We propose certain non-Turing models of computation, but our intent is not to advocate models that surpass the power of Turing Machines (TMs), but to defend the need for models with orthogonal notions of power. We review the nature of models and argue that they are relative to a domain of application and are ill-suited to use outside that domain. Hence we review the presuppositions and context of the TM model and show that it is unsuited to natural computation (computation occurring in or inspired by nature). Therefore we must consider an expanded definition of computation that includes alternative (especially analog) models as well as the TM. Finally we present an alternative model, of continuous computation, more suited to natural computation. We conclude with remarks on the expressivity of formal mathematics. Key words: analog computation, analog computer, biocomputation, computability, computation on reals, continuous computation, formal system, hypercomputation,

Hypercomputation or super-Turing computation is a “computation ” that transcends the limit imposed by Turing’s model of computability. The field still faces some basic questions, technical (can we mathematically and/or physically build a hypercomputer?), cognitive (can hypercomputers realize the AI dream?), philosophical (is thinking more than computing?). The aim of this paper is to address the question: can we mathematically build a hypercomputer? We will discuss the solutions of the Infinite Merchant Problem, a decision problem equivalent to the Halting Problem, based on results obtained in [9, 2]. The accent will be on the new computational technique and results rather than formal proofs. 1

Although analog computation was eclipsed by digital computation in the second half of the twentieth century, it is returning as an important alternative computing technology. Indeed, as explained in this report, theoretical results imply that analog computation can escape from the limitations of digital computation. Furthermore, analog computation has emerged as an important theoretical framework for discussing computation in the brain and other natural systems. The report (1) summarizes the fundamentals of analog computing, starting with the continuous state space and the various processes by which analog computation can be organized in time; (2) discusses analog computation in nature, which provides models and inspiration for many contemporary uses of analog computation, such as neural networks; (3) considers general-purpose analog computing, both from a theoretical perspective and in terms of practical general-purpose analog computers; (4) discusses the theoretical power of

“Hypercomputation ” is often defined as transcending Turing computation in the sense of computing a larger class of functions than can Turing machines. While this possibility is important and interesting, this paper argues that there are many other important senses in which we may “transcend Turing computation. ” Turing computation, like all models, exists in a frame of relevance, which underlies the assumptions on which it rests and the questions that it is suited to answer. Although appropriate in many circumstances, there are other important applications of the idea of computation for which this model is not relevant. Therefore we should supplement it with new models based on different assumptions and suited to answering different questions. In alternative frames of relevance, including natural computation and nanocomputation, the central issues include real-time response, continuity, indeterminacy, and parallelism. Once we understand computation in a broader sense, we can see new possibilities for using physical processes to achieve computational goals, which will increase in importance as we approach the limits of electronic binary logic. Key words: hypercomputation, Church-Turing thesis, natural computation, theory of computation, model of computation, Turing computation,

www.cs.utk.edu/~mclennan/UPIM

Post-Moore’s Law computing will require an assimilation between computational processes and their physical realizations, both to achieve greater speeds and densities and to allow computational processes to assemble and control matter at the nanoscale. Therefore, we need to investigate “embodied computing, ” which addresses the essential interrelationships of information processing and physical processes in the system and its environment in ways that are parallel to those in the theory of embodied cognition. We briefly discuss matters of function and structure, regulation and causation, and the definition of computation. We address both the challenges and opportunities of embodied computation. Analysis is more difficult because physical effects must be included, but information processing may be simplified by dispensing with explicit representations and allowing massively parallel physical processes to process information. Nevertheless, in order to fully exploit embodied computation, we need robust and powerful theoretical tools, but we argue that the theory of Church-Turing computation is not suitable for the task. 1. Post-Moore’s Law Computation Although estimates differ, it is clear that the end of Moore’s Law is in sight; there are physical limits to the density of binary logic devices and to their speed of operation. This will require us to approach computation

My goal in this report is to recontextualize the concept of computation. I review the historical roots of Church-Turing computation to show that the theory exists in a frame of relevance, which underlies the assumptions on which it rests and the questions it is suited to answer. Although this frame of relevance is appropriate in many circumstances, there are many important applications of the idea of computation for which it is not relevant. These include natural computation (computation occurring in or inspired by nature), nanocomputation (computation based on nanoscale objects and processes), and computation based on quantum theory. As a consequence we need, not so much to abandon the Church-Turing model of computation, as to supplement it with new models based on different assumptions and suited to answering different questions. Therefore I will discuss alternative frames of relevance more suited to the interrelated application areas of natural computation, emergent computation, and nanocomputation. Central issues include continuity, indeterminacy, and parallelism. Finally, I will argue that once we understand computation in a broader sense than the Church-Turing model, we begin to see new possibilities for using natural processes to achieve our computational goals. These possibilities will increase in importance as we approach the limits of electronic binary logic as a basis for computation. They will also help us to understand computational processes in nature. * This report is based on an invited presentation at the workshop “Natural Processes & Models of