Incomputability at the Foundations of Physics (A Study in the Philosophy of Science)

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On discovering scientific laws

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Solving Smullyan puzzles with formal systems

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Three forms of physical measurement and their computability

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East-West Paths to Unconventional Computing

Unconventional computing is about breaking boundaries in thinking, acting and computing. Typical topics of this non-typical field include, but are not limited to physics of computation, non-classical logics, new complexity measures, novel hardware, mechanical, chemical and quantum computing. Unconventional computing encourages a new style of thinking while practical applications are obtained from uncovering and exploiting principles and mechanisms of information processing in and functional properties of, physical, chemical and living systems; in particular, efficient algorithms are developed, (almost) optimal architectures are designed and working prototypes of future computing devices are manufactured. This article includes idiosyncratic accounts of ‘unconventional computing’ scientists reflecting on their personal experiences, what attracted them to the field, their inspirations and discoveries.info:eu-repo/semantics/publishedVersio

AN ANALOGUE-DIGITAL CHURCH-TURING THESIS

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Classifying the computational power of stochastic physical oracles

Consider a computability and complexity theory in which the classical set-theoretic oracle to a Turing machine is replaced by a physical process, and oracle queries return measurements of physical behaviour. The idea of such physical oracles is relevant to many disparate situations, but research has focussed on physical oracles that were classic deterministic experiments which measure physical quantities. In this paper, we broaden the scope of the theory of physical oracles by tackling non-deterministic systems. We examine examples of three types of non-determinism, namely systems that are: (1) physically nondeterministic, as in quantum phenomena; (2) physically deterministic but whose physical theory is non-deterministic, as in statistical mechanics; and (3) physically deterministic but whose computational theory is non-deterministic caused by error margins. Physical oracles that have probabilistic theories we call stochastic physical oracles. We propose a set SPO of axioms for a basic form of stochastic oracles. We prove that Turing machines equipped with a physical oracle satisfying the axioms SPO compute precisely the non-uniform complexity class BPP//log* in polynomial time. This result of BPP/log* is a computational limit to a great range of classical and non-classical measurement, and of analogue-digital computation in polynomial time under general conditions.info:eu-repo/semantics/publishedVersio

Oracles that measure thresholds: The Turing machine and the broken balance

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A hierarchy for BPP//log* based on counting calls to an oracle

Algorithms whose computations involve making physical measurements can be modelled by Turing machines with oracles that are physical systems and oracle queries that obtain data from observation and measurement. The computational power of many of these physical oracles has been established using non-uniform complexity classes; in particular, for large classes of deterministic physical oracles, with fixed error margins constraining the exchange of data between algorithm and oracle, the computational power has been shown to be the non-uniform class BPP//log⋆ . In this paper, we consider non-deterministic oracles that can be modelled by random walks on the line. We show how to classify computations within BPP//log⋆ by making an infinite non-collapsing hierarchy between BPP//log⋆ and BPP . The hierarchy rests on the theorem that the number of calls to the physical oracle correlates with the size of the responses to queries.info:eu-repo/semantics/publishedVersio

Building Neural Net Software

In a recent paper [Neto et al. 97] we showed that programming languages can be translated on recurrent (analog, rational weighted) neural nets. The goal was not efficiency but simplicity. Indeed we used a number-theoretic approach to machine programming, where (integer) numbers were coded in a unary fashion, introducing a exponential slow down in the computations, with respect to a two-symbol tape Turing machine. Implementation of programming languages in neural nets turns to be not only theoretical exciting, but has also some practical implications in the recent efforts to merge symbolic and subsymbolic computation. To be of some use, it should be carried in a context of bounded resources. Herein, we show how to use resource boundedness to speed up computations over neural nets, through suitable data type coding like in the usual programming languages. We introduce data types and show how to code and keep them inside the information flow of neural nets. Data types and control structures are part of a suitable programming language called netdef. Each netdef program has a specific neural net that computes it. These nets have a strong modular structure and a synchronisation mechanism allowing sequential or parallel execution of subnets, despite the massive parallel feature of neural nets. Each instruction denotes an independent neural net. There are constructors for assignment, conditional and loop instructions. Besides the language core, many other features are possible using the same method. There is also a netdef compiler, available at www.di.fc.ul.pt/~jpn/netdef/netdef.ht