Related papers: The provably total recursive functions and the MRD…
Induction is typically formalized as a rule or axiom extension of the LK-calculus. While this extension of the sequent calculus is simple and elegant, proof transformation and analysis can be quite difficult. Theories with an induction…
In this article, we will show that uncomputability is a relative property not only of oracle Turing machines, but also of subrecursive classes. We will define the concept of a Turing submachine, and a recursive relative version for the Busy…
An approach to build Probabilistic Arithmetic in which initial values of all correlated random variables are known, but with varying degrees of accuracy. As a result of the proposed Probabilistic Arithmetic operations, variable values,…
The representation of mathematical objects in terms of (more) basic ones is part and parcel of (the foundations of) mathematics. In the usual foundations of mathematics, i.e. $\textsf{ZFC}$ set theory, all mathematical objects are…
I deal with two approaches to proof-theoretic semantics: one based on argument structures and justifications, which I call reducibility semantics, and one based on consequence among (sets of) formulas over atomic bases, called base…
In this paper we provide a semantic and syntactic analysis of parametrised natural numbers object in coherent categories, or pr-coherent categories. Semantically, we show the definable functions in the initial pr-coherent category are…
Classical theory proves that every primitive recursive function is strongly representable in PA; that formal Peano Arithmetic, PA, and formal primitive recursive arithmetic, PRA, can both be interpreted in Zermelo-Fraenkel Set Theory, ZF;…
Primitive recursion is a mature, well-understood topic in the theory and practice of programming. Yet its dual, primitive corecursion, is underappreciated and still seen as exotic. We aim to put them both on equal footing by giving a…
We present a unified theory for formal mathematical systems including recursive systems closely related to formal grammars, including the predicate calculus as well as a formal induction principle. We introduce recursive systems generating…
A longstanding open problem is whether there exists a non-syntactical model of untyped lambda-calculus whose theory is exactly the least equational lambda-theory (=Lb). In this paper we make use of the Visser topology for investigating the…
We illustrate the use of the notion of derived recurrences introduced earlier to evaluate the algebraic entropy of self-maps of projective spaces. We in particular give an example, where a complete proof is still awaited, but where…
We explore recursive programming with extensible data types. Row types make the structure of data types first class, and can express a variety of type system features including record subtyping and combination of case branches. Our goal is…
In this paper we outline the most general and universal algorithmic approach to reduction of loop integrals to basic integrals. The approach is based on computation of Groebner bases for recurrence relations derived from the integration by…
Moore introduced a class of real-valued "recursive" functions by analogy with Kleene's formulation of the standard recursive functions. While his concise definition inspired a new line of research on analog computation, it contains some…
Calculi with control operators have been studied as extensions of simple type theory. Real programming languages contain datatypes, so to really understand control operators, one should also include these in the calculus. As a first step in…
In this note we axiomatize the classes of rudimentary functions, primitive recursive functions, safe recursive set functions, and predicatively computable functions.
We extend two well-known results on primitive ideals in enveloping algebras of semisimple Lie algebras, the `Irreducibility theorem' and `Duflo theorem', to much wider classes of algebras. Our general version of Irreducibility theorem says…
In this work we propose a multi-valued extension of logic programs under the stable models semantics where each true atom in a model is associated with a set of justifications, in a similar spirit than a set of proof trees. The main…
While the theory of matrix-weighted function spaces is well established, the majority of previous results in the infinite-dimensional operator-valued setting deal with "no go" theorems, showing the impossibility of some prospective…
We prove recursive formulas involving sums of divisors and sums of triangular numbers and give a variety of identities relating arithmetic functions to divisor functions providing inductive identities for such arithmetic functions.