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splaystyle e f displaystyle f g displaystyle g and h displaystyle h a function that returns the value of g displaystyle g when e f displaystyle e leq f and the value of h displaystyle h otherwise is primitive recursive operations on integers and rational numbers edit 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 edit the following examples and definitions are from kleene 1974 pp 222 231 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 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 a b 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 a 1 a n maximum a 1 a n 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 p i the i 1th prime number a i exponent of p i in a the unique x such that p i x a not p i x 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 b x a else 0 in the following the abbreviation x def x 1 x n subscripts may be applied if the meaning requires a a function φ definable explicitly from functions ψ and constants q 1 q n is primitive recursive in ψ b the finite sum σ y z ψ x y and product π y z ψ x y are primitive recursive in ψ c a predicate p obtained by substituting functions χ 1 χ m for the respective variables of a predicate q is primitive recursive in χ 1 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 q 1 q m are mutually exclusive predicates or ψ x shall have the value given by the first clause that applies is primitive recursive in φ 1 q 1 q m φ x φ 1 x if q 1 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 course φ y x 2 x n x 2 x n then φ is primitive recursive in χ the value course φ y x 2 to n of the course of values function encodes the sequence of values φ 0 x 2 to n φ y 1 x 2 to n of the original function relationship to recursive functions edit 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 9 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 f can be explicitly constructed by iteratively repeating all possible ways of creating primitive recursive functions thus it is provably total one can use a diagonalization argument to show that f is not recursive primitive in itself had it been such so would be h n f n n 1 but if this equals some primitive recursive function there is an m such that h n f m n for all n and then h m f m m leading to contradiction however the set of primitive recursive functions is not the largest recursively enumerable subset of the set of all total recursive functions for example the set of provably total functions in peano arithmetic is also recursively enumerable as one can enumerate all the proofs of the theory while all primitive recursive functions are provably total the converse is not true limitations edit 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 n displaystyle n th definition of a primitive recursive function in this enumeration can be effectively determined from n displaystyle n indeed if one uses some gödel numbering to encode definitions as numbers then this n displaystyle n th definition in the list is computed by a primitive recursive function of n displaystyle n let f n displaystyle f_ n denote the unary primitive recursive function given by this definition now define the evaluator function e v displaystyle ev with two arguments by e v i j f i j displaystyle ev i j f_ i j clearly e v displaystyle ev is total and computable since one can effectively determine the definition of f i displaystyle f_ i and being a primitive recursive function f i displaystyle f_ i is itself total and computable so f i j displaystyle f_ i j is always defined and effectively computable however a diagonal argument will show that the function e v displaystyle ev of two arguments is not primitive recursive suppose e v displaystyle ev were primitive recursive then the unary function g displaystyle g defined by g i s e v i i displaystyle g i s ev i i would also be primitive recursive as it is defined by composition from the successor function and e v displaystyle ev but then g displaystyle g occurs in the enumeration so there is some number n displaystyle n such that g f n displaystyle g f_ n but now g n s e v n n s f n n s g n displaystyle g n s ev n n s f_ n n s g n gives a contradiction this argument can be applied to show that the class of total computable functions cannot be enumerated in this way in particular it shows that total turing machines cannot be enumerated 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 the sudan function the goodstein function variants edit constant functions edit instead of c n k displaystyle c_ n k alternative definitions use just one 0 ary zero function c 0 0 displaystyle c_ 0 0 as a primitive function that always returns zero and build the constant functions from the zero function the successor function and the composition operator citation needed iterative functions edit robinson 10 considered various restrictions of the recursion rule one is the so called iteration rule where the function h does not have access to the parameters x i in this case we may assume without loss of generality that the function g is just the identity as the general case can be obtained by substitution f 0 x x f s y x h y f y x displaystyle begin aligned f 0 x x f s y x h y f y x end aligned he proved that the class of all primitive recursive functions can still be obtained in this way pure recursion edit another restriction considered by robinson 10 is pure recursion where h does not have access to the induction variable y f 0 x 1 x k g x 1 x k f s y x 1 x k h f y x 1 x k x 1 x k displaystyle begin aligned f 0 x_ 1 ldots x_ k g x_ 1 ldots x_ k f s y x_ 1 ldots x_ k h f y x_ 1 ldots x_ k x_ 1 ldots x_ k end aligned gladstone 11 proved that this rule is enough to generate all primitive recursive functions gladstone 12 improved this so that even the combination of these two restrictions i e the pure iteration rule below is enough f 0 x x f s y x h f y x displaystyle begin aligned f 0 x x f s y x h f y x end aligned further improvements are possible severin 13 prove that even the pure iteration rule without parameters namely f 0 0 f s y h f y displaystyle begin aligned f 0 0 f s y h f y end aligned suffices to generate all unary primitive recursive functions if we extend the set of initial functions with truncated subtraction x y we get all primitive recursive functions if we additionally include as an initial function additional primitive recursive forms edit 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 computer language definition edit 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 with neither i nor n modifiable by the loop body no control structures of greater generality such as while loops or if then plus goto are admitted in a primitive recursive language the loop language introduced in a 1967 paper by albert r meyer and dennis m ritchie 14 is such a language its computing power coincides with the primitive recursive functions a variant of the loop language is douglas hofstadter s bloop in gödel escher bach adding unbounded loops while goto makes the language general recursive and turing complete as are all real world computer programming languages the definition of primitive recursive functions implies that their computation halts on every input after a finite number of steps on the other hand the halting problem is undecidable for general recursive functions finitism and consistency results edit 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 fun...