Related papers: A mathematical commitment without computational st…
We give another proof of a theorem of Fife - understood broadly as providing a finite automaton that gives a complete description of all infinite binary overlap-free words. Our proof is significantly simpler than those in the literature. As…
To enable the study of open sets in computational approaches to mathematics, lots of extra data and structure on these sets is assumed. For both foundational and mathematical reasons, it is then a natural question, and the subject of this…
A constructive proof of the Goedel-Rosser incompleteness theorem has been completed using the Coq proof assistant. Some theory of classical first-order logic over an arbitrary language is formalized. A development of primitive recursive…
This is an exposition of facts about Arithmetic with an approach via mathematical logic. In Section 1 we present Peano Arithmetic, PA, and the complete theory of $\mathbb{N}$, and we show that $\mathbb{N}$ is a prime model of the theory of…
The prevalent interpretation of G\"odel's Second Theorem states that a sufficiently adequate and consistent theory does not prove its consistency. It is however not entirely clear how to justify this informal reading, as the formulation of…
We show that it is possible to define a realizability interpretation for the $\Sigma_2$-fragment of classical Analysis using G\"odel's System T only. This supplements a previous result of Schwichtenberg regarding bar recursion at types 0…
We propose an axiomatic foundation of mathematics based on the finite sequence as the foundational concept, rather than based on logic and set, as in set theory, or based on type as in dependent type theories. Finite sequences lead to a…
We reconstruct quantum theory starting from the premise that, as Asher Peres remarked, "Unperformed experiments have no results." The tools of modern quantum information theory, and in particular the symmetric informationally complete (SIC)…
Armstrong's axioms of functional dependency form a well-known logical system that captures properties of functional dependencies between sets of database attributes. This article assumes that there are costs associated with attributes and…
In synthetic computability, pioneered by Richman, Bridges, and Bauer, one develops computability theory without an explicit model of computation. This is enabled by assuming an axiom equivalent to postulating a function $\phi$ to be…
Although the categorical arithmetic is not effectively axiomatizable, the belief that the incompleteness Theorems can be apply to it is fairly common. Furthermore, the so-called "essential" (or "inherent") semantic incompleteness of the…
Algorithmic meta-theorems state that problems definable in a fixed logic can be solved efficiently on structures with certain properties. An example is Courcelle's Theorem, which states that all problems expressible in monadic second-order…
Hoare's logic is an axiomatic system of proving programs correct, which has been extended to be a separation logic to reason about mutable heap structure. We develop the most fundamental logical structure of strongest postcondition of…
e use Prolog as a flexible meta-language to provide executable specifications of some fundamental mathematical objects and their transformations. In the process, isomorphisms are unraveled between natural numbers and combinatorial objects…
This talk describes how a combination of symbolic computation techniques with first-order theorem proving can be used for solving some challenges of automating program analysis, in particular for generating and proving properties about the…
In this memoir, we seek to construct a constructive theory that is as complete as possible to describe the algebraic properties of the real number field in constructive mathematics without a dependent choice axiom. To this purpose, we use a…
Courcelle's celebrated theorem states that all MSO-expressible properties can be decided in linear time on graphs of bounded treewidth. Unfortunately, the hidden constant implied by this theorem is a tower of exponentials whose height…
We give an axiomatization of confidence transfer - a known conditioning scheme - from the perspective of expectation-based inference in the sense of Gardenfors and Makinson. Then, we use the notion of belief independence to "filter out"…
Pearl and Verma developed d-separation as a widely used graphical criterion to reason about the conditional independencies that are implied by the causal structure of a Bayesian network. As acyclic ground probabilistic logic programs…
One of the fundamental results in computability is the existence of well-defined functions that cannot be computed. In this paper we study the effects of data representation on computability; we show that, while for each possible way of…