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Hypercomputation and the Physical ChurchTuring Thesis
, 2003
"... A version of the ChurchTuring Thesis states that every e#ectively realizable physical system can be defined by Turing Machines (`Thesis P'); in this formulation the Thesis appears an empirical, more than a logicomathematical, proposition. We review the main approaches to computation beyond Turing ..."
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A version of the ChurchTuring Thesis states that every e#ectively realizable physical system can be defined by Turing Machines (`Thesis P'); in this formulation the Thesis appears an empirical, more than a logicomathematical, proposition. We review the main approaches to computation beyond Turing definability (`hypercomputation'): supertask, nonwellfounded, analog, quantum, and retrocausal computation. These models depend on infinite computation, explicitly or implicitly, and appear physically implausible; moreover, even if infinite computation were realizable, the Halting Problem would not be a#ected. Therefore, Thesis P is not essentially di#erent from the standard ChurchTuring Thesis.
Theory of real computation according to EGC
 In Proceedings of the Dagstuhl Seminar on Reliable Implementation of Real Number Algorithms: Theory and Practice, Lecture Notes in Computer Science
, 2006
"... The Exact Geometric Computation (EGC) mode of computation has been developed over the last decade in response to the widespread problem of numerical nonrobustness in geometric algorithms. Its technology has been encoded in libraries such as LEDA, CGAL and Core Library. The key feature of EGC is the ..."
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Cited by 7 (2 self)
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The Exact Geometric Computation (EGC) mode of computation has been developed over the last decade in response to the widespread problem of numerical nonrobustness in geometric algorithms. Its technology has been encoded in libraries such as LEDA, CGAL and Core Library. The key feature of EGC is the necessity to decide zero in its computation. This paper addresses the problem of providing a foundation for the EGC mode of computation. This requires a theory of real computation that properly addresses the Zero Problem. The two current approaches to real computation are represented by the analytic school and algebraic school. We propose a variant of the analytic approach based on real approximation. • To capture the issues of representation, we begin with a reworking of van der Waerden’s idea of explicit rings and fields. We introduce explicit sets and explicit algebraic structures. • Explicit rings serve as the foundation for real approximation: our starting point here is not R, but F ⊆ R, an explicit ordered ring extension of Z that is dense in R. We develop the approximability of real functions within standard Turing machine computability, and show its connection to the analytic approach. • Current discussions of real computation fail to address issues at the intersection of continuous and discrete computation. An appropriate computational model for this purpose is obtained by extending Schönhage’s pointer machines to support both algebraic and numerical computation. • Finally, we propose a synthesis wherein both the algebraic and the analytic models coexist to play complementary roles. Many fundamental questions can now be posed in this setting, including transfer theorems connecting algebraic computability with approximability. 1
Toward accurate polynomial evaluation in rounded arithmetic
 In Foundations of computational mathematics, Santander 2005, volume 331 of London Math. Soc. Lecture Note Ser
, 2006
"... Given a multivariate real (or complex) polynomial p and a domain D, we would like to decide whether an algorithm exists to evaluate p(x) accurately for all x ∈ D using rounded real (or complex) arithmetic. Here “accurately ” means with relative error less than 1, i.e., with some correct leading digi ..."
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Given a multivariate real (or complex) polynomial p and a domain D, we would like to decide whether an algorithm exists to evaluate p(x) accurately for all x ∈ D using rounded real (or complex) arithmetic. Here “accurately ” means with relative error less than 1, i.e., with some correct leading digits. The answer depends on the model of rounded arithmetic: We assume that for any arithmetic operator op(a, b), for example a+b or a·b, its computed value is op(a, b)·(1+δ), where δ  is bounded by some constant ǫ where 0 < ǫ ≪ 1, but δ is otherwise arbitrary. This model is the traditional one used to analyze the accuracy of floating point algorithms. Our ultimate goal is to establish a decision procedure that, for any p and D, either exhibits an accurate algorithm or proves that none exists. In contrast to the case where numbers are stored and manipulated as finite bit strings (e.g., as floating point numbers or rational numbers) we show that some polynomials p are impossible to evaluate accurately. The existence of an accurate algorithm will depend not just on p and D, but on which arithmetic operators and constants are available to the algorithm and whether branching is permitted in the algorithm. Toward this goal, we present necessary conditions on p for it to be accurately evaluable on open real or complex domains D. We also give sufficient conditions, and describe progress toward a complete decision procedure. We do present a complete decision procedure for homogeneous polynomials p with integer coefficients, D = C n, using only arithmetic operations +, − and ·. 1
Abstract computability and algebraic specification
 ACM Transactions on Computational Logic
, 2002
"... Abstract computable functions are defined by abstract finite deterministic algorithms on manysorted algebras. We show that there exist finite universal algebraic specifications that specify uniquely (up to isomorphism) (i) all abstract computable functions on any manysorted algebra; (ii) all functi ..."
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Abstract computable functions are defined by abstract finite deterministic algorithms on manysorted algebras. We show that there exist finite universal algebraic specifications that specify uniquely (up to isomorphism) (i) all abstract computable functions on any manysorted algebra; (ii) all functions effectively approximable by abstract computable functions on any metric algebra. We show that there exist universal algebraic specifications for all the classically computable functions on the set R of real numbers. The algebraic specifications used are mainly bounded universal equations and conditional equations. We investigate the initial algebra semantics of these specifications, and derive situations where algebraic specifications precisely define the computable functions.
1996], On reality and models
 Boundaries and Barriers: On the Limits to Scientific Knowledge (J.l. Casti and
, 1996
"... Recently, I heard a researcher present a colloquium on computational aspects of proteinfolding. Although this man was obviously an expert on the topic, he casually mentioned in passing that, of course, ``proteinfolding is NPcomplete''. ..."
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Recently, I heard a researcher present a colloquium on computational aspects of proteinfolding. Although this man was obviously an expert on the topic, he casually mentioned in passing that, of course, ``proteinfolding is NPcomplete''.
The Nature of Computation and The Development of Computational Models
"... Abstract. This paper presents a study in the nature of computation, contributing with computation typologies: essential, spatial, temporal, representational and hierarchylevel based. Drawing from the historical development of the idea of number we argue that the concept of computation necessarily m ..."
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Abstract. This paper presents a study in the nature of computation, contributing with computation typologies: essential, spatial, temporal, representational and hierarchylevel based. Drawing from the historical development of the idea of number we argue that the concept of computation necessarily must develop. We thus address the development of models of computation, with emphasis on natural/physical/embodied computation and unconventional computing. Our analysis suggests that much better understanding of computation is needed than we have today. Finally, we propose possible directions for future research. 1
Mach Learn DOI 10.1007/s109940060219y Online calibrated forecasts: Memory efficiency versus universality for learning in games
, 2006
"... Abstract We provide a simple learning process that enables an agent to forecast a sequence of outcomes. Our forecasting scheme, termed tracking forecast, is based on tracking the past observations while emphasizing recent outcomes. As opposed to other forecasting schemes, we sacrifice universality i ..."
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Abstract We provide a simple learning process that enables an agent to forecast a sequence of outcomes. Our forecasting scheme, termed tracking forecast, is based on tracking the past observations while emphasizing recent outcomes. As opposed to other forecasting schemes, we sacrifice universality in favor of a significantly reduced memory requirements. We show that if the sequence of outcomes has certain properties—it has some internal (hidden) state that does not change too rapidly—then the tracking forecast is weakly calibrated so that the forecast appears to be correct most of the time. For binary outcomes, this result holds without any internal state assumptions. We consider learning in a repeated strategic game where each player attempts to compute some forecast of the opponent actions and play a best response to it. We show that if one of the players uses a tracking forecast, while the other player uses a standard learning algorithm (such as exponential regret matching or smooth fictitious play), then the player using the tracking forecast obtains the best response to the actual play of the other players. We further show that if both players use tracking forecast, then under certain conditions on the game matrix, convergence to a Nash