Related papers: The computational content of classical arithmetic
The theory of finite term algebras provides a natural framework to describe the semantics of functional languages. The ability to efficiently reason about term algebras is essential to automate program analysis and verification for…
Several different proof translations exist between classical and intuitionistic logic (negative translations), and intuitionistic and linear logic (Girard translations). Our aims in this paper are (1) to consider extensions of…
This survey provides a practical and algorithmic perspective on Drinfeld modules over $\mathbb F_q[T]$. Starting with the construction of the Carlitz module, we present Drinfeld modules in any rank and some of their arithmetic properties.…
We extend the theoretical framework of proof mining by establishing general logical metatheorems that allow for the extraction of the computational content of theorems with prima facie "non-computational" proofs from probability theory,…
This work proposes a complete algebraic model for classical information theory. As a precursor the essential probabilistic concepts have been defined and analyzed in the algebraic setting. Examples from probability and information theory…
The purpose of this thesis is to make an analysis of the concept of Hypercomputation and of some hypermachines. This thesis is separated in three main parts. We start in the first chapter with an analysis of the concept of Classical…
Verifying software correctness has always been an important and complicated task. Recently, formal proofs of critical properties of algorithms and even implementations are becoming practical. Currently, the most powerful automated proof…
Quantum theory (QT) has been confirmed by numerous experiments, yet we still cannot fully grasp the meaning of the theory. As a consequence, the quantum world appears to us paradoxical. Here we shed new light on QT by having it follow from…
Quantum computations operate in the quantum world. For their results to be useful in any way, there is an intrinsic necessity of cooperation and communication controlled by the classical world. As a consequence, full formal descriptions of…
A provenance analysis for a query evaluation or a model checking computation extracts information on how its result depends on the atomic facts of the model or database. Traditional work on data provenance was, to a large extent, restricted…
Brouwer's constructivist foundations of mathematics is based on an intuitively meaningful notion of computation shared by all mathematicians. Martin-L\"of's meaning explanations for constructive type theory define the concept of a type in…
We devise a classical algorithm which efficiently computes the quantum expectation values arising in a class of continuous variable quantum circuits wherein the final quantum observable | after the Heisenberg evolution associated with the…
The book "A Course in Constructive Algebra" (1988) shows the way of understanding classical basic algebra in a constructive style similar to Bishop's Constructive Mathematics. Classical theorems are revisited, with a new flavour, and become…
Teaching proofs is a crucial component of any undergraduate-level program that covers formal reasoning. We have developed a calculational reasoning format and refined it over several years of teaching a freshman-level course, "Logic and…
The likelihood of an automated reasoning program being of substantial assistance for a wide spectrum of applications rests with the nature of the options and parameters it offers on which to base needed strategies and methodologies. This…
Given a computable sequence of natural numbers, it is a natural task to find a G\"odel number of a program that generates this sequence. It is easy to see that this problem is neither continuous nor computable. In algorithmic learning…
This introduction to arithmetic coding is divided in two parts. The first explains how and why arithmetic coding works. We start presenting it in very general terms, so that its simplicity is not lost under layers of implementation details.…
There are several approaches for using computers in deriving mathematical proofs. For their illustration, we provide an in-depth study of using computer support for proving one complex combinatorial conjecture -- correctness of a strategy…
The model theory of a first-order logic called N^4 is introduced. N^4 does not eliminate double negations, as classical logic does, but instead reduces fourfold negations. N^4 is very close to classical logic: N^4 has two truth values;…
Glivenko's theorem says that, in propositional logic, classical provability of a formula entails intuitionistic provability of double negation of that formula. We generalise Glivenko's theorem from double negation to an arbitrary nucleus,…