Related papers: Godel Implication on Finite Chains: Truth Tables a…
We consider a large family of theories of equivalence relations, each with finitely many classes, and assuming the existence of an $\omega$-Erdos cardinal, we determine which of these theories are Borel complete. We develop machinery,…
Purpose: This study extends the structural theory of finite commutative ternary $\Gamma$-semirings into a computational and categorical framework for explicit classification and constructive reasoning. Methods: Constraint-driven enumeration…
Drawing inspiration from Emmy Noether'set-theoretic foundations for algebra and Charles Ehresmann's topology without points, we adopt a new order-theoretic approach to ideal theory. For this we emphasize the order of divisibility in…
G\"odel's second incompleteness theorem is standardly understood as showing that no sufficiently strong, consistent theory of arithmetic can prove its own consistency, a result typically interpreted against a model-theoretic background in…
This thesis introduces the "method of structural refinement", which serves as a means of transforming the relational semantics of a modal and/or constructive logic into an 'economical' proof system by connecting two proof-theoretic…
This paper is a continuation of our papers [EK1, EK2]. In [EK2] we showed that for the root system A_n-1 one can obtain Macdonald's polynomials - a new interesting class of symmetric functions recently defined by I. Macdonald {M1] - as…
<p>We address the general problem of determining the validity of boolean combinations of equalities and inequalities between real-valued expressions. In particular, we consider methods of establishing such assertions using only restricted…
Nicholas Pippenger and Kristin Schleich have recently given a combinatorial interpretation for the second-order super-Catalan numbers (u_{n})_{n>=0}=(3,2,3,6,14,36,...): they count "aligned cubic trees" on n internal vertices. Here we give…
Real-valued logics underlie an increasing number of neuro-symbolic approaches, though typically their logical inference capabilities are characterized only qualitatively. We provide foundations for establishing the correctness and power of…
We consider the untyped lambda calculus with constructors and recursively defined constants. We construct a domain-theoretic model such that any term not denoting bottom is strongly normalising provided all its `stratified approximations'…
We extend coded distributed computing over finite fields to allow the number of workers to be larger than the field size. We give codes that work for fully general matrix multiplication and show that in this case we serendipitously have…
In this paper we investigate the algebraic structure of the truth tables of all bracketed formulae with n distinct variables connected by the binary connective of implication.
This paper presents a two-phase algorithm for computing exact Catalan numbers at an unprecedented scale. The method is demonstrated by computing $C(n)$ for $n = 2,050,572,903$ yielding a result with a targeted $1,234,567,890$ decimal…
We extend the threshold resummation exponents G^N in Mellin-N space to the fourth logarithmic (N^3LL) order collecting the terms alpha_s^2 (alpha_s ln N)^n to all orders in the strong coupling constant as. Comparing the results to our…
In this paper we count the number of rows yn with the value "false" in the truth tables of all bracketed formulae with n distinct variables connected by the binary connective of 2modified-implication". We find a recurrence and an asymptotic…
The solution to the general univariate polynomial equation has been sought for centuries. It is well known there is no general solution in radicals for degrees five and above. The hyper-Catalan numbers $C[m_2,m_3,m_4,\ldots]$ count the ways…
This paper is a continuation of our papers \cite{EK1, EK2}. In \cite{EK2} we showed that for the root system $A_{n-1}$ one can obtain Macdonald's polynomials as weighted traces of intertwining operators between certain finite-dimensional…
We describe arithmetic algorithms on a canonical number representation based on the Catalan family of combinatorial objects specified as a Haskell type class. Our algorithms work on a {\em generic} representation that we illustrate on…
We classify cuts in (totally) ordered abelian groups $\g$ and compute the coinitiality and cofinality of all cuts in case $\g$ is divisible, in terms of data intrinsically associated to the invariance group of the cut. We relate cuts with…
The theorem of factorisation forests shows the existence of nested factorisations -- a la Ramsey -- for finite words. This theorem has important applications in semigroup theory, and beyond. The purpose of this paper is to illustrate the…