Related papers: Recursion does not always help
In Chapter 3 of his Notes on constructive mathematics, Martin-L{\"o}f describes recursively constructed ordinals. He gives a constructively acceptable version of Kleene's computable ordinals. In fact, the Turing definition of computable…
We define two extensions of the typed linear lambda-calculus that yield minimal Turing-complete systems. The extensions are based on unbounded recursion in one case, and bounded recursion with minimisation in the other. We show that both…
In this note, we establish some new results on some special types of function algebras and also give new proofs to some existing ones
We prove that Tietze Extension does not always exist in constructive mathematics if closed sets on which the function we are extending are defined as sequentially closed sets. Firstly, we take a discrete metric space as our topological…
For $\lambda$ inaccessible, we may consider $(< \lambda)$-support iteration of some specific $(<\lambda)$-complete $\lambda^+$-c.c. forcing notion. But this fails a "preservation by restricting to a sub-sequence of the forcing, we "correct"…
If the result of an expensive computation is invalidated by a small change to the input, the old result should be updated incrementally instead of reexecuting the whole computation. We incrementalize programs through their derivative. A…
Incremental computation has recently been studied using the concepts of change structures and derivatives of programs, where the derivative of a function allows updating the output of the function based on a change to its input. We…
It is well known that any power series over a finite field represents a rational function if and only if its sequence of coefficients is ultimately periodic. The famous Christol's Theorem states that a power series over a finite field is…
The categorical models of the differential lambda-calculus are additive categories because of the Leibniz rule which requires the summation of two expressions. This means that, as far as the differential lambda-calculus and differential…
A finite-dimensional unital and associative algebra over $\mathbb{R}$, or what we shall call simply "an algebra" in this paper for short, generalities the construction by which we derive the complex numbers by "adjoining an element $i$" to…
The logical technique of focusing can be applied to the $\lambda$-calculus; in a simple type system with atomic types and negative type formers (functions, products, the unit type), its normal forms coincide with $\beta\eta$-normal forms.…
We derive recursions for the probability distribution of random sums by computer algebra. Unlike the well-known Panjer-type recursions, they are of finite order and thus allow for computation in linear time. This efficiency is bought by the…
Delimited control operator shift0 exhibits versatile capabilities: it can express layered monadic effects, or equivalently, algebraic effects. Little did we know it can express lambda calculus too! We present $ \Lambda_\$ $, a call-by-value…
Most of theoretical physics is based on the mathematics of functions of a real or a complex variable; yet we frequently are drawn to try extending our reach to include quaternions. The non-commutativity of the quaternion algebra poses…
Using large language models (LMs) for query or document expansion can improve generalization in information retrieval. However, it is unknown whether these techniques are universally beneficial or only effective in specific settings, such…
We employ the notions of `sequential function' and `interrogation' (dialogue) in order to define new partial combinatory algebra structures on sets of functions. These structures are analyzed using J. Longley's preorder-enriched category of…
We provide a computational definition of the notions of vector space and bilinear functions. We use this result to introduce a minimal language combining higher-order computation and linear algebra. This language extends the Lambda-calculus…
The cogrowth of a subgroup is defined as the growth of a set of coset representatives which are of minimal length. A subgroup is essential if it intersects non-trivially every non-trivial subgroup. The main result of this paper is that…
We study a recursion that generates real sequences depending on a parameter $x$. Given a negative $x$ the growth of the sequence is very difficult to estimate due to canceling terms. We reduce the study of the recursion to a problem about a…
A typical way of analyzing the time complexity of functional programs is to extract a recurrence expressing the running time of the program in terms of the size of its input, and then to solve the recurrence to obtain a big-O bound. For…