Related papers: Kleene's Two Kinds of Recursion
Representation theorems relate seemingly complex objects to concrete, more tractable ones. In this paper, we take advantage of the abstraction power of category theory and provide a general representation theorem for a wide class of…
Resolution modulo is a first-order theorem proving method that can be applied both to first-order presentations of simple type theory (also called higher-order logic) and to set theory. When it is applied to some first-order presentations…
Strict-Tolerant Logic (ST) underpins naive theories of truth and vagueness (respectively including a fully disquotational truth predicate and an unrestricted tolerance principle) without jettisoning any classically valid laws. The classical…
Recently, in Axioms 10(2): 119 (2021), a nonclassical first-order theory T of sets and functions has been introduced as the collection of axioms we have to accept if we want a foundational theory for (all of) mathematics that is not weaker…
Sharpness is an almost generic assumption in continuous optimization that bounds the distance from minima by objective function suboptimality. It facilitates the acceleration of first-order methods through restarts. However, sharpness…
Building a search relevance model that achieves both low latency and high performance is a long-standing challenge in the search industry. To satisfy the millisecond-level response requirements of online systems while retaining the…
Let $F$ be a probability measure on $\mathbb{R}$ in the domain of attraction of a stable law with exponent $\alpha\in (0, 1)$. We establish integral criteria on $F$ that significantly expand the probabilistic approach to Strong Renewal…
Classical results in computability theory, notably Rice's theorem, focus on the extensional content of programs, namely, on the partial recursive functions that programs compute. Later and more recent work investigated intensional…
Many theorems of mathematics have the form that for a certain problem, e.g. a differential equation or polynomial (in)equality, there exists a solution. The sequential version then states that for a sequence of problems, there is a sequence…
In the theory of conditional sets, many classical theorems from areas such as functional analysis, probability theory or measure theory are lifted to a conditional framework, often to be applied in areas such as mathematical economics or…
This work is motivated by the problem of finding the limit of the applicability of the first incompleteness theorem ($\sf G1$). A natural question is: can we find a minimal theory for which $\sf G1$ holds? We examine the Turing degree…
Criteria for defining errors of a physical theory are formulated. It is shown that the Special Theory of Relativity (STR) has a solid mathematical basis. An enormous amount of experiments carried out in particle physics use beams of…
Definite descriptions are first-order expressions that denote unique objects. In this paper, we propose a second-order counterpart, designed to refer to unique relations between objects. We investigate this notion within the framework of…
The Ran-Reurings fixed point theorem [Proc. Amer. Math. Soc., 132 (2004), 1435-1443] is but a particular case of Maia's [Rend. Sem. Mat. Univ. Padova, 40 (1968), 139-143]. A "functional" version of this last result is then provided, in a…
We study versions of the tree pigeonhole principle, $\mathsf{TT}^1$, in the context of Weihrauch-style computable analysis. The principle has previously been the subject of extensive research in reverse mathematics. Two outstanding…
We discuss the (first- and second-order) optimality conditions for nonlinear programming under the relaxed constant rank constraint qualification. This condition generalizes the so-called linear independence constraint qualification.…
We consider a class of formula equations in first-order logic, Horn formula equations, which are defined by a syntactic restriction on the occurrences of predicate variables. Horn formula equations play an important role in many…
Possibility theory offers a framework where both Lehmann's "preferential inference" and the more productive (but less cautious) "rational closure inference" can be represented. However, there are situations where the second inference does…
The concepts of rough and definite objects are relatively more determinate than those of granules and granulation in general rough set theory (RST) [1]. Representation of rough objects can however depend on the dialectical relation between…
The standard reinforcement learning (RL) formulation considers the expectation of the (discounted) cumulative reward. This is limiting in applications where we are concerned with not only the expected performance, but also the distribution…