## Three Paths to Effectiveness

Citations: | 4 - 4 self |

### BibTeX

@MISC{Boker_threepaths,

author = {Udi Boker and Nachum Dershowitz},

title = {Three Paths to Effectiveness},

year = {}

}

### OpenURL

### Abstract

For Yuri, profound thinker, esteemed expositor, and treasured friend. Abstract. Over the past two decades, Gurevich and his colleagues have developed axiomatic foundations for the notion of algorithm, be it classical, interactive, or parallel, and formalized them in a new framework of abstract state machines. Recently, this approach was extended to suggest axiomatic foundations for the notion of effective computation over arbitrary countable domains. This was accomplished in three different ways, leading to three, seemingly disparate, notions of effectiveness. We show that, though having taken different routes, they all actually lead to precisely the same concept. With this concept of effectiveness, we establish that there is – up to isomorphism – exactly one maximal effective model across all countable domains.

### Citations

1316 |
On computable numbers, with an application to the \Entscheidungsproblem
- Turing
- 1936
(Show Context)
Citation Context ...sive effective model, up to isomorphism. However, this claim only speaks about the extensional power of an effective computational model, not about its internal mechanism. Turing, in his seminal work =-=[16]-=-, justified the thesis by arguing that every “purely mechanical” human computation can be represented by a Turing machine whose steps more or less correspond to the manual computation. Indeed, Turing’... |

892 | The Theory of Recursive Functions and Effective Computability - Rogers - 1987 |

119 | Sequential Abstract State Machines capture sequential algorithms
- Gurevich
(Show Context)
Citation Context ...lgorithm. Sequential algorithms – that is, deterministic algorithms without unbounded parallelism or (intra-step) interaction with the outside world – have been analyzed and formalized by Gurevich in =-=[6]-=-. There it was proved that any algorithm satisfying three natural formal postulates (given below) can be emulated, step by step, by a program in a very general model of computation, called “abstract s... |

71 |
Computable algebra, general theory, and theory of computable fields
- Rabin
- 1960
(Show Context)
Citation Context ...sive functions. – An algorithm is computable if all its initial states are. – A model is computable if all its algorithms are, via the same encoding. This is a standard notion of “computable algebra” =-=[8,11,7,15]-=-, adopted by [5] (which, however, considers only a single algorithm and total functions). The choice of the partial recursive functions as the starting point for defining effective algorithms over arb... |

23 | A natural axiomatization of computability and proof of Church’s Thesis
- Dershowitz, Gurevich
(Show Context)
Citation Context ...tibility. Initial data are inductively defined to be effective if they only contain a Herbrand universe, in addition to finite data and functions that can be shown constructible in the same way. – In =-=[5]-=-, Dershowitz and Gurevich require an injective mapping between the arbitrary domain and the natural numbers. Initial data are effective if they are tracked – under that representation – by recursive f... |

14 |
Some philosophical implications of mathematical logic. I. Three classes of ideas
- Myhill
- 1952
(Show Context)
Citation Context ...cannot all be effectively represented and manipulated. We adopt the necessary point of view that effectiveness is a notion applicable to collections of functions, rather than to single functions (cf. =-=[10]-=-). A single function over an arbitrary domain cannot be classified as effective or ineffective [9,14], since its effectiveness depends on the context. A detailed discussion of this issue can be found ... |

12 | The Church-Turing thesis over arbitrary domains
- Boker, Dershowitz
(Show Context)
Citation Context ...A single function over an arbitrary domain cannot be classified as effective or ineffective [9,14], since its effectiveness depends on the context. A detailed discussion of this issue can be found in =-=[3]-=-. To capture what it is that makes a sequential algorithm mechanically computable, three different generic formalizations of effectiveness have recently been suggested: – In [3], the authors base thei... |

7 |
Acceptable notation. Notre Dame
- Shapiro
- 1982
(Show Context)
Citation Context ...fectiveness is a notion applicable to collections of functions, rather than to single functions (cf. [10]). A single function over an arbitrary domain cannot be classified as effective or ineffective =-=[9,14]-=-, since its effectiveness depends on the context. A detailed discussion of this issue can be found in [3]. To capture what it is that makes a sequential algorithm mechanically computable, three differ... |

6 |
The computable kernel of Abstract State Machines
- Reisig
- 2008
(Show Context)
Citation Context ...main and the natural numbers. Initial data are effective if they are tracked – under that representation – by recursive functions, as in the traditional definition of “computable” algebras [15]. – In =-=[12]-=-, Reisig bases effectiveness on the natural congruence relation between vocabulary terms that arises in the theory of ASMs. Initial data are effective if the induced congruence between terms is Turing... |

4 |
A notion of effectiveness in arbitrary structures
- Lambert
- 1968
(Show Context)
Citation Context ...sive functions. – An algorithm is computable if all its initial states are. – A model is computable if all its algorithms are, via the same encoding. This is a standard notion of “computable algebra” =-=[8,11,7,15]-=-, adopted by [5] (which, however, considers only a single algorithm and total functions). The choice of the partial recursive functions as the starting point for defining effective algorithms over arb... |

4 |
Constructive algebras I
- Mal’tsev
- 1961
(Show Context)
Citation Context ...sive functions. – An algorithm is computable if all its initial states are. – A model is computable if all its algorithms are, via the same encoding. This is a standard notion of “computable algebra” =-=[8,11,7,15]-=-, adopted by [5] (which, however, considers only a single algorithm and total functions). The choice of the partial recursive functions as the starting point for defining effective algorithms over arb... |

3 | The influence of domain interpretations on computational models
- Boker, Dershowitz
- 2009
(Show Context)
Citation Context ...veness was established in Theorem 1. As for their maximality, the partial recursive functions are “interpretation-complete”, in the sense that they cannot simulate a more inclusive model, as shown in =-=[2,4]-=-. By Theorem 2, they can simulate every effective model, leading to the conclusion that there is no effective model more inclusive than the partial recursive functions. Next, we show that the partial ... |

2 | Exact exploration
- Blass, Dershowitz, et al.
- 2009
(Show Context)
Citation Context ...ormal postulates (given below) can be emulated, step by step, by a program in a very general model of computation, called “abstract state machines” (ASMs). This formalization was recently extended in =-=[1]-=- to handle partial functions. But an algorithm, or abstract state machine program, need not yield an effective function. Gaussian elimination, for example, is a perfectly well-defined algorithm over t... |

2 |
Towards a general theory of computability. Synthese 12
- Montague
- 1960
(Show Context)
Citation Context ...fectiveness is a notion applicable to collections of functions, rather than to single functions (cf. [10]). A single function over an arbitrary domain cannot be classified as effective or ineffective =-=[9,14]-=-, since its effectiveness depends on the context. A detailed discussion of this issue can be found in [3]. To capture what it is that makes a sequential algorithm mechanically computable, three differ... |

1 |
J.V.: 4. In: Effective Algebra
- Stoltenberg-Hansen, Tucker
- 1995
(Show Context)
Citation Context ...rbitrary domain and the natural numbers. Initial data are effective if they are tracked – under that representation – by recursive functions, as in the traditional definition of “computable” algebras =-=[15]-=-. – In [12], Reisig bases effectiveness on the natural congruence relation between vocabulary terms that arises in the theory of ASMs. Initial data are effective if the induced congruence between term... |