相关论文: Hoops, Coops and the Algebraic Semantics of Contin…

We present a novel definition of an algorithm and its corresponding algorithm language called CoLweb. The merit of CoLweb [1] is that it makes algorithm design so versatile. That is, it forces us to a high-level, proof-carrying,…

Sequential propositional logic deviates from ordinary propositional logic by taking into account that during the sequential evaluation of a propositional statement,atomic propositions may yield different Boolean values at repeated…

The article explores the arithmetic of multiplication as a model of many valued projective logic. It is demonstrated that closed numerical intervals within this framework constitute Heyting algebras. The conditions for these algebras to be…

Since the discovery of differential calculus by Newton and Leibniz and the subsequent continuous growth of its applications to physics, mechanics, geometry, etc, it was observed that partial derivatives in the study of various natural…

We study the cardinality of classes of equational theories (varieties) and logics by applying descriptive set theory. We affirmatively solve open problems raised by Jackson and Lee [Trans. Am. Math. Soc. 370 (2018), pp. 4785-4812] regarding…

The role of coalgebras as well as algebraic groups in non-commutative probability has long been advocated by the school of von Waldenfels and Sch\"urmann. Another algebraic approach was introduced more recently, based on shuffle and pre-Lie…

The approach taken by Gheorghiu, Gu and Pym in their paper on giving a Base-extension Semantics for Intuitionistic Multiplicative Linear Logic is an interesting adaptation of the work of Sandqvist for IPL to the substructural setting. What…

A combinatorial formula of G.-C. Rota and J.A. Stein is taken to perform Wick re-ordering in quantum field theory. Wick's theorem becomes a Hopf algebraic identity called Cliffordization. The combinatorial method relying on Hopf algebras is…

Game semantics allows us to look at basic logical concepts from another side. This approach to logic has a long history, there are plenty of different types of games: provability games, semantic games, etc. And there is an interesting type…

In this paper we present some results on the variety of divisible MV-algebras. Any free divisible MV-algebra is an algebra of continuous piecewise linear functions with rational coefficients. Correspondingly, Rational {\L}ukasiewicz logic…

Dynamical systems are abstract models of interaction between space and time. They are often used in fields such as physics and engineering to understand complex processes, but due to their general nature, they have found applications for…

The notion of a Hopf module over a Hopf (co)quasigroup is introduced and a version of the fundamental theorem for Hopf (co)quasigroups is proven.

A detailed exposition of foundations of a logic-algebraic model for reasoning with knowledge bases specified by propositional (Boolean) logic is presented. The model is conceived from the logical translation of usual derivatives on…

We describe a graded extension of the usual Hecke algebra: it acts in a graded fashion on the cohomology of an arithmetic group $\Gamma$. Under favorable conditions, the cohomology is freely generated in a single degree over this graded…

Latent Chain-of-Thought (Latent-CoT) aims to enable step-by-step computation without emitting long rationales, yet its mechanisms remain unclear. We study CODI, a continuous-thought teacher-student distillation model, on strictly sequential…

We study various formulations of the completeness of first-order logic phrased in constructive type theory and mechanised in the Coq proof assistant. Specifically, we examine the completeness of variants of classical and intuitionistic…

Mathematical theorems are human knowledge able to be accumulated in the form of symbolic representation, and proving theorems has been considered intelligent behavior. Based on the BHK interpretation and the Curry-Howard isomorphism, proof…

We introduce a notion of algorithmic randomness for algebraic fields. We prove the existence of a continuum of algebraic extensions of $\mathbb{Q}$ that are random according to our definition. We show that there are noncomputable algebraic…

Balloons are two-dimensional spheres. Hoops are one dimensional loops. Knotted Balloons and Hoops (KBH) in 4-space behave much like the first and second homotopy groups of a topological space - hoops can be composed as in \pi_1, balloons as…

We propose a generalization of first-order logic originating in a neglected work by C.C. Chang: a natural and generic correspondence language for any types of structures which can be recast as Set-coalgebras. We discuss axiomatization and…