It is shown that all the provably total functions of Basic Arithmetic , a theory introduced by Ruitenburg based on Predicate Basic Calculus, are primitive recursive. Along the proof a new kind of primitive recursive realizability to which is sound, is introduced. This realizability is similar to Kleene’s recursive realizability, except that recursive functions are restricted to primitive recurs...

We study the space of binary cubic and quadratic forms over the ring of integers O of an algebraic number field k. By applying the theory of prehomogeneous vector spaces founded by M. Sato and T. Shintani, we can associate the zeta functions for these spaces. Applying these zeta functions, we derive some density theorems on the distributions of discriminants of cubic algebras of O. In the case ...

We present an illative system Is of classical higher-order logic with subtyping and basic inductive types. The system Is allows for direct definitions of partial and general recursive functions, and provides means for handling functions whose termination has not been proven. We give examples of how properties of some recursive functions may be established in our system. In a technical appendix ...

We present a model that simplifies the formulation of a wide range of complex, mainly selection-based, objectoriented queries, including linear recursive queries. They are complex because it is almost impossible for naive users to predict the formulation of their predicate expressions. Naive users are mainly decision makers who are most probably not computer professionals. Therefore, it is nece...

The design of programs as the composition of smaller ones is a wide spread approach to programming. In functional programming, this approach raises the necessity of creating a good amount of intermediate data structures with the only aim of passing data from one function to another. Using program fusion techniques, it is possible to eliminate many of those intermediate data structures by an app...

We study the computability-theoretic complexity and proof-theoretic strength of the following statements: (1) “If X is a well-ordering, then so is εX ”, and (2) “If X is a well-ordering, then so is φ(α,X )”, where α is a fixed computable ordinal and φ represents the two-placed Veblen function. For the former statement, we show that ω iterations of the Turing jump are necessary in the proof and ...

A theorem published by A. Grzegorczyk in 1955 states a certain kind of effective uniform continuity of computable functionals that transform systems of total functions in the set of the natural numbers and of natural numbers into natural numbers. Namely, for any such functional a computable functional with one function-argument and the same number-arguments exists such that the values of the fi...

B functions are a class of relatively complicated exponentially decaying basis functions. Since the molecular multicenter integrals of the much simpler Slater-type functions are notoriously difficult, it is not at all obvious why B functions should offer any advantages. However, B functions have Fourier transforms of exceptional simplicity, which greatly simplifies many of their molecular multi...

1. Several procedures have been devised to establish the existence of an invariant mean for almost periodic functions on a group (see [l; 3; 4; 5]).1 The object of the present short note is to prove this basic result concerning such functions in another way. Throughout what follows the formula (x): A—*B regarding a function (x) will be used to imply that <p(x) is defined on a set A and ha...

Several problems which we studied under “pure graph problems” in Chapter (???) will now be generalized. Connectivity becomes considerably more interesting when we introduce cost functions. Connectivity has to do with paths. In the presence of cost functions, these paths has associated costs. We now study various problems related to paths of minimum cost. It is interesting to note that shortest ...