Deviant encodings and Turing’s analysis of computability B. Jack Copeland, Diane Proudfoot Philosophy Department, School of Humanities, University of Canterbury, Private Bag 4800, Christchurch, New Zealand article info Keywords: Systematic procedure Turing machine Church–Turing thesis Deviant encoding Acceptable encoding Turing’s analysis of computability Turing’s Notational Thesis abstract Turing’s analysis of computability has recently been challenged; it is claimed that it is circular to analyse the intuitive concept of numerical computability in terms of the Turing machine. This claim threatens the view, canonical in mathematics and cognitive science, that the concept of a systematic procedure or algo- rithm is to be explicated by reference to the capacities of Turing machines. We defend Turing’s analysis against the challenge of ‘deviant encodings’. Ó 2010 Elsevier Ltd. All rights reserved. When citing this paper, please use the full journal title Studies in History and Philosophy of Science 1. The problem of deviant encodings In this paper we outline a number of related techniques—which we call collectively ‘deviant encodings’ 1 —that appear to enable Tur- ing machines to perform ‘impossible’ tasks, such as solving the halt- ing problem. In each case, the Turing machine’s problematic behaviour results from the encoding scheme employed in preparing or reading the machine’s tape. The basic idea of a deviant encoding is easily illustrated. You can give the correct answer to any Yes/No question that I ask you, if it is arranged in advance that I will wink at you (as I ask the question) if and only if the correct answer is Yes. Likewise, a computer is able to answer any and every question if the programmer is permitted to code the answer into the presen- tation of the question. In setting up a Turing machine to perform a particular job, the programmer faces a wide range of notational choices. For exam- ple, numbers may be represented on the machine’s tape in stan- dard binary notation, or in unary notation (where 1 is |, 2 is ||, and so on). There are many other possibilities. To mention two exotica from the history of computing: in the world’s first elec- tronic stored-program computer, at the University of Manchester, Turing and his colleagues used backwards base 32 encoding, where the least significant digit of the representation of the number appeared on the left (rather than the right, as usual); and in the early SEAC computer, Ralph Slutz used a form of binary encoding in which ‘1’ represented zero and ‘0’ represented unity. (Slutz’s encoding scheme enabled a gain in processing speed, but had to be abandoned because users of the machine found the scheme excessively perplexing.) Depending on which encoding scheme is employed by the programmer or user, the same string of digits—the same input string or output string—means different things. For example, if Slutz’s encoding scheme is employed, an output of ‘0111’ says that the answer to the problem set is 8; whereas, if the encoding scheme is standard binary, the output ‘0111’ says that the answer is 7. We discuss encoding schemes that illicitly boost the logical power of Turing machines. Why ‘illicitly’? If deviant encodings are permitted, tasks or functions that are not computable in the intuitive sense become computable by Turing machine. Hence, if the Turing machine is to be used—as Turing wished to use it—to analyse our intuitive concept of computability, deviant encodings cannot be permitted. Or to put the difficulty another way: if computability is defined formally by reference to Turing machines (as is now standard in both mathematical logic and cognitive sci- ence) and deviant encodings are allowed, then tasks or functions that are not intuitively computable are computable according to 0039-3681/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.shpsa.2010.07.010 E-mail addresses: jack.copeland@canterbury.ac.nz, diane.proudfoot@canterbury.ac.nz 1 A version of this paper, entitled ‘Deviant encodings and the Turing definition of computability’, was read at a Philosophy seminar at La Trobe University, Australia in December 2003. The expressions ‘deviant encodings’ and ‘the problem of deviant encodings’ have recently been employed in the same sense by Michael Rescorla (2007), p. 266. Studies in History and Philosophy of Science 41 (2010) 247–252 Contents lists available at ScienceDirect Studies in History and Philosophy of Science journal homepage: www.elsevier.com/locate/shpsa