The set theory of arithmetic decomposition

Journal ArticleThe Set Theory of Arithmetic Decomposition is a method for designing complex addition/ subtraction circuits at any radix using strictly positional, sign-local number systems. The specification of an addition circuit is simply an equation that describes the inputs and the outputs as weighted digit sets. Design is done by applying a set of rewrite rules known as decomposition operators to the equation. The order in which and weight at which each operator is applied maps directly to a physical implementation, including both multiple-level logic and connectivity. The method is readily automated and has been used to design some higher radix arithmetic circuits. It is possible to compute the cost of a given adder before the detailed design is complete

Probabilistic Aspects of Dirichlet Series

We investigate and generalise some properties of a family of probability distributions closely related to the Riemann zeta function. Random variables that have the property that divisibility by a set of distinct primes occurs as a set of independent events are characterised in terms of functions that are well known in number theory. We refer to random variables with this independence property as Khinchin random variables. In characterising the collection of Khinchin random variables, we make a connection between the probabilistic theory of discrete distributions and the number-theoretic concept of Dirichlet series. We outline some interesting correspondences between discrete probability distributions and arithmetic functions. A subset of the Khinchin random variables have infinitely divisible logarithms. We establish the necessity of a condition, already known to be sufficient, that ensures infinite divisibility. Some Khinchin random variables admit a multiplicative decomposition into a product of random prime numbers. The number of terms in such a product follows a Poisson distribution. We explore two instances of this decomposition: one related to the zeta distribution, and the other related to the so-called prime zeta function. We use the zeta distribution to derive known results from number theory via probabilistic methods, and provide a generalisation of the distribution for other unique factorisation domains

Sharply o-minimal structures and sharp cellular decomposition

Sharply o-minimal structures (denoted \so-minimal) are a strict subclass of the o-minimal structures, aimed at capturing some finer features of structures arising from algebraic geometry and Hodge theory. Sharp o-minimality associates to each definable set a pair of integers known as \emph{format} and \emph{degree}, similar to the ambient dimension and degree in the algebraic case; gives bounds on the growth of these quantities under the logical operations; and allows one to control the geometric complexity of a set in terms of its format and degree. These axioms have significant implications on arithmetic properties of definable sets -- for example, \so-minimality was recently used by the authors to settle Wilkie's conjecture on rational points in $\mathbb{R}_{{\exp}}$-definable sets. In this paper we develop some basic theory of sharply o-minimal structures. We introduce the notions of reduction and equivalence on the class of \so-minimal structures. We give three variants of the definition of \so-minimality, of increasing strength, and show that they all agree up to reduction. We also consider the problem of ``sharp cell decomposition'', i.e. cell decomposition with good control on the number of the cells and their formats and degrees. We show that every \so-minimal structure can be reduced to one admitting sharp cell decomposition, and use this to prove bounds on the Betti numbers of definable sets in terms of format and degree

Global semantic typing for inductive and coinductive computing

Inductive and coinductive types are commonly construed as ontological (Church-style) types, denoting canonical data-sets such as natural numbers, lists, and streams. For various purposes, notably the study of programs in the context of global semantics, it is preferable to think of types as semantical properties (Curry-style). Intrinsic theories were introduced in the late 1990s to provide a purely logical framework for reasoning about programs and their semantic types. We extend them here to data given by any combination of inductive and coinductive definitions. This approach is of interest because it fits tightly with syntactic, semantic, and proof theoretic fundamentals of formal logic, with potential applications in implicit computational complexity as well as extraction of programs from proofs. We prove a Canonicity Theorem, showing that the global definition of program typing, via the usual (Tarskian) semantics of first-order logic, agrees with their operational semantics in the intended model. Finally, we show that every intrinsic theory is interpretable in a conservative extension of first-order arithmetic. This means that quantification over infinite data objects does not lead, on its own, to proof-theoretic strength beyond that of Peano Arithmetic. Intrinsic theories are perfectly amenable to formulas-as-types Curry-Howard morphisms, and were used to characterize major computational complexity classes Their extensions described here have similar potential which has already been applied