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As one of the examples used to demonstrate such reasoning, Manna's book includes a tail-recursive algorithm equivalent to the nested-recursive 91 function. Many of the papers that report an "automated verification" (or termination proof ) of the 91 function only handle the tail-recursive version.
Chains of three or more functions are possible; for example, function 1 calls function 2, function 2 calls function 3, and function 3 calls function 1 again. Indirect recursion is also called mutual recursion , which is a more symmetric term, though this is simply a difference of emphasis, not a different notion.
The μ-recursive functions (or general 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 minimization operator μ .
In this case the tree function calls the forest function by single recursion, but the forest function calls the tree function by multiple recursion. Using the Standard ML datatype above, the size of a tree (number of nodes) can be computed via the following mutually recursive functions: [5]
In mathematical logic, Gödel's β function is a function used to permit quantification over finite sequences of natural numbers in formal theories of arithmetic. The β function is used, in particular, in showing that the class of arithmetically definable functions is closed under primitive recursion, and therefore includes all primitive recursive functions.
The leaves of the tree are the base cases of the recursion, the subproblems (of size less than k) that do not recurse. The above example would have a child nodes at each non-leaf node. Each node does an amount of work that corresponds to the size of the subproblem n passed to that instance of the recursive call and given by (). The total amount ...
In functional programming, fold (also termed reduce, accumulate, aggregate, compress, or inject) refers to a family of higher-order functions that analyze a recursive data structure and through use of a given combining operation, recombine the results of recursively processing its constituent parts, building up a return value.
Since many Scheme compilers use C as an intermediate target code, the tail recursion must be encoded in C without growing the stack, even if the C compiler does not optimize tail calls. Many implementations achieve this by using a device known as a trampoline, a piece of code that repeatedly calls functions. All functions are entered via the ...