This paper presents general methods for studying the problems of translatability between classes of schemes and equivalence of schemes in a given class. There are four methods: applying the theory of formal languages, programming, measuring the complexity of a computation, and "cutting and pasting." These methods are used to answer several questions of translatability and equivalence for classe...

α-recursion lifts classical recursion theory from the first transfinite ordinal ω to an arbitrary admissible ordinal α [13]. Turing machine models for α-recursion and other types of transfinite computation have been proposed and studied [5] and [7] and are applicable in computational approaches to the foundations of logic and mathematics [11]. They also provide a natural setting for modeling ex...

We describe how to perform monadic computations over recursive data structures with fine grained control over the evaluation strategy. This solves the issue that the definition of a recursive monadic function already determines the evaluation strategy due to the necessary sequencing of the monadic operations. We show that compositional data types already provide the structure needed in order to...

We present a high-level programming abstraction which extends the concept of collection and array. The new construct is based on an algebra of index sets, called a shape, and a functional extension of the array type, the eld type. Shape constructions are based on group theory and we introduce the neighborhood compliant recursive deenition of elds. Then we show how to implement the computation o...

The spreadsheet metaphor has, over the years, proved itself valuable for the definition and use of computations by non-programmers. However, the computation model adopted in commercial spreadsheets is still limited to non-recursive computations and lacks abstraction mechanisms that would provide modularization and better reuse (beyond copy and paste). We investigate these problems by identifyin...

The minimum mean square error (MMSE) estimation problem for a discrete-index reciprocal Gaussian process impaired by additive white Gaussian noise is completely solved in the general case of noisily observed Dirichlet random boundary conditions. Finite sets of recursive equations are obtained for the computation of the filtered sequence and of the fixed-point, fixed-interval, and fixed-lag smoo...

In 2000, Wu presented two new types of generalized Ball curves, one of which is called an NB1 curve located between the Wang–Ball curve and the Said–Ball curve. In this article, the authors aim to discuss properties of NB1 curves and surfaces, including the recursive algorithms, conversion algorithms between NB1 and Bézier curves and surfaces, etc. In addition the authors compare the computatio...

A polarized strong category consists of a cartesian category, X, and a category Y, together with a module M : X × Y −→ Y equipped with a strong composition and identities. These categories can be used to provide an abstract setting for investigating computational setting with complexity below primitive recursive. This paper develops the theory of polarized strong categories, explains how they r...