Related papers: Aspects of an internal logic for persistence
In this short note we relate some known properties of propositional calculus to purely algebraic considerations of a Boolean algebra. Classes of formulas of propositional calculus are considered as elements of a Boolean algebra. As such…
Non-classical negations may fail to be contradictory-forming operators in more than one way, and they often fail also to respect fundamental meta-logical properties such as the replacement property. Such drawbacks are witnessed by intricate…
We extend the meet-implication fragment of propositional intuitionistic logic with a meet-preserving modality. We give semantics based on semilattices and a duality result with a suitable notion of descriptive frame. As a consequence we…
We define a simple, explicit map sending a morphism $f:M \rightarrow N$ of pointwise finite dimensional persistence modules to a matching between the barcodes of $M$ and $N$. Our main result is that, in a precise sense, the quality of this…
Spatial conjunction is a powerful construct for reasoning about dynamically allocated data structures, as well as concurrent, distributed and mobile computation. While researchers have identified many uses of spatial conjunction, its…
In the present paper, as we did previously in [7], we investigate the relations between the geometric properties of tilings and the algebraic properties of associated relational structures. Our study is motivated by the existence of…
We study the algebraic theory of computable functions, which can be viewed as arising from possibly non-halting computer programs or algorithms, acting on some state space, equipped with operations of composition, {\em if-then-else} and…
In this work, a novel quaternary algebra has been proposed that can be used to implement an arbitrary quaternary logic function in more than one systematic ways. The proposed logic has evolved from and is closely related to the Boolean…
We consider evolution algebras and their related substructures: evolution ideals and evolution subalgebras. After exposing some of the concepts related to them in the literature, we explore the order structures that arise in the sets of…
A plausible definition of "reasoning" could be "algebraically manipulating previously acquired knowledge in order to answer a new question". This definition covers first-order logical inference or probabilistic inference. It also includes…
We aim at studying collections of algebraic structures defined over a commutative ring and investigating the complexity of significant constructions carried out on these objects. The assignment of measures of size, via a multiplicity…
The relationship between algebraic geometry and the inferential framework of the Bayesian Networks with hidden variables has now been fruitfully explored and exploited by a number of authors. More recently the algebraic formulation of…
Coherence phenomena appear in two different situations. In the context of category theory the term `coherence constraints' refers to a set of diagrams whose commutativity implies the commutativity of a larger class of diagrams. In the…
The relationship between Heyting algebras (HA) and semirings is explored. A new class of HAs called Symmetric Heyting algebras (SHAs) is proposed, and a necessary condition on SHAs to be consider semirings is given. We define a new…
We generalize the construction of multitildes in the aim to provide multitilde operators for regular languages. We show that the underliying algebraic structure involves the action of some operads. An operad is an algebraic structure that…
Starting from a description of various generalized function algebras based on sequence spaces, we develop the general framework for considering linear problems with singular coefficients or non linear problems. Therefore, we prove…
We develop some aspects of the homological algebra of persistence modules, in both the one-parameter and multi-parameter settings, considered as either sheaves or graded modules. The two theories are different. We consider the graded module…
While modern deep networks have demonstrated remarkable versatility, their training dynamics remain poorly understood--often driven more by empirical tweaks than architectural insight. This paper investigates how internal structural choices…
Statistics and Optimization are foundational to modern Machine Learning. Here, we propose an alternative foundation based on Abstract Algebra, with mathematics that facilitates the analysis of learning. In this approach, the goal of the…
While topological data analysis has emerged as a powerful paradigm for structural inference, its foundational tools, notably persistent homology and the persistent Laplacian, are frequently insensitive to localized structural fluctuations…