Bennett / Hoffman / Prakash Observer Mechanics

A Formal Theory of Perception
1. Auflage 2014
ISBN: 978-1-4832-6313-7
Verlag: Elsevier Science & Techn.
Format: EPUB
Kopierschutz: 6 - ePub Watermark

E-Book, Englisch, 298 Seiten

Observer Mechanics: A Formal Theory of Perception provides information pertinent to the fundamental aspects of perception. This book provides an approach to the study of perception that attempts to be both general and rigorous. Organized into 10 chapters, this book begins with an overview of the structure of perceptual capacity. This text then presents the relationship between observers and Turing machines. Other chapters provide a formal framework in which to describe an observer and its objects of perception, and then develop from this framework a perceptual dynamics. This book discusses as well the conditions in which an observer may be said to perceive truly and discusses how stabilities in perceptual dynamics might permit the genesis of higher level observers. The final chapter deals with the relationship between the formalisms of quantum mechanics and observer mechanics. This book is a valuable resource for physicists, psychophysicists, philosophers, cognitive scientists, and perceptual psychologists.

Autoren/Hrsg.

Bennett, Bruce M.

Hoffman, Donald D.

Prakash, Chetan

Leseproben

CHAPTER TWO

DEFINITION OF OBSERVER

In this chapter we define the concept The previous chapter introduced this notion by concrete examples. We now abstract from these examples a formal definition. We discuss the definition, discuss under what conditions an observer is ideal, and give an example.

1 Mathematical notation and terminology

The definition of observer given in the next section makes use of several mathematical concepts from probability and measure theory. In this section we collect basic terminology and notation from these fields for the convenience of the reader.1

Let be an arbitrary abstract space, namely a nonempty set of elements called “points.” Points are often denoted generically by A collection of subsets of is called a s- if it contains itself and is closed under the set operations of complementation and countable union (and is therefore closed under countable intersection as well). The pair (X, ) is called a and any set in is called an If (X, ) is a measurable space and ? is any subset, we define a s-algebra on as follows: = { n | ? }. This measurable structure on is called the A map p from a measurable space (X, ) to another measurable space (Y, ), p: ? , is said to be if p-1() is in for each in ; this is indicated by writing p ? /. In this case the set s(p) = {p-1 () | ? } is a subs-algebra of , called the s- p. It is also denoted p*. A measurable function p is said to be if, moreover, p() is in for all ? . A measurable function whose range is R or is also called a ; the symbol also denotes the random variables on (The s-algebra on R or is described in the next paragraph.) A on the measurable space (X, ) is a map from to R ? {8}, such that the measure of a countable union of disjoint sets in is the sum of their individual measures. A measure is if the range of lies in the closed interval [0, 8]. A measure is called s- if the space is a countable union of events in , each having finite measure. A property is said to hold “µ almost surely” (abbreviated a.s.) or “µ almost everywhere” (µ a.e.) if it holds everywhere except at most on a set of -measure zero. A of a measure is any measurable set with the property that its complement has measure zero. If is a discrete set whose s-algebra is the collection of all its subsets, then is the measure defined by ({}) = 1 for all ? A is a measure whose range is the closed interval [0, 1] and that satisfies () = 1. A is a probability measure supported on a single point. If and are two measures defined on the same measurable space, we say that is (written ) on a measurable set if () = 0 for every ? with () = 0. A on (X, ) is an equivalence class of positive measures on under the equivalence relation of mutual absolute continuity. Given a measure space (X, , ) and a mapping from (X, , ) to a measurable space (Y, ), one can induce a measure *µ on (Y, ) by (*µ)() = (-1()). Then *µ is called the , or the , or the .

If and are two topological spaces, a map : ? is if -1() is an open set of whenever is an open set of . A continuous is a if it has a continuous inverse. A for a topology is any collection of sets that are open and such that any open set is a union of sets in the basis. A topological space is called if it has a countable basis. The smallest s-algebra containing the open sets of a topology (and therefore also the closed sets) is called the s- or the A on a set is a function : × ? R+ = [0, 8) such that for all ? () = 0 iff = , (, ) = (, ), and (, ) + (, ) = (). Given ? > 0, the set (, ?) = {|() < ?} is called the ?- A topological space is if there is a metric on the space such that the open balls in the metric are a basis for the topology. A is a separable metrizable topological space with a s-algebra generated by the topology. The topology on R or is here taken to be that generated by the open intervals. The associated measurable structure constitutes the ? is the unique measure on the Borel structure such that ?((, )) = - for = . The is the smallest s-algebra containing all Borel sets and all subsets of measure zero Borel sets. Lebesgue measure ? then extends to a measure with the same name on the Lebesgue structure.

Let be a finite measure on . Let denote the set of functions from to . The relation ~ on defined by ~ iff = , -almost everywhere, is an equivalence relation. Let be the collection of equivalence classes of under ~. is a vector space which has a distinguished subspace 1 (, ) and a linear function

1(X,µ)?Rf??fdµ

with the following three properties (by an abuse of notation we do not distinguish between functions and their equivalence classes):

(i) 1(, ) contains all indicator functions 1, for ? ;

(ii) For all ? , ? 1 = ();

(iiii) If {} is an increasing sequence of nonnegative functions in 1 (, ) and if () = ?8(), then ? 1(, ) iff ?8 ? < 8. In that case ? = ?8 ? .

Let (X, ), (Y, ) be measurable spaces. A or a × is a mapping : × ? R ? {8}, such that

(i) for every in , the mapping ? () is a measure on , denoted by (, ·);

(ii) for every in , the mapping ? () is a measurable function on , denoted by (·, ).

is called if its range is in [0, 8] and if it is positive and, for all ? , (, ) = 1. If = we simply say that is a In what follows, unless otherwise stated. If is a kernel on ×...