this paper we outline the general characteristics of continuous formal systems

The ‘hard problem’ is hard because of the special epistemological status of consciousness, which does not, however, preclude its scientific investigation. Data from phenomenologically trained observers can be combined with neurological investigations to establish the relation between experience and neurodynamics. Although experience cannot be reduced to physical phenomena, parallel phenomenological and neurological analyses allow the structure of experience to be related to the structure of the brain. Such an analysis suggests a theoretical entity, an elementary unit of experience, the protophenomenon, which corresponds to an activity site (such as a synapse) in the brain. The structure of experience is determined by connections (e.g. dendrites) between these activity sites; the connections correspond to temporal patterns among the elementary units of experience, which can be expressed mathematically. This theoretical framework illuminates several issues, including degrees of consciousness, nonbiological consciousness, sensory inversions, unity of consciousness and the unconscious mind.

The `hard problem' is hard because of the special epistemological status of consciousness, which does not, however, preclude its scientific investigation. Data from phenomenologically trained observers can be combined with neurological investigations to establish the relation between experience and neurodynamics. Although experience cannot be reduced to physical phenomena, parallel phenomenological and neurological analyses allow the structure of experience to be related to the structure of the brain. Such an analysis suggests a theoretical entity, an elementary unit of experience, the protophenomenon, which corresponds to an activity site (such as a synapse) in the brain. The structure of experience is determined by connections (e.g. dendrites) between these activity sites; the connections correspond to temporal patterns among the elementary units of experience, which can be expressed mathematically. This theoretical framework illuminates several issues, including degrees of conscious...

“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,

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

Abstract. This paper deals with a dynamical system of the form ˙ A = [[N, AT + A], A] + ν[[AT, A], A], where A is an n × n real matrix, N is a constant n × n real matrix, ν is a positive constant and [A, B] = AB − BA. In particular, the purpose of this paper is to establish a sorting behavior of the dynamical system and to represent it in a general Lie algebraic setting. Moreover, some applications of the dynamical system are presented.