are the ones you want on your aspect. Additionally they present humbleness and might relate with the typical Singaporean higher. Relentlessly pursuing any deal, calling prospects even without being prompted. Even if they get rejected a hundred times, they still come again for more. These are the property brokers who will find consumers what they need eventually, and who would be the most successful in what they do. 4. Honesty and Integrity

This feature is suitable for you who need to get the tax deductions out of your PIC scheme to your property agency firm. It's endorsed that you visit the correct site for filling this tax return software. This utility must be submitted at the very least yearly to report your whole tax and tax return that you're going to receive in the current accounting 12 months. There may be an official website for this tax filling procedure. Filling this tax return software shouldn't be a tough thing to do for all business homeowners in Singapore.

A wholly owned subsidiary of SLP Worldwide, SLP Realty houses 900 associates to service SLP's fast rising portfolio of residential tasks. Real estate is a human-centric trade. Apart from offering comprehensive coaching applications for our associates, SLP Realty puts equal emphasis on creating human capabilities and creating sturdy teamwork throughout all ranges of our organisational hierarchy. Worldwide Presence At SLP International, our staff of execs is pushed to make sure our shoppers meet their enterprise and investment targets. Under is an inventory of some notable shoppers from completely different industries and markets, who've entrusted their real estate must the expertise of SLP Worldwide.

If you're looking for a real estate or Singapore property agent online, you merely need to belief your instinct. It is because you don't know which agent is sweet and which agent will not be. Carry out research on a number of brokers by looking out the internet. As soon as if you find yourself certain that a selected agent is dependable and trustworthy, you'll be able to choose to utilize his partnerise find you a house in Singapore. More often than not, a property agent is considered to be good if she or he places the contact data on his web site. This is able to imply that the agent does not thoughts you calling them and asking them any questions regarding properties in Singapore. After chatting with them you too can see them of their office after taking an appointment.

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This feature is suitable for you who need to get the tax deductions out of your PIC scheme to your property agency firm. It's endorsed that you visit the correct site for filling this tax return software. This utility must be submitted at the very least yearly to report your whole tax and tax return that you're going to receive in the current accounting 12 months. There may be an official website for this tax filling procedure. Filling this tax return software shouldn't be a tough thing to do for all business homeowners in Singapore.

A wholly owned subsidiary of SLP Worldwide, SLP Realty houses 900 associates to service SLP's fast rising portfolio of residential tasks. Real estate is a human-centric trade. Apart from offering comprehensive coaching applications for our associates, SLP Realty puts equal emphasis on creating human capabilities and creating sturdy teamwork throughout all ranges of our organisational hierarchy. Worldwide Presence At SLP International, our staff of execs is pushed to make sure our shoppers meet their enterprise and investment targets. Under is an inventory of some notable shoppers from completely different industries and markets, who've entrusted their real estate must the expertise of SLP Worldwide.

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Professional agents are readily available once you need to discover an condominium for hire in singapore In some cases, landlords will take into account you more favourably in case your agent comes to them than for those who tried to method them by yourself. You need to be careful, nevertheless, as you resolve in your agent. Ensure that the agent you are contemplating working with is registered with the IEA – Institute of Estate Brokers. Whereas it might sound a hassle to you, will probably be worth it in the end. The IEA works by an ordinary algorithm and regulations, so you'll protect yourself in opposition to probably going with a rogue agent who prices you more than they should for his or her service in finding you an residence for lease in singapore.

There isn't any deal too small. Property agents who are keen to find time for any deal even if the commission is small are the ones you want on your aspect. Additionally they present humbleness and might relate with the typical Singaporean higher. Relentlessly pursuing any deal, calling prospects even without being prompted. Even if they get rejected a hundred times, they still come again for more. These are the property brokers who will find consumers what they need eventually, and who would be the most successful in what they do. 4. Honesty and Integrity

This feature is suitable for you who need to get the tax deductions out of your PIC scheme to your property agency firm. It's endorsed that you visit the correct site for filling this tax return software. This utility must be submitted at the very least yearly to report your whole tax and tax return that you're going to receive in the current accounting 12 months. There may be an official website for this tax filling procedure. Filling this tax return software shouldn't be a tough thing to do for all business homeowners in Singapore.

A wholly owned subsidiary of SLP Worldwide, SLP Realty houses 900 associates to service SLP's fast rising portfolio of residential tasks. Real estate is a human-centric trade. Apart from offering comprehensive coaching applications for our associates, SLP Realty puts equal emphasis on creating human capabilities and creating sturdy teamwork throughout all ranges of our organisational hierarchy. Worldwide Presence At SLP International, our staff of execs is pushed to make sure our shoppers meet their enterprise and investment targets. Under is an inventory of some notable shoppers from completely different industries and markets, who've entrusted their real estate must the expertise of SLP Worldwide.

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Professional agents are readily available once you need to discover an condominium for hire in singapore In some cases, landlords will take into account you more favourably in case your agent comes to them than for those who tried to method them by yourself. You need to be careful, nevertheless, as you resolve in your agent. Ensure that the agent you are contemplating working with is registered with the IEA – Institute of Estate Brokers. Whereas it might sound a hassle to you, will probably be worth it in the end. The IEA works by an ordinary algorithm and regulations, so you'll protect yourself in opposition to probably going with a rogue agent who prices you more than they should for his or her service in finding you an residence for lease in singapore.

There isn't any deal too small. Property agents who are keen to find time for any deal even if the commission is small are the ones you want on your aspect. Additionally they present humbleness and might relate with the typical Singaporean higher. Relentlessly pursuing any deal, calling prospects even without being prompted. Even if they get rejected a hundred times, they still come again for more. These are the property brokers who will find consumers what they need eventually, and who would be the most successful in what they do. 4. Honesty and Integrity

This feature is suitable for you who need to get the tax deductions out of your PIC scheme to your property agency firm. It's endorsed that you visit the correct site for filling this tax return software. This utility must be submitted at the very least yearly to report your whole tax and tax return that you're going to receive in the current accounting 12 months. There may be an official website for this tax filling procedure. Filling this tax return software shouldn't be a tough thing to do for all business homeowners in Singapore.

A wholly owned subsidiary of SLP Worldwide, SLP Realty houses 900 associates to service SLP's fast rising portfolio of residential tasks. Real estate is a human-centric trade. Apart from offering comprehensive coaching applications for our associates, SLP Realty puts equal emphasis on creating human capabilities and creating sturdy teamwork throughout all ranges of our organisational hierarchy. Worldwide Presence At SLP International, our staff of execs is pushed to make sure our shoppers meet their enterprise and investment targets. Under is an inventory of some notable shoppers from completely different industries and markets, who've entrusted their real estate must the expertise of SLP Worldwide.

If you're looking for a real estate or Singapore property agent online, you merely need to belief your instinct. It is because you don't know which agent is sweet and which agent will not be. Carry out research on a number of brokers by looking out the internet. As soon as if you find yourself certain that a selected agent is dependable and trustworthy, you'll be able to choose to utilize his partnerise find you a house in Singapore. More often than not, a property agent is considered to be good if she or he places the contact data on his web site. This is able to imply that the agent does not thoughts you calling them and asking them any questions regarding properties in Singapore. After chatting with them you too can see them of their office after taking an appointment.

Another method by way of which you could find out whether the agent is sweet is by checking the feedback, of the shoppers, on the website. There are various individuals would publish their comments on the web site of the Singapore property agent. You can take a look at these feedback and the see whether it will be clever to hire that specific Singapore property agent. You may even get in contact with the developer immediately. Many Singapore property brokers know the developers and you may confirm the goodwill of the agent by asking the developer. is itself total and computable, so Buying, selling and renting HDB and personal residential properties in Singapore are simple and transparent transactions. Although you are not required to engage a real property salesperson (generally often known as a "public listed property developers In singapore agent") to complete these property transactions, chances are you'll think about partaking one if you are not accustomed to the processes concerned.

Professional agents are readily available once you need to discover an condominium for hire in singapore In some cases, landlords will take into account you more favourably in case your agent comes to them than for those who tried to method them by yourself. You need to be careful, nevertheless, as you resolve in your agent. Ensure that the agent you are contemplating working with is registered with the IEA – Institute of Estate Brokers. Whereas it might sound a hassle to you, will probably be worth it in the end. The IEA works by an ordinary algorithm and regulations, so you'll protect yourself in opposition to probably going with a rogue agent who prices you more than they should for his or her service in finding you an residence for lease in singapore.

There isn't any deal too small. Property agents who are keen to find time for any deal even if the commission is small are the ones you want on your aspect. Additionally they present humbleness and might relate with the typical Singaporean higher. Relentlessly pursuing any deal, calling prospects even without being prompted. Even if they get rejected a hundred times, they still come again for more. These are the property brokers who will find consumers what they need eventually, and who would be the most successful in what they do. 4. Honesty and Integrity

This feature is suitable for you who need to get the tax deductions out of your PIC scheme to your property agency firm. It's endorsed that you visit the correct site for filling this tax return software. This utility must be submitted at the very least yearly to report your whole tax and tax return that you're going to receive in the current accounting 12 months. There may be an official website for this tax filling procedure. Filling this tax return software shouldn't be a tough thing to do for all business homeowners in Singapore.

A wholly owned subsidiary of SLP Worldwide, SLP Realty houses 900 associates to service SLP's fast rising portfolio of residential tasks. Real estate is a human-centric trade. Apart from offering comprehensive coaching applications for our associates, SLP Realty puts equal emphasis on creating human capabilities and creating sturdy teamwork throughout all ranges of our organisational hierarchy. Worldwide Presence At SLP International, our staff of execs is pushed to make sure our shoppers meet their enterprise and investment targets. Under is an inventory of some notable shoppers from completely different industries and markets, who've entrusted their real estate must the expertise of SLP Worldwide.

If you're looking for a real estate or Singapore property agent online, you merely need to belief your instinct. It is because you don't know which agent is sweet and which agent will not be. Carry out research on a number of brokers by looking out the internet. As soon as if you find yourself certain that a selected agent is dependable and trustworthy, you'll be able to choose to utilize his partnerise find you a house in Singapore. More often than not, a property agent is considered to be good if she or he places the contact data on his web site. This is able to imply that the agent does not thoughts you calling them and asking them any questions regarding properties in Singapore. After chatting with them you too can see them of their office after taking an appointment.

Another method by way of which you could find out whether the agent is sweet is by checking the feedback, of the shoppers, on the website. There are various individuals would publish their comments on the web site of the Singapore property agent. You can take a look at these feedback and the see whether it will be clever to hire that specific Singapore property agent. You may even get in contact with the developer immediately. Many Singapore property brokers know the developers and you may confirm the goodwill of the agent by asking the developer. is always defined and effectively computable. However a diagonal argument will show that the function Template:Mvar of two arguments is not primitive recursive.

Suppose Template:Mvar were primitive recursive, then the unary function Template:Mvar defined by Buying, selling and renting HDB and personal residential properties in Singapore are simple and transparent transactions. Although you are not required to engage a real property salesperson (generally often known as a "public listed property developers In singapore agent") to complete these property transactions, chances are you'll think about partaking one if you are not accustomed to the processes concerned.

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This feature is suitable for you who need to get the tax deductions out of your PIC scheme to your property agency firm. It's endorsed that you visit the correct site for filling this tax return software. This utility must be submitted at the very least yearly to report your whole tax and tax return that you're going to receive in the current accounting 12 months. There may be an official website for this tax filling procedure. Filling this tax return software shouldn't be a tough thing to do for all business homeowners in Singapore.

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There isn't any deal too small. Property agents who are keen to find time for any deal even if the commission is small are the ones you want on your aspect. Additionally they present humbleness and might relate with the typical Singaporean higher. Relentlessly pursuing any deal, calling prospects even without being prompted. Even if they get rejected a hundred times, they still come again for more. These are the property brokers who will find consumers what they need eventually, and who would be the most successful in what they do. 4. Honesty and Integrity

This feature is suitable for you who need to get the tax deductions out of your PIC scheme to your property agency firm. It's endorsed that you visit the correct site for filling this tax return software. This utility must be submitted at the very least yearly to report your whole tax and tax return that you're going to receive in the current accounting 12 months. There may be an official website for this tax filling procedure. Filling this tax return software shouldn't be a tough thing to do for all business homeowners in Singapore.

A wholly owned subsidiary of SLP Worldwide, SLP Realty houses 900 associates to service SLP's fast rising portfolio of residential tasks. Real estate is a human-centric trade. Apart from offering comprehensive coaching applications for our associates, SLP Realty puts equal emphasis on creating human capabilities and creating sturdy teamwork throughout all ranges of our organisational hierarchy. Worldwide Presence At SLP International, our staff of execs is pushed to make sure our shoppers meet their enterprise and investment targets. Under is an inventory of some notable shoppers from completely different industries and markets, who've entrusted their real estate must the expertise of SLP Worldwide.

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This argument can be applied to any class of computable (total) functions that can be enumerated in this way, as explained in the article Machines that always halt. Note however that the partial computable functions (those that need not be defined for all arguments) can be explicitly enumerated, for instance by enumerating Turing machine encodings.

Other examples of total recursive but not primitive recursive functions are known:

• The function that takes m to Ackermann(m,m) is a unary total recursive function that is not primitive recursive.
• The Paris–Harrington theorem involves a total recursive function that is not primitive recursive. Because this function is motivated by Ramsey theory, it is sometimes considered more "natural" than the Ackermann function.
• The Sudan function
• The Goodstein function

Some common primitive recursive functions

The following examples and definitions are from Kleene (1952) pp. 223-231 — many appear with proofs. Most also appear with similar names, either as proofs or as examples, in Boolos-Burgess-Jeffrey 2002 pp. 63-70; they add #22 the logarithm lo(x, y) or lg(x, y) depending on the exact derivation.

In the following we observe that primitive recursive functions can be of four types:

1. functions for short: "number-theoretic functions" from { 0, 1, 2, ...} to { 0, 1, 2, ...},
2. predicates: from { 0, 1, 2, ...} to truth values { t =true, f =false },
3. propositional connectives: from truth values { t, f } to truth values { t, f },
4. representing functions: from truth values { t, f } to { 0, 1, 2, ... }. Many times a predicate requires a representing function to convert the predicate's output { t, f } to { 0, 1 } (note the order "t" to "0" and "f" to "1" matches with ~(sig( )) defined below). By definition a function φ(x) is a "representing function" of the predicate P(x) if φ takes only values 0 and 1 and produces 0 when P is true".

In the following the mark " ' ", e.g. a', is the primitive mark meaning "the successor of", usually thought of as " +1", e.g. a +1 =_def a'. The functions 16-21 and #G are of particular interest with respect to converting primitive recursive predicates to, and extracting them from, their "arithmetical" form expressed as Gödel numbers.

1. Addition: a+b
2. Multiplication: a×b
3. Exponentiation: a^b,
4. Factorial a! : 0! = 1, a'! = a!×a'
5. pred(a): Decrement: "predecessor of a" defined as "If a> 0 then a-1 → a_new else 0 → a."
6. Proper subtraction: a ∸ b defined as "If a ≥ b then a-b else 0."
7. Minimum (a₁, ... a_n)
8. Maximum (a₁, ... a_n)
9. Absolute value: | a-b | =_defined (a ∸ b) + (b ∸ a)
10. ~sg(a): NOT[signum(a)]: If a=0 then sg(a)=1 else if a>0 then sg(a)=0
11. sg(a): signum(a): If a=0 then sg(a)=0 else if a>0 then sg(a)=1
12. "b divides a" [ a | b ]: If the remainder ( a, b )=0 then [ a | b ] else b does not divide a "evenly"
13. Remainder ( a, b ): the leftover if b does not divide a "evenly". Also called MOD(a, b)
14. a = b: sg | a - b |
15. a < b: sg( a' ∸ b )
16. Pr(a): a is a prime number Pr(a) =_def a>1 & NOT(Exists c)_1<c<a [ c|a ]
17. P_i: the i+1-st prime number
18. (a)_i : exponent a_i of p_i =_def μx [ _x<a [p_i^x|a & NOT(p_i ^x'|a ]; μx is the minimization operator described in #E below.
19. lh(a): the "length" or number of non-vanishing exponents in a
20. a×b: given the expression of a and b as prime factors then a×b is the product's expression as prime factors
21. lo(x, y): logarithm of x to the base y

In the following, the abbreviation x =_def x_i, ... x_n; subscripts may be applied if the meaning requires.

• #A: A function φ definable explicitly from functions Ψ and constants q₁, ... q_n is primitive recursive in Ψ.
• #B: The finite sum Σ_y<z ψ(x, y) and product Π_y<zψ(x, y) are primitive recursive in ψ.
• #C: A predicate P obtained by substituting functions χ₁,..., χ_m for the respective variables of a predicate Q is primitive recursive in χ₁,..., χ_m, Q.
• #D: The following predicates are primitive recursive in Q and R:

• NOT_Q(x) .
• Q OR R: Q(x) V R(x),
• Q AND R: Q(x) & R(x),
• Q IMPLIES R: Q(x) → R(x)
• Q is equivalent to R: Q(x) ≡ R(x)

• #E: The following predicates are primitive recursive in the predicate R:

• (Ey)_y<z R(x, y) where (Ey)_y<z denotes "there exists at least one y that is less than z such that"
• (y)_y<z R(x, y) where (y)_y<z denotes "for all y less than z it is true that"
• μy_y<z R(x, y). The operator μy_y<z R(x, y) is a bounded form of the so-called minimization- or mu-operator: Defined as "the least value of y less than z such that R(x, y) is true; or z if there is no such value."

• #F: Definition by cases: The function defined thus, where Q₁, ..., Q_m are mutually exclusive predicates (or "ψ(x) shall have the value given by the first clause that applies), is primitive recursive in φ₁, ..., Q₁, ... Q_m:

φ(x) =

• φ₁(x) if Q₁(x) is true,
• . . . . . . . . . . . . . . . . . . .
• φ_m(x) if Q_m(x) is true
• φ_m+1(x) otherwise

• #G: If φ satisfies the equation:

φ(y,x) = χ(y, NOT-φ(y; x₂, ... x_n ), x₂, ... x_n then φ is primitive recursive in χ. 'So, in a sense the knowledge of the value NOT-φ(y; x_{2 to n} ) of the course-of-values function is equivalent to the knowledge of the sequence of values φ(0,x_{2 to n}), ..., φ(y-1,x_{2 to n}) of the original function."

Additional primitive recursive forms

Some additional forms of recursion also define functions that are in fact primitive recursive. Definitions in these forms may be easier to find or more natural for reading or writing. Course-of-values recursion defines primitive recursive functions. Some forms of mutual recursion also define primitive recursive functions.

The functions that can be programmed in the LOOP programming language are exactly the primitive recursive functions. This gives a different characterization of the power of these functions. The main limitation of the LOOP language, compared to a Turing-complete language, is that in the LOOP language the number of times that each loop will run is specified before the loop begins to run.

Finitism and consistency results

The primitive recursive functions are closely related to mathematical finitism, and are used in several contexts in mathematical logic where a particularly constructive system is desired. Primitive recursive arithmetic (PRA), a formal axiom system for the natural numbers and the primitive recursive functions on them, is often used for this purpose.

PRA is much weaker than Peano arithmetic, which is not a finitistic system. Nevertheless, many results in number theory and in proof theory can be proved in PRA. For example, Gödel's incompleteness theorem can be formalized into PRA, giving the following theorem:

If T is a theory of arithmetic satisfying certain hypotheses, with Gödel sentence G_T, then PRA proves the implication Con(T)→G_T.

Similarly, many of the syntactic results in proof theory can be proved in PRA, which implies that there are primitive recursive functions that carry out the corresponding syntactic transformations of proofs.

In proof theory and set theory, there is an interest in finitistic consistency proofs, that is, consistency proofs that themselves are finitistically acceptable. Such a proof establishes that the consistency of a theory T implies the consistency of a theory S by producing a primitive recursive function that can transform any proof of an inconsistency from S into a proof of an inconsistency from T. One sufficient condition for a consistency proof to be finitistic is the ability to formalize it in PRA. For example, many consistency results in set theory that are obtained by forcing can be recast as syntactic proofs that can be formalized in PRA.

See also

• Course-of-values recursion
• Grzegorczyk hierarchy
• Machine that always halts
• Recursion (computer science)
• Primitive recursive functional
• Double recursion

References

• Brainerd, W.S., Landweber, L.H. (1974), Theory of Computation, Wiley, ISBN 0-471-09585-0
• Robert I. Soare, Recursively Enumerable Sets and Degrees, Springer-Verlag, 1987. ISBN 0-387-15299-7
• Stephen Kleene (1952) Introduction to Metamathematics, North-Holland Publishing Company, New York, 11th reprint 1971: (2nd edition notes added on 6th reprint). In Chapter XI. General Recursive Functions §57
• George Boolos, John Burgess, Richard Jeffrey (2002), Computability and Logic: Fourth Edition, Cambridge University Press, Cambridge, UK. Cf pp. 70–71.
• Robert I. Soare 1995 Computability and Recursion http://www.people.cs.uchicago.edu/~soare/History/compute.pdf
• Daniel Severin 2008, Unary primitive recursive functions, J. Symbolic Logic Volume 73, Issue 4, pp. 1122–1138 arXiv projecteuclid

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