## Effectiveness (2011)

### BibTeX

@MISC{Dershowitz11effectiveness,

author = {Nachum Dershowitz and Evgenia Falkovich},

title = {Effectiveness },

year = {2011}

}

### OpenURL

### Abstract

We describe axiomatizations of several aspects of effectiveness: effectiveness of transitions; effectiveness relative to oracles; and absolute effectiveness, as posited by the Church-Turing Thesis.

### Citations

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Citation Context ...orithm. 5 Effectiveness Having axiomatized algorithmic processes, we turn to the question of how to describe them by finite means. 5.1 Algorithms Gurevich [23] showed that his abstract state machines =-=[22]-=-, constitute a most general model of computation, one that can precisely describe effective transitions of any classical algorithm, on any desired level of abstraction of data structures and native op... |

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Citation Context ...s from all the above that any model purporting to be hypercomputational model, that computes all the Turing-computable functions and then some, be they (idealized) humans (as claimed, for example, in =-=[39, 41, 6]-=-), theoretical contrivance (e.g. [20, 43, 7]), or hypothetical (or idealized) physical apparatus (as proposed, for instance, in [15, 28, 54]), must violate one of our postulates. Note that, to be trul... |

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Citation Context .... Likewise, processes in which states evolve continuously (as in analog processes, like the position of a bouncing ball), rather than discretely, are eschewed. Yuri Gurevich’s “sequential postulates” =-=[23]-=- characterize algorithmicity in its classical sense. They assert that states are first-order structures and that transitions respect isomorphisms (see Section 3). An algorithm provides a prescription ... |

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Citation Context ...e nondeterministic choice, probabilistic choice, or interaction with the environment, one would need to modify the above notion of transition. 5This postulate is meant to exclude formalisms, such as =-=[20, 43]-=-, in which the result of a computation—or the continuation of a computation—may depend on (the limit of) an infinite sequence of preceding (finite or infinitesimal) steps. Likewise, processes in which... |

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Citation Context ...e nondeterministic choice, probabilistic choice, or interaction with the environment, one would need to modify the above notion of transition. 5This postulate is meant to exclude formalisms, such as =-=[20, 43]-=-, in which the result of a computation—or the continuation of a computation—may depend on (the limit of) an infinite sequence of preceding (finite or infinitesimal) steps. Likewise, processes in which... |

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Citation Context ...he same oracle. And a model is relatively computable if all its algorithms are, via the same encoding and same oracle. Sans oracles, we call it computable. This is akin to a computable algebra, as in =-=[17, 40, 44, 55]-=-, but we are not placing restrictions on the injective encoding. Were we not to require the encoding to be an injection, we could trivially simulate everything by encoding everything by a single const... |

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Citation Context ...and functions, thus providing a dramatic ‘proof’ of Church’s Thesis.” We discuss such an axiomatization of effectiveness in Sections 2–5. Unlike Turing’s analysis [56], and subsequent generalizations =-=[37, 38, 18, 49, 52, 50, 53]-=-, our axioms of effective computation are, at the same time, both formal and generic. They are formal, in that they may be cast as precise mathematical statements [4, 14]; they are generic, in that th... |

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Citation Context ...ng’s analysis of “the sorts of operations which a human computer could perform, working according to preassigned instructions” showed that these can be simulated by Turing machines. Gödel is reported =-=[12]-=- to have believed “that it might be possible . . . to state a set of axioms which would embody the generally accepted properties of [effective calculability], and to do something on that basis”. As ex... |

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Citation Context ...izations [37, 38, 18, 49, 52, 50, 53], our axioms of effective computation are, at the same time, both formal and generic. They are formal, in that they may be cast as precise mathematical statements =-=[4, 14]-=-; they are generic, in that they apply to computations with arbitrary states (Section 3) and arbitrary programmable transitions (Section 4). Computability is a more general notion than recursiveness o... |

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Citation Context ...ess stored in the current state. State-transitions are deterministic. Classical algorithms in fact never leave room for choices, nor do they involve any sort of interaction with the 3 An exception is =-=[41]-=-, which argues that algorithms should be equated with recursive definitions. 5environment to determine the next step. To incorporate nondeterministic choice, probabilistic choice, or interaction with... |

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Citation Context ...er 1 Introduction In 1900, David Hilbert posed, among other problems, the research challenge of how to effectively determine whether any given polynomial with rational coefficients has rational roots =-=[25]-=-: 1 [Probleme] 10. Entscheidung der Lösbarkeit einer Diophantischen Gleichung. Eine Diophantische Gleichung mit irgend welchen Unbekannten und mit ganzen rationalen Zahlencoefficienten sei vorgelegt: ... |

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Citation Context ...bularies are finite, since an algorithm must be describable in finite terms, so can only refer explicitly to finitely many operations. Hence, an algorithm cannot, for instance, involve all of Knuth’s =-=[36]-=- arrow operations, ↑, ↑↑, ↑↑↑, etc. Instead one could employ a ternary operation λxyz. x ↑z y. In restricting structures to be “first-order”, we are limiting the syntax to be first-order. This preclud... |

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Citation Context ...or of [21] wrote, “applicable to any continuous function” f over the reals—including ones that are not programmable. This program cannot be considered effective; indeed its domain is uncountable. See =-=[45]-=- for examples of geometric constructions with compass and straightedge. 5.2 Effective Algorithms The sequential postulates limit transitions to be effective, in the sense of being amenable to finite d... |

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Citation Context ...izations [37, 38, 18, 49, 52, 50, 53], our axioms of effective computation are, at the same time, both formal and generic. They are formal, in that they may be cast as precise mathematical statements =-=[4, 14]-=-; they are generic, in that they apply to computations with arbitrary states (Section 3) and arbitrary programmable transitions (Section 4). Computability is a more general notion than recursiveness o... |

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Citation Context ...ugh some interesting partial results have been obtained). This challenge of proving the Church-Turing Thesis is first in Richard Shore’s list of “pie-in-the-sky problems” for the twenty-first century =-=[8]-=-. Indeed, Harvey Friedman [16] has predicted that sometime in this century, “There will be an unexpected striking discovery that any model of computation satisfying certain remarkably weak conditions ... |

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Citation Context ...the algorithm. It follows that an algorithm should work equally well in isomorphic worlds. Compare [18, p. 128]. In this sense states are “abstract”. These considerations lead to the second postulate =-=[23, 1]-=-: Postulate (State) States of an algorithm are (first-order) structures over a finite vocabulary, closed under isomorphism of domains, such that initial states and terminal states are also closed unde... |

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Citation Context ... states with the same constructors and oracles. One can give an alternate characterization of effective state, one that is based on oracular Turing machines, extending a suggestion of Wolfgang Reisig =-=[46]-=-. Lemma A state X is effective relative to a set of oracles if and only if there is a Turing machine with the same oracles that can semi-decide the congruence induced by X. In other words, given two t... |

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Citation Context ...s from all the above that any model purporting to be hypercomputational model, that computes all the Turing-computable functions and then some, be they (idealized) humans (as claimed, for example, in =-=[39, 41, 6]-=-), theoretical contrivance (e.g. [20, 43, 7]), or hypothetical (or idealized) physical apparatus (as proposed, for instance, in [15, 28, 54]), must violate one of our postulates. Note that, to be trul... |

4 | Three paths to effectiveness
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Citation Context ... by X, false when both are defined but not equal, and diverges otherwise. Input and output for the machines’s oracles is via constructor terms. The proof is along the lines of the non-oracular one in =-=[5]-=-. 4 Transitions For a process, effective or not, to be deemed algorithmic, it must be possible to express the transition rules for going from state to state in some finite fashion. Kleene stressed thi... |

4 |
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Citation Context ...wers. See Sections 3.3 and 5.3. We draw some conclusions in the final section. 2 Discrete Algorithms By an algorithm, one invariably means some type of state-transition system. As Donald Knuth writes =-=[35]-=-, for example: Algorithms are concepts which have existence apart from any programming language. . . . I believe algorithms were present long before Turing et al. formulated them, just as the concept ... |

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Citation Context ...and functions, thus providing a dramatic ‘proof’ of Church’s Thesis.” We discuss such an axiomatization of effectiveness in Sections 2–5. Unlike Turing’s analysis [56], and subsequent generalizations =-=[37, 38, 18, 49, 52, 50, 53]-=-, our axioms of effective computation are, at the same time, both formal and generic. They are formal, in that they may be cast as precise mathematical statements [4, 14]; they are generic, in that th... |

3 |
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Citation Context ...ng to be hypercomputational model, that computes all the Turing-computable functions and then some, be they (idealized) humans (as claimed, for example, in [39, 41, 6]), theoretical contrivance (e.g. =-=[20, 43, 7]-=-), or hypothetical (or idealized) physical apparatus (as proposed, for instance, in [15, 28, 54]), must violate one of our postulates. Note that, to be truly hypercomputational, it is crucial that a m... |

3 |
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Citation Context ...he same oracle. And a model is relatively computable if all its algorithms are, via the same encoding and same oracle. Sans oracles, we call it computable. This is akin to a computable algebra, as in =-=[17, 40, 44, 55]-=-, but we are not placing restrictions on the injective encoding. Were we not to require the encoding to be an injection, we could trivially simulate everything by encoding everything by a single const... |

3 |
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Citation Context ...s from all the above that any model purporting to be hypercomputational model, that computes all the Turing-computable functions and then some, be they (idealized) humans (as claimed, for example, in =-=[39, 41, 6]-=-), theoretical contrivance (e.g. [20, 43, 7]), or hypothetical (or idealized) physical apparatus (as proposed, for instance, in [15, 28, 54]), must violate one of our postulates. Note that, to be trul... |

2 |
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Citation Context ...results have been obtained). This challenge of proving the Church-Turing Thesis is first in Richard Shore’s list of “pie-in-the-sky problems” for the twenty-first century [8]. Indeed, Harvey Friedman =-=[16]-=- has predicted that sometime in this century, “There will be an unexpected striking discovery that any model of computation satisfying certain remarkably weak conditions must stay within the recursive... |

2 |
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Citation Context ...ondition is true. This program describes the standard bisection search for the root of a function, as described in [21, Algorithm #4]. The point is that this abstract formulation is, as the author of =-=[21]-=- wrote, “applicable to any continuous function” f over the reals—including ones that are not programmable. This program cannot be considered effective; indeed its domain is uncountable. See [45] for e... |

2 |
Konstruktivnyye algyebry. 1, Uspekhi Matematicheskikh Nauk
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Citation Context ...he same oracle. And a model is relatively computable if all its algorithms are, via the same encoding and same oracle. Sans oracles, we call it computable. This is akin to a computable algebra, as in =-=[17, 40, 44, 55]-=-, but we are not placing restrictions on the injective encoding. Were we not to require the encoding to be an injection, we could trivially simulate everything by encoding everything by a single const... |

1 |
Comparing computational power. Logic Journal of the IGPL 14(5):633–648, 2006. doi:10.1007/ 978-3-540-78127-1. Cf. [11, p. 1]: “Hypercomputation is the computation of functions or numbers that cannot be computed
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Citation Context ... is an injective encoding ρ of E into D such that for every partial function g of B there is a partial function f of A, such that g = ρ−1 ◦ f ◦ ρ. A detailed discussion of simulations may be found in =-=[2]-=-. So, a state X over vocabulary F and arbitrary domain D is computable over oracles O if there is an encoding of D into the naturals and a recursive structure Y with domain N over oracles ρ ◦ o ◦ ρ−1 ... |

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