Chapter 13 - The Structure of Computability in Analysis and Physical Theory: An Extension of Church's Thesis
Introduction
This paper is devoted to a survey of computability in analysis and physical theory. Since computers are playing an ever increasing role in solving problems in these and related fields, it is useful to know – at least theoretically – which processes are computable and which are not.
A major consequence of the research described here is the extension of the notion of computability to computability on a Banach space. Recall that Banach space theory is a fundamental tool in analysis, physics and engineering – from the solutions of differential equations in the classical theory to the study of quantum theory. As examples of Banach spaces which are particularly useful we have Hilbert space (more generally Lp-spaces), Sobolev spaces, and others. The Banach space C[a, b], of continuous functions on [a, b], is of particular interest to logicians. Computable continuous functions of a real variable were defined and studied in the 1950's by Grzegorczyk [1957] and Lacombe [1955a, 1955b, 1957]. Thus, the work presented here may be regarded as a generalization of the work begun in the 1950's. It is worth remarking that Grzegorczyk and Lacombe gave several definitions for the computability of continuous functions of a real variable. All of these were proved to be equivalent. The proofs were complicated. We will see that the equivalence of these definitions is an immediate consequence of the notion of a “computability structure” on a Banach space. No additional proof is necessary.
The notion of a computability structure on a Banach space is of central importance in this paper. It accords well with the intuitive notion of computability. The definition is given axiomatically. The axioms provide for the interaction of elementary recursion theory with the basic tenets of Banach space theory. The concept which is axiomatized is “computable sequence of elements” of the Banach space. A point x in the Banach space is computable if the constant sequence x, x, x, … is. It is both necessary and natural to work with sequences of elements rather than individual elements. There are two reasons for this – topological and recursion – theoretic. Since a Banach space is a metric space, the topology can be given by sequences. In recursion theory one of basic notions is the notion of a recursively enumerable set – a set whose elements can be arranged in a computable sequence.
The computability structure is characterized by three axioms. This is not surprising, since the Banach space, itself, is defined by three properties. It is a linear space, with a norm, which is complete in the norm. It is then shown that, under general conditions which in practice are always satisfied, the computability structure on a Banach space is unique. Thus, we are able to achieve the intrinsic quality we associate with the notion of computability. The situation is reminiscent of the one in ordinary recursion theory, when the various definitions, proposed by Turing, Herbrand/Gödel, Church, Post and others, all intuitively convincing, were proved to be equivalent. The notion of a computability structure acts as a unifying concept, since seemingly different definitions of computability, are, in fact, equivalent because of this unicity. Thus, we have a “Church's Thesis” for the given Banach space.
It should be remarked that the notion of computability which appears in the axioms for a computability structure depends ultimately on elementary recursive function theory – i.e. the notion of a recursive function mapping N into N and of a recursively enumerable set. These are combined with the basic facts of Banach space theory to produce the axioms referred to above. No prior knowledge of Banach space theory is presupposed in this paper. Any facts which are required – and they are all quite elementary – will be stated precisely.
It may be useful to comment further on the computability structure. As remarked above, the structure is characterized by three axioms. There is one axiom for each of the basic concepts of Banach space theory – linearity, norm and limit. The axiom provides for the interaction between the associated concept and recursive function theory. When viewed in this light the computability structure is minimal – just sufficient for the fundamental notions of recursive function theory to interact with the basic concepts of Banach space theory. As stated earlier, it will be shown that under very natural conditions the computability structure is not merely minimal, but also maximal – in fact, unique.
We will return to a more detailed discussion of the computability structure and its applications later. We now give a summary of the contents of this paper.
The paper is divided into two parts. Part I is, in essence, a primer on computable analysis. Part II contains the main results. It is suggested that the reader glance briefly at Part I, and go directly to Part II.
Part I begins with the definition of a computable real. We note that the computable reals form a field which contains all algebraic numbers and also the well-known transcendentals – e.g., π and e. This is followed by an account of computability for C[a, b], where a and b are computable reals. (Recall that C[a, b] is the set of continuous functions on [a, b]. It is a Banach space with ||f|| = supa ≤ x ≤ b |f(x)|.) The discussion includes integration, differentiation, the max.–min. theorem and the intermediate value theorem, all from the viewpoint of computability. Of particular interest is the relation of computability to physical theory. For example, the propagation of waves need not be computable, even if the initial conditions which determine the wave propagation uniquely are computable. However, heat dissipation is always computable if the initial conditions are.
Part I concludes with a section on computability for Lp-spaces.
We now turn to Part II and the principal results of this work. They are contained in three general theorems: The First Main Theorem, The Second Main Theorem, and the Eigenvector Theorem. The notion of a computability structure plays a fundamental role in the formulation of these theorems, as we now explain.
Why is it the case that wave propagation can be noncomputable even if the initial conditions which uniquely determine the propagation are computable? However, the dissipation of heat is always computable whenever the initial conditions are. The answer is given by the First Main Theorem. Roughly the First Main Theorem states: under general conditions which in practice are always satisfied
Bounded linear operators from one Banach space with a computability structure to another preserve computability, unbounded linear operators do not.
Wave propagation is associated with an unbounded linear operator, whereas the dissipation of heat is associated with a bounded operator. Thus waves can propagate noncomputably even though the initial conditions are computable. However, heat will always dissipate computably.
The First Main Theorem can be applied to a host of examples: integration, differentiation, Fourier series, Fourier transform and many others. We will discuss this briefly in Part II. The fact that the First Main Theorem is so widely applicable is a consequence of two other facts. First, the Theorem is easy to apply. Second, essentially all of the well-known operators in analysis are linear. Indeed, mathematicians are just beginning to study nonlinearity.
We now turn to the Second Main Theorem. It is concerned with the computability/noncomputability of eigenvalues. Since eigenvalues are the quantities which are measured in experiments, it is of some interest to determine whether or not they are computable.
The setting for the Second Main Theorem is self-adjoint linear operators on Hilbert space. Thus the eigenvalues are real numbers. The operators may be bounded or unbounded. We are led to the following questions: are the eigenvalues – i.e. the quantities which are measured – computable real numbers? Can the eigenvalues be arranged in a computable sequence? The Second Main Theorem answers these questions and the answers are quite easy to state. Under mild side conditions which in practice are always satisfied, each eigenvalue is a computable real number. However, in general, the eigenvalues cannot be arranged in a computable sequence.
Incidentally, linear operators which satisfy these mild side conditions are referred to as “effectively determined”. All of the standard operators of analysis and physics are effectively determined.
The Second Main Theorem has many corollaries. Some of these will be discussed in Part II. The Theorem can be extended to bounded normal operators.
The Eigenvector Theorem, our third major result, is concerned with the computability/noncomputability of eigenvectors. Recall that eigenvectors have some physical significance: they are associated with the “state of the system”. Hence it is of interest to determine whether or not they are computable. The Eigenvector Theorem asserts that there exists an effectively determined, compact, self-adjoint operator such that 0 is an eigenvalue. However, none of the eigenvectors corresponding to zero is computable.
We now turn to Part I.
Section snippets
Addendum: Open Problems
There are many open problems. Some were discussed in Pour-El and Richards [1989, pp. 192–194], In recent years progress has been made in two of the seven problem areas listed in Pour-El and Richards [1989].
The first is concerned with Problem 5, the recursive topology for Rn. The notion of an r.e. open set is well-known. Using the definitions of an r.e. closed set and of a recursive closed set formulated by Pour-El and Richards in the 1980's, Qing Zhou investigated the recursive topology of Rn
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