# Not All Computational Methods Are Effective Methods

## Abstract

**:**

## 1. Introduction

## 2. Distinguishing Features of This Argument

#### 2.1. No Dependence on Hypercomputation

“… ‘effective computation’ (i.e., calculation by means of effective procedures) encompasses a wide, and an important, class of computations, but not necessarily all computations … none of the hyper-machines described in the literature computes by means of effective procedures.”Shagrir and Pitowsky [6] (p. 94)

“It is perhaps surprising that not all classical algorithms are manual methods. That this is in fact the case has emerged from recent work on quantum computation … Algorithms for quantum Turing machines are not in general manual methods, since not all the primitive operations made available by the quantum hardware can be performed by a person unaided by machinery.”Copeland and Sylvan [2] (p. 55)

#### 2.2. Computations Should Be Individuated by Their Internal Workings

## 3. What Is an Effective Method?

“A method, or procedure, M, for achieving some desired result is called ‘effective’ (or ‘systematic’ or ‘mechanical’) just in case:

- M is set out in terms of a finite number of exact instructions (each instruction being expressed by means of a finite number of symbols);
- M will, if carried out without error, produce the desired result in a finite number of steps;
- M can (in practice or in principle) be carried out by a human being unaided by any machinery except paper and pencil;
- M demands no insight, intuition, or ingenuity, on the part of the human being carrying out the method.” Copeland [24]

“A mathematical method is termed ‘effective’ or ‘mechanical’ if and only if it can be set out in the form of a list of instructions able to be followed by an obedient human clerk … who works with paper and pencil, reliably but without insight or ingenuity, for as long as is necessary.”Copeland [25] (p. 12)

“Turing examined … human mechanical computability and exploited, in sharp contrast to Post, limitations of the human computing agent to motivate restrictive conditions … Turing asked in the historical context in which he found himself the pertinent question, namely, what are the possible processes a human being can carry out (when computing a number or, equivalently, determining algorithmically the value of a number theoretic function)?”Sieg [26] (p. 395)

“[Computable problems are those] which can be solved by human clerical labour, working to fixed rule, and without understanding”Turing [27] (pp. 38–39)

“[With regard to what is effectively calculable] Both Church and Turing had in mind calculation by an abstract human being using some mechanical aids (such as paper and pencil).”Gandy [28] (p. 123)

“Turing’s analysis makes no reference whatsoever to calculating machines. Turing machines appear as a result, as a codification, of his analysis of calculation by humans [previously defined as ‘effective calculability’].”Gandy [29] (p. 77)

“Roughly speaking, an algorithm [previously defined as an ‘effective procedure’] is a clerical (i.e., deterministic, book-keeping) procedure which can be applied to any of a certain class of symbolic inputs and which will eventually yield, for each such input a corresponding symbolic output.”Rogers [30] (p. 1)

“Effectiveness. An algorithm is also generally expected to be effective, in the sense that its operations must all be sufficiently basic that they can in principle be done exactly and in a finite length of time by someone using pencil and paper.”Knuth [31] (p. 6)

“[an effective procedure is] a list of instructions … that in principle make it possible to determine the value $f\left(n\right)$ for any argument n … The instructions must be completely definite and explicit. They should tell you at each step what to do, not tell you to go ask someone else what to do, or to figure out for yourself what to do: the instructions should require no external sources of information, and should require no ingenuity to execute …”Boolos et al. [32] (p. 23)

## 4. Didn’t Turing Define “Effective Method”?

“Turing’s work is a paradigm of philosophical analysis: it shows that what appears to be a vague intuitive notion has in fact a unique meaning which can be stated with complete precision.”Gandy [29] (p. 79)

“Church’s thesis is the proposal to identify an intuitive notion with a precise, formal, definition.”Folina [42] (p. 311)

“In 1928, the notion of an algorithm [effective method] was pretty vague. Up to that point, algorithms were often carried out by human beings using paper and pencil … Attacking Hilbert’s problem forced Turing to make precise exactly what was meant by an algorithm. To do this, Turing described what we now call a Turing machine.”Matuschak and Nielsen [43]

“If Turing’s thesis is correct, then talk about the existence and non-existence of effective methods can be replaced throughout mathematics, logic and computer science by talk about the existence or non-existence of Turing machine programs.”Copeland [24]

“… one can reduce it [the definition of a solvable puzzle] to the definition of ‘computable function’ or ‘systematic [effective] procedure’. A definition of any one of these would define all the rest. Since 1935 a number of definitions have been given [Turing machines, the $\lambda $-calculus, the $\mu $-recursive functions, etc.], explaining in detail the meaning of one or other of these terms, and these have all been proved equivalent to one another …”Turing [44] (p. 589)

## 5. All Computational Methods Are Effective Methods

“An algorithm or effective method … is a procedure for correctly calculating the values of a function or solving a class of problems that can be executed in a finite time and mechanically—that is, without the exercise of intelligence or ingenuity or creativity … A computation is anything that … calculates the values of a function or solves a problem by following an algorithm or effective method.”Burkholder [46] (p. 47)

“The logician Turing proposed (and solved) the problem of giving a characterization of computing machines in the widest sense—mechanisms for solving problems by effective series of logical operations.”Oppenheim and Putnam [47] (p. 19)

“We have assumed the reader’s understanding of the general notion of effectiveness, and indeed it must be considered as an informally familiar mathematical notion, since it is involved in mathematical problems of a frequently occurring kind, namely, problems to find a method of computation, i.e., a method by which to determine a number, or other thing, effectively. We shall not try to give here a rigorous definition of effectiveness, the informal notion being sufficient to enable us, in the cases we shall meet, to distinguish methods as effective or non-effective … The notion of effectiveness may also be described by saying that an effective method of computation, or algorithm, is one for which it would be possible to build a computing machine.”Church [48] (p. 52)

“Sometimes computers are called information processors … How they process or manipulate is by carrying out effective procedures … Computation [means] the use of an algorithm … also called an ‘effective method’ or a ‘mechanical procedure’ … to calculate the value of a function.”Crane [49] (pp. 102, 233)

“The functional organisation of mental processes can be characterized in terms of effective procedures, since the mind’s ability to construct working models is a computational process.”Johnson-Laird [50] (pp. 9–10)

“… [a] procedure admissible as an ‘ultimate’ procedure in a psychological theory [will fall] well within the intuitive boundaries of the ‘computable’ or ‘effective’ as these terms are presumed to be used in Church’s Thesis.”Dennett [51] (p. 83)

## 6. Quantum Computations That Are Not Effective Methods

“In addition to classical sequential algorithms, in use from antiquity, we have now parallel, interactive, distributed, real-time, analog, hybrid, quantum, etc. algorithms. New kinds of numbers and algorithms may be introduced. The notions of numbers and algorithms have not crystallized (and maybe never will) to support rigorous definitions.”Gurevich [64] (p. 32)

## 7. Simulating the Quantum System by Hand

## 8. Conclusions

- If two putative computational methods have different worst-case complexity profile, then they are genuinely different computational methods.
- There are abstract quantum computational methods that have different worst-case complexity profiles to that of any known effective method.

- 3.
- There are computational methods that are not effective methods.

“The real question at issue is ‘What are the possible processes which can be carried out in computing a number?’ ”Turing [45] (p. 249)

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Notes

1 | See Copeland and Proudfoot [1]. |

2 | |

3 | |

4 | I do not claim that quantum cases are the only examples of non-effective computational methods; see the end of Section 6 for discussion of other possible examples. |

5 | |

6 | In other words, by the function that they compute, where “function” is understood in a purely extensional way, i.e., as a set or ordered pairs corresponding to the overall input and output. |

7 | |

8 | See Block [15] for a classic discussion of this. |

9 | Worst-case measures of space or time complexity are not the only ones used to describe this resource usage, but they are the most commonly employed. Thanks to an anonymous reviewer for this point. |

10 | In this notation, n is the size of the list and $O\left(g\right(n\left)\right)$ provides an asymptotic upper bound on the resource consumption: for large enough n, resource consumption is always less than or equal to some constant times the $g\left(n\right)$ function named inside the $O(\xb7)$. For more on complexity theory and use of big-O notation to measure resource usage, see Papadimitriou [16]. |

11 | |

12 | For a helpful analysis of these two problems in relation to creating a general theory, see Blass et al. [21]. |

13 | See Knuth [22], (p. 97), who suggests that a distinguishing feature of computer science is that algorithms should be individuated by their complexity class. He argues that this “algorithmic” mode of thinking separates the thought processes of earlier mathematicians from those of later computer scientists (pp. 96–98). See Dean [17], (pp. 20–29); Shagrir [23] for further discussion of how and why complexity profiles matter to the individuation of computational methods. |

14 | Some critics of Turing argued that his human-centric characterisation of an effective method was not too narrow (as the authors above suggest), but too broad: the definition should be narrowed by adding a requirement that the number of steps taken by the human clerk should be somehow bounded or determinable in advance. For criticism of such proposals, see Gandy [29] (pp. 59–60); Mendelson [37] (p. 202); Rogers [30] (p. 5). |

15 | |

16 | |

17 | Copeland [25,52] criticises a number of the same authors for committing what he calls the “Church–Turing fallacy”. The fallacy is to assume that any possible physical mechanism could be simulated by some Turing machine. My claim is that the authors make a second mistake in that they assume that any possible computational method is also an effective method. Copeland argues that although “effective” and “mechanical” sometimes appear to be synonyms in mathematical logic, the relationship between them should be handled with caution. “Mechanical” should be understood as a term of art and defined in the way described in Section 3. It does not correspond in any straightforward way to the concept of a physical mechanism. |

18 | See Searle [54] (p. 202) and Searle (personal correspondence). |

19 | |

20 | See Lycan [58] for the name “homuncular functionalism” and a clear reconstruction of the view. |

21 | See Shagrir [61] for a helpful analysis and criticism of this second claim about the multiple realisability of computation. |

22 | See also Blass and Gurevich [65]: “In fact the notion of algorithm is richer these days than it was in Turing’s days. And there are algorithms … not covered directly by Turing’s analysis, for example, algorithms that interact with their environments, algorithms whose inputs are abstract structures, and geometric or, more generally, non-discrete algorithms.” (p. 31) |

23 | |

24 | If a superposition state $\alpha |0\rangle +\beta |1\rangle $ is measured, then the result is 0 with probability ${\left|\alpha \right|}^{2}$, and 1 with probability ${\left|\beta \right|}^{2}$, with ${\left|\alpha \right|}^{2}+{\left|\beta \right|}^{2}=1$. A $\sqrt{\mathrm{NOT}}$ gate performs the operation on the quantum state vector $\left(\right)$ described by the complex-valued matrix $\frac{1}{\sqrt{2}}\left(\right)open="("\; close=")">\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\begin{array}{cc}1\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}& \phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}-i\\ -i\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}& \phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}1\end{array}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}$. |

25 | Cf. Nielsen and Chuang [69] (p. 50): “… it is tempting to dismiss quantum computation as yet another technological fad … This is a mistake, since quantum computation is an abstract paradigm for information processing that may have many different implementations in technology.” |

26 | More accurately, a unitary (reversible) operator ${U}_{f}$ is applied to the input, ${U}_{f}$: $|x,y\rangle \to |x,y\oplus f(x)\rangle $, where ⊕ indicates addition modulo 2. ${U}_{f}$ is used because there is no guarantee that an arbitrary f itself is unitary, and the evolution of a quantum mechanical system must be governed by unitary operators. This modification does not affect the point above. |

27 | See Nielsen and Chuang [69] (pp. 30–32). |

28 | Strictly, a pair of values can be recovered, $x,f\left(x\right)$. The output is a pair because the evolution of the quantum state is governed by unitary operators (quantum computations must be reversible). |

29 | |

30 | A Hadamard gate is a quantum operator that works in a similar way to Deutsch’s $\sqrt{\mathrm{NOT}}$ operator, but defined over the real numbers. The transformation provided by a Hadamard gate is given by the real-valued matrix $\frac{1}{\sqrt{2}}\left(\right)open="("\; close=")">\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\begin{array}{cc}1\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}& \phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}1\\ 1\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}& \phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}-1\end{array}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}$. Like Deutsch’s $\sqrt{\mathrm{NOT}}$, a Hadamard gate may be physically implemented with half-silvered mirrors; see Barz [73]. |

31 | See Nielsen and Chuang [69] (pp. 32–36) for the details of the algorithm. |

32 | Ibid. |

33 | For further discussion of this point, see Nielsen and Chuang [69], pp. 30–34. |

34 | See Ekert and Jozsa [75] for algorithms that use quantum entanglement, and Bennett et al. [76], Gottesman and Chuang [77] for algorithms that use teleportation. Counterfactual computation is a counterintuitive method in which the intermediate steps of the computations do not take place in the actual world (according to measurement), yet the desired output is still produced; for a proposed application, see Hosten et al. [78]. |

35 | Shagrir [40], pp. 46–47. |

36 | I am assuming the methods in question have the same overall input–output profile and that one is trying to individuate them based on their internal workings. As discussed in Section 2.1, I am setting aside the use of hypercomputers for establishing the claim that not all computational methods are effective methods. |

37 | Nielsen and Chuang [69] (pp. 48, 204–206). |

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Sprevak, M.
Not All Computational Methods Are Effective Methods. *Philosophies* **2022**, *7*, 113.
https://doi.org/10.3390/philosophies7050113

**AMA Style**

Sprevak M.
Not All Computational Methods Are Effective Methods. *Philosophies*. 2022; 7(5):113.
https://doi.org/10.3390/philosophies7050113

**Chicago/Turabian Style**

Sprevak, Mark.
2022. "Not All Computational Methods Are Effective Methods" *Philosophies* 7, no. 5: 113.
https://doi.org/10.3390/philosophies7050113