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Maths › 03. Logic and Foundations ›

10.5 Equivalence

The previous section introduced several formal models of computation and stated that they compute the same class of functions. In this section, we give a more systematic account of this equivalence by presenting explicit constructions that translate computations from one model into another. The central result is that all standard models define exactly the same class of partial functions: $$ f : \mathbb{N}^n \rightharpoonup \mathbb{N}. $$ This class is called...
logic computability equivalence turing machines recursive functions May 2, 2026
Maths › 03. Logic and Foundations ›

10.3 Church Turing

The purpose of computability theory is to make precise the idea of an effective procedure. Informally, an effective procedure is a method that can be followed step by step, without guesswork, and which produces an answer after finitely many steps whenever it is meant to halt. The central question is whether this informal idea can be captured by a formal mathematical definition. Recursive functions give one such definition. Turing machines...
logic computability church turing thesis turing machines recursive functions May 2, 2026
Maths › 03. Logic and Foundations ›

10. Computable Functions

Computability theory studies which functions can be calculated by a precise mechanical procedure, and it gives a mathematical meaning to the informal idea of an effective algorithm. The chapter begins with recursive functions, where computation is described by basic initial functions together with operations that generate more complex functions, such as composition, primitive recursion, and minimization. Partial and total functions are then distinguished, since some computations produce an output for...
logic computability recursive functions computation May 2, 2026
Maths › 03. Logic and Foundations ›

10.1 Recursive Functions

Recursive function theory gives a mathematical definition of computable numerical functions by specifying basic functions and closure operations that generate increasingly complex functions from simple ones, and this framework allows one to reason about computation purely within arithmetic. All functions considered in this section have inputs and outputs in the natural numbers: $$ \mathbb{N} = {0,1,2,\dots}. $$ An $n$ ary function is written: $$ f : \mathbb{N}^n \to \mathbb{N}, $$...
logic computability recursive functions primitive recursion May 2, 2026