Related papers: Kleene, Rogers and Rice Theorems Revisited in C an…
We present and analyze a natural hierarchy of weak theories, develop analysis in them, and show that they are interpretable in bounded quantifier arithmetic $\text{I}\Delta_0$ (and hence in Robinson arithmetic Q). The strongest theories…
A series of works has established rewriting as an essential tool in order to prove coherence properties of algebraic structures, such as MacLane's coherence theorem for monoidal categories, based on the observation that, under reasonable…
Constructive type theory combines logic and programming in one language. This is useful both for reasoning about programs written in type theory, as well as for reasoning about other programming languages inside type theory. It is…
This thesis is devoted to the study of a calculus that describes the application of conditional rewriting rules and the obtained results at the same level of representation. We introduce the rewriting calculus, also called the rho-calculus,…
We describe a numerical algorithm for evaluating the numbers of roots minus the number of poles contained in a region based on the argument principle with the function of interest being written as a Mellin transformation of a usually…
Applying machine learning to mathematical terms and formulas requires a suitable representation of formulas that is adequate for AI methods. In this paper, we develop an encoding that allows for logical properties to be preserved and is…
When the Canonical Ramsey's Theorem by Erd\H{o}s and Rado is applied to regressive functions one obtains the Regressive Ramsey's Theorem by Kanamori and McAloon. Taylor proved a "canonical" version of Hindman's Theorem, analogous to the…
In this note we consider several kind of partition functions of one-dimensional models with nearest - neighbor interactions $I_n, n\in \mathbf{Z}$ and spin values $\pm 1$. We derive systems of recursive equations for each kind of such…
In this paper we study a new approach to classify mathematical theorems according to their computational content. Basically, we are asking the question which theorems can be continuously or computably transferred into each other? For this…
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…
The Ackermann function is a famous total recursive binary function on the natural numbers. It is the archetypal example of such a function that is not primitive recursive, in the sense of classical recursion theory. However, and in seeming…
We study Basic Arithmetic, BA introduced by W. Ruitenburg. BA is an arithmetical theory based on basic logic which is weaker than intuitionistic logic. We show that the class of the provably total recursive functions of BA is a proper…
This paper presents a theory of non-linear integer/real arithmetic and algorithms for reasoning about this theory. The theory can be conceived as an extension of linear integer/real arithmetic with a weakly-axiomatized multiplication…
In 2008, Ben-Amram, Jones and Kristiansen showed that for a simple "core" programming language - an imperative language with bounded loops, and arithmetics limited to addition and multiplication - it was possible to decide precisely whether…
`What more than its truth do we know if we have a proof of a theorem in a given formal system?' We examine Kreisel's question in the particular context of program termination proofs, with an eye to deriving complexity bounds on program…
This thesis consists of two independent parts: random matrices, which form the first one-third of this thesis, and machine learning, which constitutes the remaining part. The main results of this thesis are as follows: a necessary and…
User defined recursive types are a fundamental feature of modern functional programming languages like Haskell, Clean, and the ML family of languages. Properties of programs defined by recursion on the structure of recursive types are…
Revised proofs of Kenneth Arrow's impossibility theorem have been presented in prose form, incorporating novel ideas such as decisive sets and pivotal voters. This study develops another approach to proving the theorem. Using a proof…
Statistical properties of random cross-correlated sequences constructed by the convolution method (likewise referred to as the Rice's or the inverse Fourier transformation) are examined. Algorithms for their generation are discussed. They…
Recursive analysis was introduced by A. Turing [1936], A. Grzegorczyk [1955], and D. Lacombe [1955]. It is based on a discrete mechanical framework that can be used to model computation over the real numbers. In this context the…