Complexity of term representations of finitary functions

The clone of term operations of an algebraic structure consists of all operations that can be expressed by a term in the language of the structure. We consider bounds for the length and the height of the terms expressing these functions, and we show that these bounds are often robust against the change of the basic operations of the structure

Hilbert's Program Then and Now

Hilbert's program was an ambitious and wide-ranging project in the philosophy and foundations of mathematics. In order to "dispose of the foundational questions in mathematics once and for all, "Hilbert proposed a two-pronged approach in 1921: first, classical mathematics should be formalized in axiomatic systems; second, using only restricted, "finitary" means, one should give proofs of the consistency of these axiomatic systems. Although Godel's incompleteness theorems show that the program as originally conceived cannot be carried out, it had many partial successes, and generated important advances in logical theory and meta-theory, both at the time and since. The article discusses the historical background and development of Hilbert's program, its philosophical underpinnings and consequences, and its subsequent development and influences since the 1930s.Comment: 43 page

Finitary and Infinitary Mathematics, the Possibility of Possibilities and the Definition of Probabilities

Some relations between physics and finitary and infinitary mathematics are explored in the context of a many-minds interpretation of quantum theory. The analogy between mathematical ``existence'' and physical ``existence'' is considered from the point of view of philosophical idealism. Some of the ways in which infinitary mathematics arises in modern mathematical physics are discussed. Empirical science has led to the mathematics of quantum theory. This in turn can be taken to suggest a picture of reality involving possible minds and the physical laws which determine their probabilities. In this picture, finitary and infinitary mathematics play separate roles. It is argued that mind, language, and finitary mathematics have similar prerequisites, in that each depends on the possibility of possibilities. The infinite, on the other hand, can be described but never experienced, and yet it seems that sets of possibilities and the physical laws which define their probabilities can be described most simply in terms of infinitary mathematics.Comment: 21 pages, plain TeX, related papers from http://www.poco.phy.cam.ac.uk/~mjd101

Synchronization and Control in Intrinsic and Designed Computation: An Information-Theoretic Analysis of Competing Models of Stochastic Computation

We adapt tools from information theory to analyze how an observer comes to synchronize with the hidden states of a finitary, stationary stochastic process. We show that synchronization is determined by both the process's internal organization and by an observer's model of it. We analyze these components using the convergence of state-block and block-state entropies, comparing them to the previously known convergence properties of the Shannon block entropy. Along the way, we introduce a hierarchy of information quantifiers as derivatives and integrals of these entropies, which parallels a similar hierarchy introduced for block entropy. We also draw out the duality between synchronization properties and a process's controllability. The tools lead to a new classification of a process's alternative representations in terms of minimality, synchronizability, and unifilarity.Comment: 25 pages, 13 figures, 1 tabl

First Order Theories of Some Lattices of Open Sets

We show that the first order theory of the lattice of open sets in some natural topological spaces is $m$-equivalent to second order arithmetic. We also show that for many natural computable metric spaces and computable domains the first order theory of the lattice of effectively open sets is undecidable. Moreover, for several important spaces (e.g., $\mathbb{R}^n$, $n\geq1$, and the domain $P\omega$) this theory is $m$-equivalent to first order arithmetic