Results 1  10
of
14
Computational complexity with experiments as oracles
, 2008
"... We discuss combining physical experiments with machine computations and introduce a form of analoguedigital Turing machine. We examine in detail a case study where an experimental procedure based on Newtonian kinematics is combined with a class of Turing machines. Three forms of analoguedigital ma ..."
Abstract

Cited by 13 (10 self)
 Add to MetaCart
We discuss combining physical experiments with machine computations and introduce a form of analoguedigital Turing machine. We examine in detail a case study where an experimental procedure based on Newtonian kinematics is combined with a class of Turing machines. Three forms of analoguedigital machine are studied, in which physical parameters can be set exactly and approximately. Using nonuniform complexity theory, and some probability, we prove theorems that show that these machines can compute more than classical Turing machines. 1
Oracles and Advice as Measurements
"... Abstract. In this paper we will try to understand how oracles and advice functions, which are mathematical abstractions in the theory of computability and complexity, can be seen as physical measurements in Classical Physics. First, we consider how physical measurements are a natural external source ..."
Abstract

Cited by 5 (4 self)
 Add to MetaCart
Abstract. In this paper we will try to understand how oracles and advice functions, which are mathematical abstractions in the theory of computability and complexity, can be seen as physical measurements in Classical Physics. First, we consider how physical measurements are a natural external source of information to an algorithmic computation. We argue that oracles and advice functions can help us to understand how the structure of space and time has information content that can be processed by Turing machines (after Cooper and Odifreddi [10] and Copeland and Proudfoot [11, 12]). We show that nonuniform complexity is an adequate framework for classifying feasible computations by Turing machines interacting with an oracle in Nature. By classifying the information content of such an oracle using Kolmogorov complexity, we obtain a hierarchical structure for advice classes. 1
Quanta in classical mechanics: uncertainty in space, time, energy (Extended Abstract)
 IN STUDIA LOGICA INTERNATIONAL CONFERENCE ON LOGIC AND THE FOUNDATIONS OF PHYSICS: SPACE, TIME AND QUANTA (TRENDS IN LOGIC VI
, 2008
"... Consider modelling an experiment intended to measure a quantity, such as position or mass in Classical Mechanics. A theoretical analysis of the process may examine the operations used in the experimental procedure and equipment, and explore relationships between the accuracy of the measurement and p ..."
Abstract

Cited by 4 (4 self)
 Add to MetaCart
Consider modelling an experiment intended to measure a quantity, such as position or mass in Classical Mechanics. A theoretical analysis of the process may examine the operations used in the experimental procedure and equipment, and explore relationships between the accuracy of the measurement and physical properties of the operations, such as their precision, and their use of time, space, and energy. How does the accuracy of measurements depend on resources? In a series of studies [5–7, 2, 1, 3] we have developed a methodology to investigate some fundamental notions about experimental methods in order to answer the question, What can one compute with a physical system? The methodology is based on defining experiments formally over some tightly specified theory T and analysing observable behaviours as the inputoutput of computations. The idea is (i) to understand the notion of computation inside a physical theory T in a way that is independent of classical computability theory of algorithms; and
Computational Models of Measurement and Hempels Axiomatization
 CAUSALITY, MEANINGFUL COMPLEXITY AND KNOWLEDGE CONSTRUCTION, VOL. 46 OF THEORY AND DECISION LIBRARY A
, 2009
"... We have developed a mathematical theory about using physical experiments as oracles to Turing machines. We suppose that an experiment makes measurements according to a physical theory and that the queries to the oracle allow the Turing machine to read the value being measured bit by bit. Using this ..."
Abstract

Cited by 4 (4 self)
 Add to MetaCart
We have developed a mathematical theory about using physical experiments as oracles to Turing machines. We suppose that an experiment makes measurements according to a physical theory and that the queries to the oracle allow the Turing machine to read the value being measured bit by bit. Using this theory of physical oracles, an experimenter performing an experiment can be modelled as a Turing machine governing an oracle that is the experiment. We consider this computational model of physical measurement in terms of the theory of measurement of Hempel and Carnap (see [16, 13]). We note that once a physical quantity is given a real value, Hempel’s axioms of measurement involve undecidabilities. To solve this problem, we introduce time into Hempel’s axiomatization. Focussing on a dynamical experiment for measuring mass, as in [1, 3, 5, 4, 6], we show that the computational model of measurement satisfies our generalization of Hempel’s axioms. Our analysis also explains undecidability in measurement and that quantities are not always measurable.
Emergence as a ComputabilityTheoretic Phenomenon
, 2008
"... In dealing with emergent phenomena, a common task is to identify useful descriptions of them in terms of the underlying atomic processes, and to extract enough computational content from these descriptions to enable predictions to be made. Generally, the underlying atomic processes are quite well un ..."
Abstract

Cited by 3 (1 self)
 Add to MetaCart
In dealing with emergent phenomena, a common task is to identify useful descriptions of them in terms of the underlying atomic processes, and to extract enough computational content from these descriptions to enable predictions to be made. Generally, the underlying atomic processes are quite well understood, and (with important exceptions) captured by mathematics from which it is relatively easy to extract algorithmic content. A widespread view is that the difficulty in describing transitions from algorithmic activity to the emergence associated with chaotic situations is a simple case of complexity outstripping computational resources and human ingenuity. Or, on the other hand, that phenomena transcending the standard Turing model of computation, if they exist, must necessarily lie outside the domain of classical computability theory. In this talk we suggest that much of the current confusion arises from conceptual gaps and the lack of a suitably fundamental model within which to situate emergence. We examine the potential for placing emergent relations in a familiar context based on Turing’s 1939 model for interactive computation over structures described in terms of reals. The explanatory power of this model is explored, formalising informal descriptions in terms of mathematical definability and invariance, and relating a range of basic scientific puzzles to results and intractable problems in computability theory. In this talk
Five views of hypercomputation
"... We overview different approaches to the study of hypercomputation and other investigations on the plausibility of the physical Church–Turing thesis. We propose five thesis to classify investigation in this area. Sly does it. Tiptoe catspaws. Slide and creep. ..."
Abstract

Cited by 3 (0 self)
 Add to MetaCart
We overview different approaches to the study of hypercomputation and other investigations on the plausibility of the physical Church–Turing thesis. We propose five thesis to classify investigation in this area. Sly does it. Tiptoe catspaws. Slide and creep.
Physical Oracles: The Turing Machine and the Wheatstone Bridge
"... Earlier, we have studied computations possible by physical systems and by algorithms combined with physical systems. In particular, we have analysed the idea of using an experiment as an oracle to an abstract computational device, such as the Turing machine. The theory of composite machines of this ..."
Abstract

Cited by 2 (2 self)
 Add to MetaCart
Earlier, we have studied computations possible by physical systems and by algorithms combined with physical systems. In particular, we have analysed the idea of using an experiment as an oracle to an abstract computational device, such as the Turing machine. The theory of composite machines of this kind can be used to understand (a) a Turing machine receiving extra computational power from a physical process, or (b) an experimenter modelled as a Turing machine performing a test of a known physical theory T. Our earlier work was based upon experiments in Newtonian mechanics. Here we extend the scope of the theory of experimental oracles beyond Newtonian mechanics to electrical theory. First, we specify an experiment that measures resistance using a Wheatstone bridge and start to classify the computational power of this experimental oracle using nonuniform complexity classes. Secondly, we show that modelling an experimenter and experimental procedure algorithmically imposes a limit on our ability to measure resistance by the Wheatstone bridge. The connection between the algorithm and physical test is mediated by a protocol controlling each query, especially the physical time taken by the experimenter. In our studies we find that physical experiments have an exponential time protocol; this we formulate as a general conjecture. Our theory proposes that measurability in Physics is subject to laws which are colateral effects of the limits of computability and computational complexity.
FROM DESCARTES TO TURING: THE COMPUTATIONAL CONTENT OF SUPERVENIENCE
"... Mathematics can provide precise formulations of relatively vague concepts and problems from the real world, and bring out underlying structure common to diverse scientific areas. Sometimes very natural mathematical concepts lie neglected and not widely understood for many years, before their fundame ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
Mathematics can provide precise formulations of relatively vague concepts and problems from the real world, and bring out underlying structure common to diverse scientific areas. Sometimes very natural mathematical concepts lie neglected and not widely understood for many years, before their fundamental relevance is recognised and their explanatory power is fully exploited. The notion of definability in a structure is such a concept, and Turing’s [77] 1939 model of interactive computation provides a fruitful context in which to exercise the usefulness of definability as a powerful and widely applicable source of understanding. In this article we set out to relate this simple idea to one of the oldest and apparently least scientifically approachable of problems — that of realistically modelling how mental properties supervene on physical ones. Mathematics can provide precise formulations of relatively vague concepts and problems from the real world, and bring out underlying structure common to diverse scientific areas. Sometimes very natural mathematical concepts lie neglected and not widely understood for many years, before their fundamental relevance is recognised and their explanatory power is fully exploited. Previously we have argued that the notion of definability in a structure is such a concept, and pointed to Turing’s [77] 1939 model of interactive computation as providing a fruitful context in which to exercise the usefulness of definability as a powerful and widely applicable source of understanding. Below, we relate this simple idea to one of the oldest and apparently least scientifically approachable of problems — that of realistically modelling how mental properties supervene on physical ones. We will first briefly review the origins with René Descartes of mindbody dualism, and the problem of mental causation. We will then summarise the subsequent difficulties encountered, and their current persistence, and the more recent usefulness of the concept of supervenience in
Series Preproceedings of the Workshop “Physics and Computation ” 2008
, 2008
"... In the 1940s, two different views of the brain and the computer were equally important. One was the analog technology and theory that had emerged before the war. The other was the digital technology and theory that was to become the main paradigm of computation. 1 The outcome of the contest between ..."
Abstract
 Add to MetaCart
In the 1940s, two different views of the brain and the computer were equally important. One was the analog technology and theory that had emerged before the war. The other was the digital technology and theory that was to become the main paradigm of computation. 1 The outcome of the contest between these two competing views derived from technological and epistemological arguments. While digital technology was improving dramatically, the technology of analog machines had already reached a significant level of development. In particular, digital technology offered a more effective way to control the precision of calculations. But the epistemological discussion was, at the time, equally relevant. For the supporters of the analog computer, the digital model — which can only process information transformed and coded in binary — wouldn’t be suitable to represent certain kinds of continuous variation that help determine brain functions. With analog machines, on the contrary, there would be few or no layers between natural objects and the work and structure of computation (cf. [4, 1]). The 1942–52 Macy Conferences in cybernetics helped to validate digital theory and logic as legitimate ways to think about the brain and the machine [4]. In particular, those conferences helped made McCullochPitts ’ digital model
On the calculating power of Laplace’s demon (Part I)
, 2006
"... We discuss several ways of making precise the informal concept of physical determinism, drawing on ideas from mathematical logic and computability theory. We outline a programme of investigating these notions of determinism in detail for specific, precisely articulated physical theories. We make a s ..."
Abstract
 Add to MetaCart
We discuss several ways of making precise the informal concept of physical determinism, drawing on ideas from mathematical logic and computability theory. We outline a programme of investigating these notions of determinism in detail for specific, precisely articulated physical theories. We make a start on our programme by proposing a general logical framework for describing physical theories, and analysing several possible formulations of a simple Newtonian theory from the point of view of determinism. Our emphasis throughout is on clarifying the precise physical and metaphysical assumptions that typically underlie a claim that some physical theory is ‘deterministic’. A sequel paper is planned, in which we shall apply similar methods to the analysis of other physical theories. Along the way, we discuss some possible repercussions of this kind of investigation for both physics and logic. 1