Theory of computation

subfield of computer science

thumb|300px|link=https://upload.wikimedia.org/wikipedia/commons/8/8a/Comparison_rounding_graphs_SMIL.svg|Graph of a function|Graphs of the result, , of rounding using different methods. For clarity, the graphs are shown displaced from integer values. In the SVG file, hover over a method to highlight it and, in SMIL-enabled browsers, click to select or deselect it.

thesis about the nature of computable functions

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

thumb|A typical parallel-scale nomogram. This example calculates the value of T when S = 7.30 and R = 1.17 are substituted into the equation. The isopleth crosses the scale for T at just under 4.65.

ability of a computing system to simulate Turing machines

total non-primitive-recursive computable function

diagram used in computer engineering and computer science

thumb|The ancient symbol Ouroboros, a dragon that continually consumes itself, denotes self-reference.

function whose values can be computed by an algorithm

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

one of several equivalent definitions of a computable function

collection of theoretical perspectives based on the premise that the universe is describable by information

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

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.

assignment of each symbol and well-formed formula of a formal language a unique natural number

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

Fault in a computer system that presents different symptoms to different observers

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.

function that can be computed with loops of bounded length

undecidable decision problem

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

number that represents the probability that a randomly constructed program will halt

thought experiment: 2 generals can talk to each other by sending a messenger through enemy territory; how can they agree on time of attack, if any messenger could be captured?

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

class of formal systems modelled visually by equal-sized squares with a color on each edge which can be arranged side by side

Hypercomputation or super-Turing computation is a set of hypothetical models of computation that can provide outputs that are not Turing-computable. For example, a machine that could solve the halting problem would be a hypercomputer; so too would one that could correctly evaluate every statement in Peano arithmetic.

form of recursion

model in computer science

highest possible rate of computation in this universe

string rewriting system that uses grammar-like rules to operate on strings of symbols

in mathematics, named after Gabriel Sudan

problem-solving procedures with certain characteristics

typed formalism that uses the lambda-symbol (λ) to denote anonymous function abstraction

programming language or computer interface that allows for flexibility in function but is difficult to learn and use because it offers little or no support for common tasks

measurement for the level of algorithmic unsolvability of a set

study of algorithms in strategic environments

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

formal system in mathematical logic

incomplete document containing the first published description of the logical design of a computer using the stored-program concept

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

rewriting system over strings from an alphabet

problem whose solution requires a computer at least the size of the Earth and a time period at least the estimated age of the Earth to be computed

physical and practical limits to the amount of computation or data storage with a given amount of mass, volume, or energy

in computing, a condition where an expression does not have a correct value

mathematical concept

hypothetical computing devices