# Primitive Recursive Functions

### Some articles on recursive functions, functions, primitive, function, recursive, primitive recursive functions:

μ-recursive Function - Definition
... The μ-recursive functions (or partial μ-recursive functions) are partial functions that take finite tuples of natural numbers and return a single natural number ... They are the smallest class of partial functions that includes the initial functions and is closed under composition, primitive recursion, and the μ operator ... The smallest class of functions including the initial functions and closed under composition and primitive recursion (i.e ...
Algorithm Characterizations - 1943, 1952 Stephen Kleene's Characterization - 1943 "Thesis I", 1952 "Church's Thesis"
... Every effectively calculable function (effectively decidable predicate) is general recursive" (First stated by Kleene in 1943 (reprinted page 274 in Davis, ed ... in Kleene (1952) p.300) In a nutshell to calculate any function the only operations a person needs (technically, formally) are the 6 primitive operators of "general" recursion (nowadays called the operators of ... calculation (decision) procedure or algorithm, for the case of a function (predicate) of natural numbers" (p ...
Grzegorczyk Hierarchy - Relation To Primitive Recursive Functions
... The definition of is the same as that of the primitive recursive functions, RP, except that recursion is limited ( for some j in ) and the functions are explicitly included in ... the Grzegorczyk hierarchy can be seen as a way to limit the power of primitive recursion to different levels ... It is clear from this fact that all functions in any level of the Grzegorczyk hierarchy are primitive recursive functions (i.e ...
Primitive Recursive Function
... The primitive recursive functions are defined using primitive recursion and composition as central operations and are a strict subset of the total µ-recursive functions ... In computability theory, primitive recursive functions are a class of functions that form an important building block on the way to a full formalization of ... These functions are also important in proof theory ...
Primitive Recursive Function - Examples - Subtraction
... Because primitive recursive functions use natural numbers rather than integers, and the natural numbers are not closed under subtraction, a limited subtraction function is studied ... This limited subtraction function sub(a,b) returns 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 ...

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