## How to compare the power of computational models (2005)

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Venue: | In Computability in Europe 2005: New Computational Paradigms |

Citations: | 3 - 3 self |

### BibTeX

@INPROCEEDINGS{Boker05howto,

author = {Udi Boker and Nachum Dershowitz},

title = {How to compare the power of computational models},

booktitle = {In Computability in Europe 2005: New Computational Paradigms},

year = {2005},

pages = {54--64},

publisher = {Springer-Verlag}

}

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### Abstract

Abstract. We argue that there is currently no satisfactory general framework for comparing the extensional computational power of arbitrary computational models operating over arbitrary domains. We propose a conceptual framework for comparison, by linking computational models to hypothetical physical devices. Accordingly, we deduce a mathematical notion of relative computational power, allowing the comparison of arbitrary models over arbitrary domains. In addition, we claim that the method commonly used in the literature for “strictly more powerful” is problematic, as it allows for a model to be more powerful than itself. On the positive side, we prove that Turing machines and the recursive functions are “complete ” models, in the sense that they are not susceptible to this anomaly, justifying the standard means of showing that a model is “hypercomputational.” 1

### Citations

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(Show Context)
Citation Context ...t. for every function f ∈ A there is f ′ ∈ B such that f ′ ◦ψ(x) = ψ◦f(x) for all x ∈ dom A. For example, Turing machines and the (untyped) λ-calculus were shown by Church [5], Kleene [8], and Turing =-=[14]-=- to embed the partial recursive functions. The reasons for the inadequacy of embedding as a generic power comparison notion are analogous to that of domain-extending. Example 3. Let RE be the recursiv... |

837 | Theory of recursive functions and effective computability - Rogers - 1967 |

348 |
Complexity and Real Computation
- Blum, Cucker, et al.
- 1997
(Show Context)
Citation Context ...ons when comparing them to Turing machines. Computing over Structures. There are models defined over structures, that is, over sets together with “built-in” functions and relations. See, for example, =-=[2, 13, 1]-=-. We consider the structure’s set as the domain, and include the structure’s functions and relations in the model. 2.2 Comparing Computational Power We generally say that a model B is at least as powe... |

263 |
An unsolvable problem of elementary number theory
- Church
(Show Context)
Citation Context ...ection ψ : dom A ↣ dom B, s.t. for every function f ∈ A there is f ′ ∈ B such that f ′ ◦ψ(x) = ψ◦f(x) for all x ∈ dom A. For example, Turing machines and the (untyped) λ-calculus were shown by Church =-=[5]-=-, Kleene [8], and Turing [14] to embed the partial recursive functions. The reasons for the inadequacy of embedding as a generic power comparison notion are analogous to that of domain-extending. Exam... |

152 | Computable Analysis, An Introduction - Weihrauch - 2000 |

141 |
Neural Networks and Analog Computation: Beyond the Turing Limit
- Siegelmann
- 1999
(Show Context)
Citation Context ...ansion of a model (additional functions) is also more powerful, that is, for B � A to imply B � A. For example, a model that computes more than Turing machines is considered more powerful (see, e.g., =-=[12]-=-). Hence, the common method of showing that a model B is more powerful than model A, for some comparison notion � ∗ , is to show that B � ∗ C � A. The Problem. Unfortunately, a proper expansion of a m... |

50 |
Lambda-definability and recursiveness
- Kleene
- 1936
(Show Context)
Citation Context ...om A ↣ dom B, s.t. for every function f ∈ A there is f ′ ∈ B such that f ′ ◦ψ(x) = ψ◦f(x) for all x ∈ dom A. For example, Turing machines and the (untyped) λ-calculus were shown by Church [5], Kleene =-=[8]-=-, and Turing [14] to embed the partial recursive functions. The reasons for the inadequacy of embedding as a generic power comparison notion are analogous to that of domain-extending. Example 3. Let R... |

29 | Iteration, inequalities, and differentiability in analog computers - Campagnolo, Moore, et al. |

28 | Abstract versus concrete computation on metric partial algebras
- Tucker, Zucker
- 2004
(Show Context)
Citation Context ...ons when comparing them to Turing machines. Computing over Structures. There are models defined over structures, that is, over sets together with “built-in” functions and relations. See, for example, =-=[2, 13, 1]-=-. We consider the structure’s set as the domain, and include the structure’s functions and relations in the model. 2.2 Comparing Computational Power We generally say that a model B is at least as powe... |

17 | Formal Languages: Automata and Structures - ENGELER - 1968 |

17 | Introduction to Computability - Hennie - 1977 |

16 | Real recursive functions and their hierarchy
- Mycka, Costa
(Show Context)
Citation Context ...unction g ∈ B onto rng ψ (g : dom B → rng ψ), such that for every function f ∈ A there is a function f ′ ∈ B such that ψ◦f(x) = f ′ ◦ψ(x) for all x ∈ dom A. Example 2. Real recursive functions (Rrec) =-=[10]-=-, are more powerful than Turing machines (TM). That is Rrec � TM. The comparison is done via the injection ψ : N ↣ R, where ψ(n) = n [10, p. 18], and the floor function (λx. ⌊x⌋) to provide the abstra... |

11 |
models
- Minsky, “Matter
- 1968
(Show Context)
Citation Context ...d 3 In Out fscan be chosen to be any set of cardinality at least as large as the cardinality of the model’s domain. The idea that a model encapsulates a point of view of the world is shared by Minsky =-=[9]-=-: We use the term “model” in the following sense: To an observer B, an object A* is a model of an object A to the extent that B can use A* to answer questions that interest him about A. The model rela... |

7 |
Computability over an arbitrary structure: Sequential and parallel polynomial time
- Bournez, Cucker, et al.
- 2003
(Show Context)
Citation Context ...ons when comparing them to Turing machines. Computing over Structures. There are models defined over structures, that is, over sets together with “built-in” functions and relations. See, for example, =-=[2, 13, 1]-=-. We consider the structure’s set as the domain, and include the structure’s functions and relations in the model. 2.2 Comparing Computational Power We generally say that a model B is at least as powe... |

4 |
Glossary of Z notation
- Bowen
- 1995
(Show Context)
Citation Context ...ercomputational. This notion provides a justification for the (otherwise improper) comparison method used in the literature for showing that a model is hypercomputational. Note. We use the Z-standard =-=[3]-=- for function arrows. For example, −↦→ denotes a partial function, → is used for a total surjective function, and ↣ is an injection. We use double-arrows for mappings (multi-valued functions). So ⇒ de... |