## WHEN ARE TWO ALGORITHMS THE SAME?

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Citations: | 8 - 1 self |

### BibTeX

@MISC{Blass_whenare,

author = {Andreas Blass and Nachum Dershowitz and Yuri Gurevich},

title = {WHEN ARE TWO ALGORITHMS THE SAME?},

year = {}

}

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### Abstract

than the programs that implement them. The natural way to formalize this idea is that algorithms are equivalence classes of programs with respect to a suitable equivalence relation. We argue that no such equivalence relation exists. 1.

### Citations

2455 | The Design and Analysis of Computer Algorithms - AHO, HOPCROFT, et al. - 1974 |

114 | Sequential abstract state machines capture sequential algorithms
- Gurevich
(Show Context)
Citation Context ...ded amount of work per step. (The bound depends on the algorithm but not on the input or on the step in the computation.) Non-interactive algorithms 1 of this sort were characterized axiomatically in =-=[9]-=-. They include traditional models of computation such as Turing machines and register machines. Most people, when they hear the word “algorithm”, think of deterministic, smallstep, non-interactive alg... |

37 |
Linear logic.Theoretical
- Girard
- 1987
(Show Context)
Citation Context ...ame issues that we have discussed in connection with algorithms. Equality of proofs has been studied in considerable detail but usually only at the level of elementary logic; see for example [12] and =-=[7]-=-. Renaming bound variables is regarded as not changing a proof; the same goes for cancelling the introduction of a connective or quantifier and an immediately following elimination of that connective ... |

31 | Ordinary interactive small-step algorithms, Part I - Blass, Gurevich - 2006 |

23 | A natural axiomatization of computability and proof of Church’s Thesis
- Dershowitz, Gurevich
(Show Context)
Citation Context ...ans small-step (also called sequential) ASMs, as presented in [12]. Remark 2. To make this paper more self-contained, we give the following brief description of sequential ASMs. For more details, see =-=[7]-=- or6 ANDREAS BLASS, NACHUM DERSHOWITZ, AND YURI GUREVICH [11] or [12]. The rest of this paper does not depend on the exact definition of ASMs; a reader willing to read ASM programs as pseudo-code sho... |

17 |
Deductive systems and categories I: Syntactic calculus and residuated categories
- Lambek
- 1968
(Show Context)
Citation Context ... of the same issues that we have discussed in connection with algorithms. Equality of proofs has been studied in considerable detail but usually only at the level of elementary logic; see for example =-=[12]-=- and [7]. Renaming bound variables is regarded as not changing a proof; the same goes for cancelling the introduction of a connective or quantifier and an immediately following elimination of that con... |

2 |
What is an algorithm?” Mathematics Unlimited
- Moschovakis
- 2001
(Show Context)
Citation Context ...fficulties would be even worse if we were to use programming languages that lack a precise semantics or programming languages that cannot work at multiple levels of abstraction. Remark 3. Moschovakis =-=[17]-=- has proposed identifying algorithms with recursors. A recursor is a monotone inductive operator whose least fixed point includes (along with some auxiliary material) the partial function computed by ... |

1 |
Eléments de Mathématique XVII, Première partie: Les structures fondamentales de l’analyse. Livre I. Théorie des ensembles, Hermann & Cie
- Bourbaki
- 1954
(Show Context)
Citation Context ...lar choice of bound variables is so unimportant that they are often eliminated altogether, for example by means of de Bruijn indices in the λ-calculus [4] or by the boxes-and-links scheme of Bourbaki =-=[3]-=-. Example 11. Sequential non-interactive ASMs, which we have chosen as our standard programs, don’t have bound variables. They can, however, and often do have variables that resemble bound variables i... |

1 |
The prospects for mathematical logic
- Buss, Kechris, et al.
- 2001
(Show Context)
Citation Context ...we consider here are small-step, non-interactive ones. Nevertheless, it seems reasonable to use the expressions “sequential algorithm” and “algorithm” to match the terminology of the sources, [9] and =-=[5]-=- respectively. Behavioral equivalence, defined in [9] (but there called simply “equivalence”) is a very fine equivalence relation on sequential algorithms, requiring that they have the same states, th... |

1 |
Evolving algebras: Lipari guide,” in Specification and Validation
- Gurevich
- 1995
(Show Context)
Citation Context ...s described by the following abstract state machine (ASM), but readers unfamiliar with ASM notation should be able to understand it as if it were pseudo-code. In accordance with the usual conventions =-=[8]-=- for ASMs, the program is to be executed repeatedly until a step leaves the state unchanged. We assume for simplicity that the numbers whose g.c.d. is sought are the values, in the initial state, of d... |

1 | Euclid’s Elements,” web page at http://aleph0.clarku.edu - Joyce |

1 | with the collaboration of Andrzej Mostowski and Raphael Robinson, Undecidable Theories, North-Holland - Tarski - 1953 |

1 |
Towards a definition of an algorithm,” preprint, arXiv:math/0602053. Mathematics Department
- Yanofsky
(Show Context)
Citation Context ... for the abstract entity captured by an equivalence class) with respect to an equivalence relation of the sort requested by Shore. Shore’s question has been addressed in a limited context by Yanofsky =-=[17]-=-. He considers programs only in the form of (fairly) standard constructions of primitive recursive functions, and he defines an equivalence relation on these programs by listing a number of situations... |

1 | Algorthms: A quest for absoute definitions - Blass, Gurevich - 2003 |

1 |
Some systems of second-order arithmetic and their use
- Friedman
- 1975
(Show Context)
Citation Context ...that each of the theorems follows easily from the other – at least more easily than proving either theorem from scratch. But “easy” and “from scratch” are subjective notions. Reverse mathematics (see =-=[9, 20]-=-) provides a notion of equivalence of theorems, namely provable equivalence over some weak base theory. This notion is, however, much coarser than the intuitive notion, since the proofs of equivalence... |

1 |
as a programming language,” Tarski Lecture 2
- Moschovakis, “English
(Show Context)
Citation Context ...ge to be a translation of a text in another language?” Is expressing the same idea a necessary condition? Is it a sufficient condition? Does it depend on the chosen notion of “same idea”? Moschovakis =-=[18]-=- has argued that the meaning of a term in English (or other natural languages) is the algorithm for computing its denotation. With this identification of meanings with algorithms, the question of when... |