Physical Computation: How General are Gandy’s Principles for Mechanisms?

What are the limits of physical computation? In his ‘Church’s Thesis and Principles for Mechanisms’, Turing’s student Robin Gandy proved that any machine satisfying four idealised physical ‘principles’ is equivalent to some Turing machine. Gandy’s four principles in effect define a class of computing machines (‘Gandy machines’). Our question is: What is the relationship of this class to the class of all (ideal) physical computing machines? Gandy himself suggests that the relationship is identity. We do not share this view. We will point to interesting examples of (ideal) physical machines that fall outside the class of Gandy machines and compute functions that are not Turing-machine computable.     

Reflections on Gödel's and Gandy's Reflections on Turing's Thesis

We sketch the historical and conceptual context of Turing's analysis of algorithmic or mechanical computation. We then discuss two responses to that analysis, by Gödel and by Gandy, both of which raise, though in very different ways. The possibility of computation procedures that cannot be reduced to the basic procedures into which Turing decomposed computation. Along the way, we touch on some of Cleland's views.     

Comments on some theories of fuzzy computation

In classical computability theory, there are several (equivalent) definitions of computable function, decidable subset and semi-decidable subset. This paper is devoted to the discussion of some proposals for extending these definitions to the framework of fuzzy set theory. The paper mainly focuses on the notions of fuzzy Turing machine and fuzzy computability by limit processes. The basic idea of this paper is that the presence of real numbers in the interval [0,1] forces us to refer to endless approximation processes (as in recursive analysis) and not to processes terminating after a finite number of steps and giving the exact output (as in recursive arithmetic). In accordance with such a point of view, an extension of the famous Church thesis is proposed.     

Turing machines, transition systems, and interaction

This paper presents persistent Turing machines (PTMs), a new way of interpreting Turing-machine computation, based on dynamic stream semantics. A PTM is a Turing machine that performs an infinite sequence of “normal” Turing machine computations, where each such computation starts when the PTM reads an input from its input tape and ends when the PTM produces an output on its output tape. The PTM has an additional worktape, which retains its content from one computation to the next; this is what we mean by persistence.A number of results are presented for this model, including a proof that the class of PTMs is isomorphic to a general class of effective transition systems called interactive transition systems; and a proof that PTMs without persistence (amnesic PTMs) are less expressive than PTMs. As an analogue of the Church-Turing hypothesis which relates Turing machines to algorithmic computation, it is hypothesized that PTMs capture the intuitive notion of sequential interactive computation.     

Physical Hypercomputation and the Church–Turing Thesis

We describe a possible physical device that computes a function that cannot be computed by a Turing machine. The device is physical in the sense that it is compatible with General Relativity. We discuss some objections, focusing on those which deny that the device is either a computer or computes a function that is not Turing computable. Finally, we argue that the existence of the device does not refute the Church–Turing thesis, but nevertheless may be a counterexample to Gandy's thesis.     

Concrete Digital Computation: What Does it Take for a Physical System to Compute?

This paper deals with the question: what are the key requirements for a physical system to perform digital computation? Time and again cognitive scientists are quick to employ the notion of computation simpliciter when asserting basically that cognitive activities are computational. They employ this notion as if there was or is a consensus on just what it takes for a physical system to perform computation, and in particular digital computation. Some cognitive scientists in referring to digital computation simply adhere to Turing??s notion of computability. Classical computability theory studies what functions on the natural numbers are computable and what mathematical problems are undecidable. Whilst a mathematical formalism of computability may perform a methodological function of evaluating computational theories of certain cognitive capacities, concrete computation in physical systems seems to be required for explaining cognition as an embodied phenomenon. There are many non-equivalent accounts of digital computation in physical systems. I examine only a handful of those in this paper: (1) Turing??s account; (2) The triviality ??account??; (3) Reconstructing Smith??s account of participatory computation; (4) The Algorithm Execution account. My goal in this paper is twofold. First, it is to identify and clarify some of the underlying key requirements mandated by these accounts. I argue that these differing requirements justify a demand that one commits to a particular account when employing the notion of computation in regard to physical systems. Second, it is to argue that despite the informative role that mathematical formalisms of computability may play in cognitive science, they do not specify the relationship between abstract and concrete computation.     

Is the Church-Turing thesis true?

The Church-Turing thesis makes a bold claim about the theoretical limits to computation. It is based upon independent analyses of the general notion of an effective procedure proposed by Alan Turing and Alonzo Church in the 1930's. As originally construed, the thesis applied only to the number theoretic functions; it amounted to the claim that there were no number theoretic functions which couldn't be computed by a Turing machine but could be computed by means of some other kind of effective procedure. Since that time, however, other interpretations of the thesis have appeared in the literature. In this paper I identify three domains of application which have been claimed for the thesis: (1) the number theoretic functions; (2) all functions; (3) mental and/or physical phenomena. Subsequently, I provide an analysis of our intuitive concept of a procedure which, unlike Turing's, is based upon ordinary, everyday procedures such as recipes, directions and methods; I call them mundane procedures. I argue that mundane procedures can be said to be effective in the same sense in which Turing machine procedures can be said to be effective. I also argue that mundane procedures differ from Turing machine procedures in a fundamental way, viz., the former, but not the latter, generate causal processes. I apply my analysis to all three of the above mentioned interpretations of the Church-Turing thesis, arguing that the thesis is (i) clearly false under interpretation (3), (ii) false in at least some possible worlds (perhaps even in the actual world) under interpretation (2), and (iii) very much open to question under interpretation (1).     

Turing's legacy for the Internet

In 1936, when the world was computerless, Alan Turing invented the first virtual machine, now called the Universal Turing Machine (A. Turing, 1936). This concept provided a common ground for a theoretical exploration of the computable. Today, in a world with millions of computers linked to form a global computing network, we are again contemplating the virtues of virtual machines. Will a virtual machine, executing on millions of physical computing devices, be as useful in computing practice as Turing's machine is in computer theory? The author argues that it will and that its coming is inevitable. These considerations pertain to the delivery systems part of the Internet architecture which is presented     

What is computation?

Three conditions are usually given that must be satisfied by a process in order for it to be called a computation, namely, there must exist a finite length algorithm for the process, the algorithm must terminate in finite time for valid inputs and return a valid output and, finally, the algorithm must never return an output for invalid inputs. These three conditions are advanced as being necessary and sufficient for the process to be computable by a universal model of computation. In fact, these conditions are neither necessary nor sufficient. On the one hand, recently defined paradigms show how certain processes that do not satisfy one or more of the aforementioned properties can indeed be carried out in principle on new, more powerful, types of computers, and hence can be considered as computations. Thus, the conditions are not necessary. On the other hand, contemporary work in unconventional computation has demonstrated the existence of processes that satisfy the three stated conditions, yet contradict the Church–Turing thesis and, more generally, the principle of universality in computer science. Thus, the conditions are not sufficient.     

Three-dimensional parallel Turing machines

Informally, the parallel Turing machine (PTM) proposed by Wiedermann is a set of identical usual sequential Turing machines (STMs) cooperating on two common tapes: storage tape and input tape. Moreover, STMs which represent the individual processors of a parallel computer can multiply themselves in the course of computation. On the other hand, during the past 25 years or so, automata on a three-dimensional tape have been proposed as computational models of three-dimensional pattern processing, and several properties of such automata have been obtained. We proposed a three-dimensional parallel Turing machine (3-PTM), and dealt with a hardware-bounded 3-PTM whose inputs are restricted to cubic ones. We believe that this machine is useful in measuring the parallel computational complexity of three-dimensional images. In this article, we continue the study of 3-PTM, whose inputs are restricted to cubic ones, and investigate some of its accepting powers. This work was presented in part at the 12th International Symposium on Artificial Life and Robotics, Oita, Japan, January 25–27, 2007     

The Interactive Nature of Computing: Refuting the Strong Church–Turing Thesis

The classical view of computing positions computation as a closed-box transformation of inputs (rational numbers or finite strings) to outputs. According to the interactive view of computing, computation is an ongoing interactive process rather than a function-based transformation of an input to an output. Specifically, communication with the outside world happens during the computation, not before or after it. This approach radically changes our understanding of what is computation and how it is modeled. The acceptance of interaction as a new paradigm is hindered by the Strong Church–Turing Thesis (SCT), the widespread belief that Turing Machines (TMs) capture all computation, so models of computation more expressive than TMs are impossible. In this paper, we show that SCT reinterprets the original Church–Turing Thesis (CTT) in a way that Turing never intended; its commonly assumed equivalence to the original is a myth. We identify and analyze the historical reasons for the widespread belief in SCT. Only by accepting that it is false can we begin to adopt interaction as an alternative paradigm of computation. We present Persistent Turing Machines (PTMs), that extend TMs to capture sequential interaction. PTMs allow us to formulate the Sequential Interaction Thesis, going beyond the expressiveness of TMs and of the CTT. The paradigm shift to interaction provides an alternative understanding of the nature of computing that better reflects the services provided by today’s computing technology.

Hopfield网的图灵等价性

本文给出了用Ｈｏｐｆｉｅｌｄ网计算部分递归函数的构造性证明．由于部分递归函数与图灵机等价，故Ｈｏｐｆｉｅｌｄ网与图灵机等价．     

有序环（域）下的程序计算与BSS机器计算

１９８９年Ｂｌｕｍ，Ｓｈｂｕ与Ｓｍａｌｅ提出了在实数域上的一个计算模型，ＢＳＳ机器计算模型主要是基于有赂图的，它很直观但没有形式化，不方便使用经典的离散计算理论中的许多成熟的工具，本文从程序设计系统出发，提出一种在任意有序与的自然的程序设计设计语言，严格定义了它的语法与语义，研究了它与     

Simplicity via provability for universal prefix-free Turing machines

Universality, provability and simplicity are key notions in computability theory. There are various criteria of simplicity for universal Turing machines. Probably the most popular one is to count the number of states/symbols. This criterion is more complex than it may appear at a first glance. In this note we propose three new criteria of simplicity for universal prefix-free Turing machines. These criteria refer to the possibility of proving various natural properties of such a machine (its universality, for example) in a formal theory, Peano arithmetic or Zermelo–Fraenkel set theory. In all cases some, but not all, machines are simple.     

Some accepting powers of three-dimensional parallel Turing machines

Informally, the parallel Turing machine (PTM) proposed by Wiedermann is a set of identical usual sequential Turing machines (STMs) cooperating on two common tapes: storage tape and input tape. Moreover, STMs which represent the individual processors of a parallel computer can multiply themselves in the course of computation. On the other hand, during the past 25 years or so, automata on a three-dimensional tape have been proposed as computational models of three-dimensional pattern processing, and several properties of such automata have been obtained. We proposed a three-dimensional parallel Turing machine (3-PTM),¹ and dealt with a hardware-bounded 3-PTM whose inputs are restricted to cubic ones. We believe that this machine is useful in measuring the parallel computational complexity of three-dimensional images. Here, we continue the study of 3-PTM, whose inputs are restricted to cubic ones, and investigate some of its accepting powers. This work was presented in part at the First European Workshop on Artificial Life and Robotics, Vienna, Austria, July 12–13, 2007     

Parallel Turing Machine,a Proposal

We have witnessed the tremendous momentum of the second spring of parallel computing in recent years. But, we should remember the low points of the field more than 20 years ago and review the lesson that has led to the question at that point whether “parallel computing will soon be relegated to the trash heap reserved for promising technologies that never quite make it” in an article entitled “the death of parallel computing” written by the late Ken Kennedy — a prominent leader of parallel computing in the world. Facing the new era of parallel computing, we should learn from the robust history of sequential computation in the past 60 years. We should study the foundation established by the model of Turing machine (1936) and its profound impact in this history. To this end, this paper examines the disappointing state of the work in parallel Turing machine models in the past 50 years of parallel computing research. Lacking a solid yet intuitive parallel Turing machine model will continue to be a serious challenge in the future parallel computing. Our paper presents an attempt to address this challenge by presenting a proposal of a parallel Turing machine model. We also discuss why we start our work in this paper from a parallel Turing machine model instead of other choices.     

基于图灵机的递归技术的实现陈晓亮[1] 卢朝辉[2] 宋文[1]

图灵机是通用的计算机模型,一般程序设计和以图灵机为机器模型的计算也是支持递归的。本文首先分析了递归的特征,利用多带图灵机作为计算模型,定义了递归技术转移 函数形式,提出了图灵机递归过程信息传递与保存的方法,给出了图灵机调用的实现,继而给出了图灵机递归技术的实现,同时证明了图灵机的调用与图灵机的递归调用是图灵可识别的。     

A survey of recursive analysis and Moore’s notion of real computation

The theory of analog computation aims at modeling computational systems that evolve in a continuous space. Unlike the situation with the discrete setting there is no unified theory of analog computation. There are several proposed theories, some of them seem quite orthogonal. Some theories can be considered as generalizations of the Turing machine theory and classical recursion theory. Among such are recursive analysis and Moore’s class of recursive real functions. Recursive analysis was introduced by Turing (Proc Lond Math Soc 2(42):230–265, 1936), Grzegorczyk (Fundam Math 42:168–202, 1955), and Lacombe (Compt Rend l’Acad Sci Paris 241:151–153, 1955). Real computation in this context is viewed as effective (in the sense of Turing machine theory) convergence of sequences of rational numbers. In 1996 Moore introduced a function algebra that captures his notion of real computation; it consists of some basic functions and their closure under composition, integration and zero-finding. Though this class is inherently unphysical, much work have been directed at stratifying, restricting, and comparing it with other theories of real computation such as recursive analysis and the GPAC. In this article we give a detailed exposition of recursive analysis and Moore’s class and the relationships between them.     

Neural and Super-Turing Computing

``Neural computing' is a research field based on perceiving the human brain as an information system. This system reads its input continuously via the different senses, encodes data into various biophysical variables such as membrane potentials or neural firing rates, stores information using different kinds of memories (e.g., short-term memory, long-term memory, associative memory), performs some operations called ``computation', and outputs onto various channels, including motor control commands, decisions, thoughts, and feelings. We show a natural model of neural computing that gives rise to hyper-computation. Rigorous mathematical analysis is applied, explicating our model's exact computational power and how it changes with the change of parameters. Our analog neural network allows for supra-Turing power while keeping track of computational constraints, and thus embeds a possible answer to the superiority of the biological intelligence within the framework of classical computer science. We further propose it as standard in the field of analog computation, functioning in a role similar to that of the universal Turing machine in digital computation. In particular an analog of the Church-Turing thesis of digital computation is stated where the neural network takes place of the Turing machine.