## A Basis For A Mathematical Theory Of Computation (1963)

### Cached

### Download Links

Venue: | Computer Programming and Formal Systems |

Citations: | 207 - 6 self |

### BibTeX

@INPROCEEDINGS{McCarthy63abasis,

author = {John McCarthy},

title = {A Basis For A Mathematical Theory Of Computation},

booktitle = {Computer Programming and Formal Systems},

year = {1963},

pages = {33--70},

publisher = {North-Holland}

}

### Years of Citing Articles

### OpenURL

### Abstract

This paper is a corrected version of the paper of the same title given at the Western Joint Computer Conference, May 1961. A tenth section discussing the relations between mathematical logic and computation has bean added. The second relevant direction of research is the theory of computability as a branch of recursive function theory. The results of the basic work in this theory, including the existence of universal machines and the existence of unsolvable problems, have established a framework in which any theory of computation must fit. Unfortunately, the general trend of research in this field has been to establish more and better unsolvability theorems, and there has been very little attention paid to positive results and none to establishing the properties of the kinds of algorithms that are actually used. Perhaps for this reason the formalisms for describing algorithms are too cumbersome to be used to describe actual algorithms. The third direction of mathematical research is the theory of finite automata. Results which use the finiteness of the number of states tend not to be very useful in dealing with present computers which have so many states that it is impossible for them to go through a substantial fraction of them in a reasonable time. The present paper is an attempt to create a basis for a mathematical theory of computation. Before mentioning what is in the paper, we shall discuss briefly what practical results can be hoped for from a suitable mathematical theory. This paper contains direct contributions towards only a few of the goals to be mentioned, but we list additional goals in order to encourage a gold rush. 1. To develop a universal programming language. We believe that this goal has been written off prematurely by a number of people. Our opini...

### Citations

1228 | On computable numbers, with an application to the Entscheidungsproblem - Turing |

333 |
The Calculi of Lambda Conversions
- Church
- 1941
(Show Context)
Citation Context ...is called the range of f . Forms and functions. In order to make properly the definitions that follow, we will distinguish between functions and expressions involving free variables. Following Church =-=[1]-=- the latter are called forms. Single letters such as f; g; h; etc. or sequences of letters such as sin are used to denote functions. Expressions such as f(x; y), f(g(x); y); x 2 + y are called forms. ... |

163 | Revised Report on the Algorithmic Language ALGOL 60 - Naur - 1960 |

117 |
Computability and Unsolvability
- DAVIS
- 1958
(Show Context)
Citation Context ...ard to show that all partial recursive functions in the sense of Church and Kleene are in Cfsucc; egg: In order to prove this we shall use the definition of partial recursive functions given by Davis =-=[3]-=-. If we modify definition 1.1 of page 41 of Davis [3] to omit reference to oracles we have the following: A function is partial recursive if it can be obtained by a finite number of applications of co... |

46 |
Recursive Predicates and Quantifiers
- KLEENE
- 1953
(Show Context)
Citation Context ...omains which are not effectively enumerable, and one may not wish to do so in domains where enumeration is unnatural. The next step is to allow quantification over functions. This gets us to Kleene's =-=[5]-=- analytic hierarchy and presumably allows the functions used in analysis. Two facts are worth noting. First 8((f); '(f)) refers to all functions on the domain and not just the computable ones. If we r... |

3 | letter to the editor - McCARTHY - 1959 |

2 |
On Operator Algorithms
- ERSHOV
(Show Context)
Citation Context ... namely flow of control is described awkwardly. The first attempt to give a formalism for describing computations that allows computations with entities from arbitrary spaces was made by A. P. Ershov =-=[4]-=-. However, his formalism uses computations with the symbolic expressions representing program steps, and this seems to be an unnecessary complication. We now discuss the relation between our formalism... |

2 | Theory of Algorithms (Russian - MARKOV - 1954 |

2 | The Logical Schemes of Algorithms, from Problems of Cybernetics I, translated from the Russian by - YANov - 1960 |