In computability theory, a primitive recursive function is, roughly speaking, a function that can be computed by a computer program whose loops are all "for" loops (that is, an upper bound of the number of iterations of every loop is fixed before entering the loop). Primitive recursive functions form a strict subset of those general recursive functions that are also total functions.
The importance of primitive recursive functions lies in the fact that most computable functions that are studied in number theory (and more generally in mathematics) are primitive recursive. For example, addition and division, the factorial and exponential function, and the function which returns the nth prime are all primitive recursive. In fact, for showing that a computable function is primitive recursive, it suffices to show that its time complexity is bounded above by a primitive recursive function of the input size. It is hence not particularly easy to devise a computable function that is not primitive recursive; some examples are shown in section below.
The set of primitive recursive functions is known as PR in computational complexity theory.
Definition
A primitive recursive function takes a fixed number of arguments, each a natural number (nonnegative integer: {0, 1, 2, ...}), and returns a natural number. If it takes n arguments, it is called n-ary.
The basic primitive recursive functions are given by these axioms:
More complex primitive recursive functions can be obtained by applying the operations given by these axioms:
The primitive recursive functions are the basic functions and those obtained from the basic functions by applying these operations a finite number of times.
Primitive-recursiveness of vector-valued functions A (vector-valued) function is primitive recursive if it can be written as
where each component is a (scalar-valued) primitive recursive function..
Examples
Addition
A definition of the 2-ary function , to compute the sum of its arguments, can be obtained using the primitive recursion operator . To this end, the well-known equations
are "rephrased in primitive recursive function terminology": In the definition of , the first equation suggests to choose to obtain ; the second equation suggests to choose to obtain . Therefore, the addition function can be defined as . As a computation example,
Doubling
Given , the 1-ary function doubles its argument,
Multiplication
In a similar way as addition, multiplication can be defined by . This reproduces the well-known multiplication equations:
and
Predecessor
The predecessor function acts as the "opposite" of the successor function and is recursively defined by the rules and . A primitive recursive definition is . As a computation example,
Truncated subtraction
The limited subtraction function (also called "monus", and denoted "") is definable from the predecessor function. It satisfies the equations
Since the recursion runs over the second argument, we begin with a primitive recursive definition of the reversed subtraction, . Its recursion then runs over the first argument, so its primitive recursive definition can be obtained, similar to addition, as . To get rid of the reversed argument order, then define . As a computation example,
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 (that is, for true, and for false), or that produce truth values as outputs. 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 with the number , and the truth value with the number . Once this identification has been made, the characteristic function of a set , which always returns or , can be viewed as a predicate that tells whether a number is in the set . Such an identification of predicates with numeric functions will be assumed for the remainder of this article.
Predicate "Is zero"
As an example for a primitive recursive predicate, the 1-ary function shall be defined such that if , and otherwise. This can be achieved by defining . Then , and e.g. .
Predicate "Less or equal"
Using the property , the 2-ary function can be defined by . Then if , and otherwise. As a computation example,
Predicate "Greater or equal"
Once a definition of is obtained, the converse predicate can be defined as . Then, is true (more precisely: has value 1) if and only if .
If-then-else
The 3-ary if-then-else operator known from programming languages can be defined by . Then, for arbitrary ,
and
That is, returns the then-part () if the if-part () is true, and the else-part () otherwise.
Junctors
Based on the function, it is easy to define logical junctors. For example, defining , one obtains , that is, is true if and only if both and are true (logical conjunction of and ).
Similarly, and lead to appropriate definitions of disjunction and negation: and .
Equality predicate
Using the above functions , and , the definition implements the equality predicate. In fact, is true if and only if equals .
Similarly, the definition implements the predicate "less-than", and implements "greater-than".
Other operations on natural numbers
Exponentiation and primality testing are primitive recursive. Given primitive recursive functions , , , and , a function that returns the value of when and the value of 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.
Some common primitive recursive functions The following examples and definitions are from . Many appear with proofs. Most also appear with similar names, either as proofs or as examples, in they add the logarithm lo(x, y) or lg(x, y) depending on the exact derivation.
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 16–20 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.
Addition: a+b Multiplication: a×b Exponentiation: ab Factorial a! : 0! = 1, a'! = a!×a' pred(a): (Predecessor or decrement): If a > 0 then a−1 else 0 Proper subtraction a ∸ b: If a ≥ b then a−b else 0 Minimum(a1, ... an) Maximum(a1, ... an) Absolute difference: | a−b | =def (a ∸ b) + (b ∸ a) ~sg(a): NOT[signum(a)]: If a=0 then 1 else 0 sg(a): signum(a): If a=0 then 0 else 1 a | b: (a divides b): If b=k×a for some k then 0 else 1 Remainder(a, b): the leftover if b does not divide a "evenly". Also called MOD(a, b) a = b: sg | a − b | (Kleene's convention was to represent true by 0 and false by 1; presently, especially in computers, the most common convention is the reverse, namely to represent true by 1 and false by 0, which amounts to changing sg into ~sg here and in the next item) a < b: sg( a' ∸ b ) Pr(a): a is a prime number Pr(a) =def a>1 & NOT(Exists c)1<c<a [ c|a ] pi: the i+1th prime number (a)i: exponent of pi in a: the unique x such that pix|a & NOT(pix'|a) lh(a): the "length" or number of non-vanishing exponents in a lo(a, b): (logarithm of a to base b): If a, b > 1 then the greatest x such that bx | a else 0
In the following, the abbreviation x =def x1, ... xn; subscripts may be applied if the meaning requires.
#A: A function φ definable explicitly from functions Ψ and constants q1, ... qn is primitive recursive in Ψ. #B: The finite sum Σy<z ψ(x, y) and product Πy<zψ(x, y) are primitive recursive in ψ. #C: A predicate P obtained by substituting functions χ1,..., χm for the respective variables of a predicate Q is primitive recursive in χ1,..., χm, 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" μyy<z R(x, y). The operator μyy<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, ..., Qm are mutually exclusive predicates (or "ψ(x) shall have the value given by the first clause that applies), is primitive recursive in φ1, ..., Q1, ... Qm: φ(x) = φ1(x) if Q1(x) is true, . . . . . . . . . . . . . . . . . . . φm(x) if Qm(x) is true φm+1(x) otherwise #G: If φ satisfies the equation: φ(y,x) = χ(y, COURSE-φ(y; x2, ... xn ), x2, ... xn then φ is primitive recursive in χ. The value COURSE-φ(y; x2 to n ) of the course-of-values function encodes the sequence of values φ(0,x2 to n), ..., φ(y-1,x2 to n) of the original function.
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 which has at most one value for each argument, but which may fail to have a value at some arguments (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 (in fact, provable total), 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.This follows from the facts that the functions of this form are the most quickly growing primitive recursive functions, and that a function is primitive recursive if and only if its time complexity is bounded by a primitive recursive function. For the former, see . For the latter, see
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 recursive function f(m,n) that enumerates the primitive recursive functions, namely: For every unary primitive recursive function g, there is an m such that g(n) = f(m,n) for all n, and For every m, the function h(n) = f(m,n) is primitive recursive. Primitive recursive functions with two or more arguments can be encoded as unary primitive recursive functions by using a primitive recursive pairing function with two primitive recursive inverses.