Computability theory

abstract computation model; mathematical model of computation that defines an abstract machine which manipulates symbols on a strip of tape according to a table of rules

formal system in mathematical logic

study of computable functions and Turing degrees

thesis about the nature of computable functions

problem of determining whether a given program will finish running or continue forever

algorithmic technique in computer science of solving a problem by reducing it to a smaller instance of the same problem

A computation is any type of arithmetic or non-arithmetic calculation that is well-defined. Common examples of computation are mathematical equation solving and the execution of computer algorithms.

yes/no problem in computer science

total non-primitive-recursive computable function

function whose values can be computed by an algorithm

measure of algorithmic complexity

Set where an algorithm can take a number as an input and can decide whether the number belongs to the set

mathematical model describing how an output of a function is computed given an input

one of several equivalent definitions of a computable function

recursive subset of the set of all possible finite sequences over the alphabet of the language

In mathematics and computer science, the ; ) is a challenge posed by David Hilbert and Wilhelm Ackermann in 1928. It asks for an algorithm that considers an inputted statement and answers "yes" or "no" according to whether it is universally valid, i.e., valid in every structure. Such an algorithm was proven to be impossible by Alonzo Church and Alan Turing in 1936.

real number that can be computed to within any desired precision by a finite, terminating algorithm

abstract machine used to study decision problems

Computability is the ability to solve a problem by an effective procedure. It is a key topic of the field of computability theory within mathematical logic and the theory of computation within computer science. The computability of a problem is closely linked to the existence of an algorithm to solve the problem.

a halting, binary-alphabet Turing machine which writes the most 1s on the tape, using only a limited set of states

undecidable decision problem

function that can be computed with loops of bounded length

theory stating that external physical reality is a mathematical structure

decision problem for which it is impossible to construct an algorithm that always leads to a correct yes-or-no answer

lemma in infinite graph theory

a set that can be output (enumerated) by an algorithm (mathematical logic, computability theory)

hierarchy which classifies certain sets based on the complexity of formulas that define them

Theorem in computability theory

problem-solving procedures with certain characteristics

measurement for the level of algorithmic unsolvability of a set

set of all total computable functions

Theorem in computability theory

ordinal-indexed family of rapidly increasing functions: ℕ→ℕ

in computability theory, the assignment of natural numbers to a set of objects

Branch of mathematical logic

complexity class, algebra

theorem

mathematical concept

concept in computability theory

hierarchy of primitive recursive functions used in computability theory

stronger, physical form of the Church–Turing thesis, that a universal Turing machine can simulate every physical process

extension of the arithmetical hierarchy

programming language

Complexity measure in computer science

an object that cannot be rewritten further