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Computational Complexity: an Empirical View Maël Pégny Computational Complexity: An Empirical View Maël Pégny To cite this version: Maël Pégny. Computational Complexity: An Empirical View. 39th Annual Convention of the Society for the Study of Artificial Intelligence and the Simulation of Behaviour (AISB 2013), Apr 2013, Exeter, United Kingdom. hal-01470374 HAL Id: hal-01470374 https://hal.archives-ouvertes.fr/hal-01470374 Submitted on 17 Feb 2017 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Computational Complexity: An Empirical View Maël Pégny 1 Abstract. Computational complexity theory (CCT) is usually con- To the extent of my knowledge, there is a striking assymetry in strued as the mathematical study of the complexity of computational the philosophical litterature2 between the treatment of computabil- problems. In recent years, research work on unconventional com- ity issues and that of complexity issues, when it comes to analyz- putational models has called into question the purely mathematical ing the role of empirical considerations. Foundational issues of com- nature of CCT, and has revealed potential relations between its sub- putability, especially the debate on the possibility of hypercomputa- ject matter and empirical sciences. In particular, recent debates sur- tion, have been the object of intensive scrutiny (see [12], [6], [27], rounding quantum computing have raised the possibility of a new [21], [9], to mention only a few references). The relations between computational model, one based on quantum mechanics, which may computational complexity and empirical considerations have been be exponentially more efficient than any previously known machine. the object of a somewhat lesser attention (see [22], [23], [16], [17], In this paper, I will show how those recent debates suggest an al- [30], [2], for a few remarkable exceptions). In particular, how this ternative, empirical view of CCT. I will then examine what are the empirical turn might affect our conception of CCT as a discipline fundamental arguments in the favor of this view, what are its con- has not been the subject of a systematic philosophical treatment. The sequences for our conception of CCT as a scientific field, and how aim of this work is to synthetise the dispersed remarks present in the further philosophical investigation should proceed. literature, and to provide such a systematic treatment. After articulating the orthodox, mathematical conception of CCT (section 1), I use recent analysis of the Extended Church-Turing The- 1 Introduction: unconventional computing and sis (ECTT) to present an empirical interpretation of CCT (section 2). philosophy I then examine how research on unconventional computational mod- In the last decades, computer scientists have been showing a growing els, especially quantum computing, can support arguments in favor interest in the study of unconventional computational models. Those of this empirical interpretation (section 3). Finally, I discuss how this models stem from a broadened understanding of computation. In his interpretation offers a new vision of the subject-matter of CCT, and original paper, A. Turing made it clear that his model was explicitly its unity as a scientific field: CCT can be seen as an empirical science, inspired by human, pen-and-paper computation (see [28], section 9). founded on empirical hypotheses (section 4) In many unconventional computational models, computation is no longer reduced to the discrete process of pen-and-paper computa- 2 The subject-matter of CCT, and the nature of tion: computation is any process that allows the encoding of an input computation value into the state of a given system, the processing of that informa- tion according to a given specification, and the reading of an output What is the object of computational complexity theory (abbreviated 3 value. Physical, chemical and biological processes can be harnessed as CCT )? In the literature, CCT is usually defined as a mathemati- to perform computations in this broadened sense. Algorithms can no cal science quantifying and studying the difficulty of mathematical, longer be conceived as abstract mathematical methods whose con- computational problems ([13], [5]). Two remarks must be made to ception and theoretical properties are completely independent of im- properly understand this definition. plementation: algorithms now become abstract models of empirical First, complexity properties are, prima facie, properties of algo- processes. rithms. They become intrinsic properties of a given problem through This new approach to computing raises new challenges for com- optimality results, which demonstrate that no algorithm can do better puter scientists. In the case of Turing machines, and other historical than a given, optimal one. It is this quantification over all possible al- models, empirical realizability was not a pending issue. The possi- gorithms, and the quest for optimality results, that distinguishes CCT bility of implementing these models by pen-and-paper computation from Algorithmic Analysis. was built into their very design. Many unconventional computational 2 By “philosophical literature”, I do not only mean the literature written by models, on the contrary, raise more serious issues of empirical real- professional philosophers, but also the foundational considerations pub- izability. Furthermore, defining the set of computable functions, or lished by physicists, computer scientists, and logicians. 3 computational costs, according to these models, might depend on I will actually focus my attention on uniform computational complexity theory. The reason for this restriction is simply that I am only interested in considerations of empirical realizability. In this broadened frame- computational models that can actually be implemented. If one thinks of the work of computation, it becomes natural to think that not only logical characterization of non-uniform complexity classes by resource-bounded and mathematical arguments, but also empirical ones, are relevant to Turing Machines receiving advice, one immediately faces the problem of the actual origin of such an advice. In full generality, it is unclear whether decide whether a given model is realistic, and what its exact compu- non-uniform computational models can be considered as empirically real- tational power is. izable models, or if they are to be considered as abstract tools used by the theorist. Even though that might be a very interesting philosophical issue, I 1 Université de Paris-1, CEA-Larsim, France, email: do not think that such a short, synthetic work is the proper place to discuss [email protected] such details. Second, and foremost, CCT as a domain rests on three distinct properties could not be attributed to computational tasks in an ab- hypotheses concerning the nature of its subject matter and the nature solute sense, but only relatively to a given model. The second reason of its results: is that the model-independence of polynomial time is one of the main arguments for the identification of polynomial-time complexity with • Hardware-independence. CCT concepts and results are indepen- tractability, and superpolynomial time-complexity with untractabil- dent of low-level hardware details. This is one of the central rea- ity (see, for instance, [19]). sons why the number of steps needed by a given algorithm is de- The obvious escape in ECTT is the adjective “reasonable.” It sug- scribed up to a linear factor, using the O notation. The exact num- gests the existence of computational models that go beyond the lim- ber of steps taken by an algorithm on a given machine could be its drawn by the ECTT, but are characterized as ’unreasonable.’ Yet sensitive to hardware details, but its gross magnitude should not without any specification of what a reasonable computational model show that sensitivity. should look like, the ECTT is not only informal, but also not well- • Model-independence. To be well defined, computational costs defined. must be considered within a given computational model, i.e. a In recent years, several computer scientists and physicist, includ- mathematical model describing precisely how algorithmic instruc- ing P. Shor, U. Vazirani, E. Bernstein and D. Aharonov suggested tions are to be written, and how they are to be executed. But the that “empirically realizable”4 should be substituted for “reasonable” main concepts and results of CCT are independent from any par- in the above phrasing (see [26], [7], [4]). The suggestion is quite ticular choice of model, e.g. Turing machine, or register machine. natural, since empirical realizability is a necessary condition of im- • Relevance for implementation. Despite hardware and model in- plementation. dependence, CCT results are relevant to determine the concrete A straightforward identification of “reasonable” with “empirically values of time-interval, and memory size, needed by a concrete realizable” might nevertheless be questionable. As P. van Emde Boas device when it executes a given algorithm. noticed in [29], a computational model might violate
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