Related papers: A functional interpretation for nonstandard arithm…
We show that the classical interpretations of Tarski's inductive definitions actually allow us to define the satisfaction and truth of the quantified formulas of the first-order Peano Arithmetic PA over the domain N of the natural numbers…
This paper relates comparative belief structures and a general view of belief management in the setting of deductively closed logical representations of accepted beliefs. We show that the range of compatibility between the classical…
We propose a generating functional for nonrelativistic gauge invariant actions. In particular, we consider actions without the usual magnetic term. Like in the Born-Infeld theory, there is an upper bound to the electric field strength in…
Cantor's ordinal numbers, a powerful extension of the natural numbers, are a cornerstone of set theory. They can be used to reason about the termination of processes, prove the consistency of logical systems, and justify some of the core…
In 1960s, Dana Scott gave a recursion theoretic characterization of standard systems of countable non-standard models of arithmetic, i.e., collections of sets of standard natural numbers coded in non-standard models. Later, Knight and Nadel…
We show some non-standard Poincar\'e type estimates in the biparametric setting with appropriate weights. We will derive these results using variants from classical estimates exploiting the interplay between maximal functions and fractional…
Expressive querying of machine learning models - viewed as a form of intentional data - enables their verification and interpretation using declarative languages, thereby making learned representations of data more accessible. Motivated by…
The principal aim of this article is to establish an iteration method on the space of resurgent functions. We discuss endless continuability of iterated convolution products of resurgent functions and derive their estimates developing the…
As a supplement to my talk at the workshop, this extended abstract motivates and summarizes my work with co-authors on problems in two separate areas: first, in the lambda-calculus with letrec, a universal model of computation, and second,…
We extend first-order logic to include variadic function symbols, and prove a substitution lemma. Two applications are given: one to bounded quantifier elimination and one to the definability of certain Borel sets.
We use fast-growing finite and infinite sequences of natural numbers and more complicated constructs to define models of hypercomputation and interpret non-arithmetic predicates, with the strongest extensions reaching full second order…
The logic embedding tool provides a procedural encoding for non-classical reasoning problems into classical higher-order logic. It is extensible and can support an increasing number of different non-classical logics as reasoning targets.…
In order to apply nonstandard methods to modern algebraic geometry, as a first step in this paper we study the applications of nonstandard constructions to category theory. It turns out that many categorial properties are well behaved under…
This is a Research and Instructional Development Project from the U. S. Naval Academy. In this monograph, the basic methods of nonstandard analysis for n-dimensional Euclidean spaces are presented. Specific rules are deveoped and these…
The idea of using unfolding as a way of computing a program semantics has been applied successfully to logic programs and has shown itself a powerful tool that provides concrete, implementable results, as its outcome is actually source…
We develop a new paradigm for finding bifurcations of solutions of nonlinear problems, which is based on the detection of extreme values of new type of variational functional associated with the considering problem. The variational…
For non-anticipative functionals, differentiable in Chitashvili's sense, the It\^o formula for cadlag semimartingales is proved. Relations between different notions of functional derivatives are established.
This paper outlines new paradigms for real analysis and computability theory in the recently proposed non-Aristotelian finitary logic (NAFL). Constructive real analysis in NAFL (NRA) is accomplished by a translation of diagrammatic concepts…
Generalizations of classical theta functions are proposed that include any even number of analytic parameters for which conditions of quasi-periodicity are fulfilled and that are representations of extended Heisenberg group. Differential…
We extend Zeilberger's approach to special function identities to cases that are not holonomic. The method of creative telescoping is thus applied to definite sums or integrals involving Stirling or Bernoulli numbers, incomplete Gamma…