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Recently a new kind of approximation to continuum topological spaces has been introduced, the approximating spaces being partially ordered sets (posets) with a finite or at most a countable number of points. The partial order endows a poset…
We show that every finite semilattice can be represented as an atomized semilattice, an algebraic structure with additional elements (atoms) that extend the semilattice's partial order. Each atom maps to one subdirectly irreducible…
On one hand, termination analysis of logic programs is now a fairly established research topic within the logic programming community. On the other hand, non-termination analysis seems to remain a much less attractive subject. If we divide…
I introduce an approach for automated reasoning in first order set theories that are not finitely axiomatizable, such as $ZFC$, and describe its implementation alongside the automated theorem proving software E. I then compare the results…
This paper explores the application of automated planning to automated theorem proving, which is a branch of automated reasoning concerned with the development of algorithms and computer programs to construct mathematical proofs. In…
A fundamental question is whether Turing machines can model all reasoning processes. We introduce an existence principle stating that the perception of the physical existence of any Turing program can serve as a physical causation for the…
The aim of the paper is to study the limit distributions and the asymptotic behavior of summation arithmetic functions. A probabilistic approach based on the use of the axioms of probability theory is used for these purposes. Sufficient…
We introduce semiframes (an algebraic structure) and investigate their duality with semitopologies (a topological one). Both semitopologies and semiframes are relatively recent developments, arising from a novel application of topological…
This dissertation is concerned with the study of program equivalence and algebraic effects as they arise in the theory of programming languages. Algebraic effects represent impure behaviour in a functional programming language, such as…
The paper considers the properties of pseudo stationarity in a broad sense and pseudo strong mixing for sequences of random variables corresponding to arithmetic functions. Assertions on this topic have been proven. The implementation of…
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…
We study the termination of rewriting modulo a set of equations in the Calculus of Algebraic Constructions, an extension of the Calculus of Constructions with functions and predicates defined by higher-order rewrite rules. In a previous…
We present a comprehensive programme analysing the decomposition of proof systems for non-classical logics into proof systems for other logics, especially classical logic, using an algebra of constraints. That is, one recovers a proof…
We prove, for stably computably enumerable formal systems, direct analogues of the first and second incompleteness theorems of G\"odel. A typical stably computably enumerable set is the set of Diophantine equations with no integer…
A valid compiler optimisation transforms a block in a program without introducing new observable behaviours to the program as a whole. Deciding which optimisations are valid can be difficult, and depends closely on the semantic model of the…
The study of semifeasible algorithms was initiated by Selman's work a quarter of century ago [Sel79,Sel81,Sel82]. Informally put, this research stream studies the power of those sets L for which there is a deterministic (or in some cases,…
We study structural limitations of purely algebraic reasoning in the analysis of arithmetic dynamical systems. Rather than addressing the truth of specific conjectures, we introduce a fragment - relative notion of algebraic refutability for…
Machine learning algorithms use error function minimization to fit a large set of parameters in a preexisting model. However, error minimization eventually leads to a memorization of the training dataset, losing the ability to generalize to…
We establish rigidity for partial transformation groupoids associated with algebraic actions of semigroups: If two such groupoids (satisfying appropriate conditions) are isomorphic, then the globalizations of the initial algebraic actions…
Parametricity states that polymorphic functions behave the same regardless of how they are instantiated. When developing polymorphic programs, Wadler's free theorems can serve as free specifications, which can turn otherwise partial…