# A report on Turing completeness

Said to be Turing-complete or computationally universal if it can be used to simulate any Turing machine (devised by English mathematician and computer scientist Alan Turing).

- Turing completeness35 related topics with Alpha

## Turing machine

7 linksMathematical model of computation describing an abstract machine that manipulates symbols on a strip of tape according to a table of rules.

Mathematical model of computation describing an abstract machine that manipulates symbols on a strip of tape according to a table of rules.

Turing completeness is the ability for a system of instructions to simulate a Turing machine.

## Programming language

7 linksAny set of rules that converts strings, or graphical program elements in the case of visual programming languages, to various kinds of machine code output.

Any set of rules that converts strings, or graphical program elements in the case of visual programming languages, to various kinds of machine code output.

XSLT, for example, is a Turing complete language entirely using XML syntax.

## Computer

7 linksDigital electronic machine that can be programmed to carry out sequences of arithmetic or logical operations automatically.

Digital electronic machine that can be programmed to carry out sequences of arithmetic or logical operations automatically.

The Engine incorporated an arithmetic logic unit, control flow in the form of conditional branching and loops, and integrated memory, making it the first design for a general-purpose computer that could be described in modern terms as Turing-complete.

## Computer program

5 linksSequence or set of instructions in a programming language for a computer to execute.

Sequence or set of instructions in a programming language for a computer to execute.

All present-day computers are Turing complete.

## Functional programming

6 linksProgramming paradigm where programs are constructed by applying and composing functions.

Programming paradigm where programs are constructed by applying and composing functions.

In 1937 Alan Turing proved that the lambda calculus and Turing machines are equivalent models of computation, showing that the lambda calculus is Turing complete.

## Halting problem

5 linksProblem of determining, from a description of an arbitrary computer program and an input, whether the program will finish running, or continue to run forever.

Problem of determining, from a description of an arbitrary computer program and an input, whether the program will finish running, or continue to run forever.

The halting problem is a decision problem about properties of computer programs on a fixed Turing-complete model of computation, i.e., all programs that can be written in some given programming language that is general enough to be equivalent to a Turing machine.

## Cellular automaton

3 linksA cellular automaton (pl.

A cellular automaton (pl.

In the 1980s, Stephen Wolfram engaged in a systematic study of one-dimensional cellular automata, or what he calls elementary cellular automata; his research assistant Matthew Cook showed that one of these rules is Turing-complete.

## Lambda calculus

5 linksFormal system in mathematical logic for expressing computation based on function abstraction and application using variable binding and substitution.

Formal system in mathematical logic for expressing computation based on function abstraction and application using variable binding and substitution.

Lambda calculus is Turing complete, that is, it is a universal model of computation that can be used to simulate any Turing machine.

## Algorithm

4 linksAlgorithm is a finite sequence of rigorous instructions, typically used to solve a class of specific problems or to perform a computation.

Algorithm is a finite sequence of rigorous instructions, typically used to solve a class of specific problems or to perform a computation.

But Minsky shows (as do Melzak and Lambek) that his machine is Turing complete with only four general types of instructions: conditional GOTO, unconditional GOTO, assignment/replacement/substitution, and HALT.

## Konrad Zuse

3 linksGerman civil engineer, pioneering computer scientist, inventor and businessman.

German civil engineer, pioneering computer scientist, inventor and businessman.

His greatest achievement was the world's first programmable computer; the functional program-controlled Turing-complete Z3 became operational in May 1941.