Related papers: Hoops, Coops and the Algebraic Semantics of Contin…
The notion of inner linear Hopf algebra is a generalization of the notion of discrete linear group. In this paper, we prove two general results that enable us to enlarge the class of Hopf algebras that are known to be inner linear: the…
We suggest an ordering for the predicates in continuous logic so that the semantics of continuous logic can be formulated as a hyperdoctrine. We show that this hyperdoctrine can be embedded into the hyperdoctrine of subobjects of a suitable…
Continuous mappings between compact Hausdorff spaces can be studied using homomorphisms between algebraic structures (lattices, Boolean algebras) associated with the spaces. This gives us more tools with which to tackle problems about these…
Both syntax-phonology and syntax-semantics interfaces in Higher Order Grammar (HOG) are expressed as axiomatic theories in higher-order logic (HOL), i.e. a language is defined entirely in terms of provability in the single logical system.…
In the same sense as classical logic is a formal theory of truth, the recently initiated approach called computability logic is a formal theory of computability. It understands (interactive) computational problems as games played by a…
We survey Hopf algebras and their generalizations. In particular, we compare and contrast three well-studied generalizations (quasi-Hopf algebras, weak Hopf algebras, and Hopf algebroids), and two newer ones (Hopf monads and hopfish…
We explore an application of homological algebra to set theoretic objects by developing a cohomology theory for Hausdorff gaps. The cohomology theory is introduced with enough generality to be applicable to other questions in set theory.…
Continuous first-order logic is used to apply model-theoretic analysis to analytic structures (e.g. Hilbert spaces, Banach spaces, probability spaces, etc.). Classical computable model theory is used to examine the algorithmic structure of…
The subject of persistent homology has vitalized applications of algebraic topology to point cloud data and to application fields far outside the realm of pure mathematics. The area has seen several fundamentally important results that are…
Opetopes are algebraic descriptions of shapes corresponding to compositions in higher dimensions. As such, they offer an approach to higher-dimensional algebraic structures, and in particular, to the definition of weak $\omega$-categories,…
This is a short paper about the relationship between logic and computation. More specifically, it is about a relationship between the completeness proof for intuitionistic propositional logic within the form of proof-theoretic semantics…
We present two embeddings of infinite-valued Lukasiewicz logic L into Meyer and Slaney's abelian logic A, the logic of lattice-ordered abelian groups. We give new analytic proof systems for A and use the embeddings to derive corresponding…
We establish a novel connection between two research areas in non-classical logics which have been developed independently of each other so far: on the one hand, input/output logic, introduced within a research program developing logical…
Using coalgebraic methods, we extend Conway's theory of games to possibly non-terminating, i.e. non-wellfounded games (hypergames). We take the view that a play which goes on forever is a draw, and hence rather than focussing on winning…
Computability logic (CoL) (see http://www.cis.upenn.edu/~giorgi/cl.html) is a recently introduced semantical platform and ambitious program for redeveloping logic as a formal theory of computability, as opposed to the formal theory of truth…
In this paper, we introduce a foundation for computable model theory of rational Pavelka logic (an extension of {\L}ukasiewicz logic) and continuous logic, and prove effective versions of some theorems in model theory. We show how to reduce…
The chain-of-though (CoT) prompting methods were successful in various natural language processing (NLP) tasks thanks to their ability to unveil the underlying complex reasoning processes. Such reasoning processes typically exhibit…
The work is devoted to Computability Logic (CoL) -- the philosophical/mathematical platform and long-term project for redeveloping classical logic after replacing truth} by computability in its underlying semantics (see…
The main aim of this paper is to show the interconnections between {\L}ukasiewicz logic and algebraic geometry using algebraic, geometric and logical instruments. We continue our investigation into a new algebraic geometry based on…
We introduce a new Hopf algebra that operates on pairs of finite interval partitions and permutations of equal length. This algebra captures vincular patterns, which involve specifying both the permutation patterns and the consecutive…