Constructive version of Boolean algebra

Ciraulo, Francesco; Maietti, Maria Emilia; Toto, Paola

doi:10.1093/jigpal/jzs021

Cited by 8 publications

(23 citation statements)

References 11 publications

(30 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In the following paragraph we will show that this definition of atom provides a solution to the problem of characterizing powersets in an intuitionistically sound way. We end this section by an elegant characterization of the notion of atom (see [4] for a detailed proof). Proposition 1.3.…”

Section: Overlap Algebrasmentioning

confidence: 99%

“…Among the consequences of the axioms O1-O4 (see [4,5]), we want to mention only one, perhaps surprising: every o-algebra is a frame, that is, binary meets distribute over arbitrary joins. In fact, for any z ∈ P, one has:…”

Section: Overlap Algebrasmentioning

confidence: 99%

“…With classical logic, x > < y becomes equivalent to x ∧ y = 0 and o-algebras turn out to be just complete Boolean algebras [4,5]. This fact suggests a general idea: in order to prove an intuitionistic version of a classical result about complete Boolean algebras, replace them by o-algebras.…”

Section: Overlap Algebrasmentioning

confidence: 99%

See 2 more Smart Citations

Intuitionistic overlap structures

Ciraulo

2013

LLP

View full text Add to dashboard Cite

We study some connections between two kinds of overlap relations: that of point-free geometries in the sense of Grzegorczyk, Whitehead and Clarke, and that recently introduced by Sambin within his constructive approach to topology. The main thesis of this paper is that the overlap relation in the latter sense is a necessary tool for a constructive and intuitionistic development of point-free geometry.

show abstract

Section: Overlap Algebrasmentioning

confidence: 99%

Section: Overlap Algebrasmentioning

confidence: 99%

See 1 more Smart Citation

Intuitionistic overlap structures

Ciraulo

2013

LLP

View full text Add to dashboard Cite

show abstract

mentioning

confidence: 99%

“…The formalization work was mainly focused on Overlap Algebras [4,3], new algebraic structures designed to ease reasoning about subsets within intuitionistic logic. The main result checked in Matita [8] is the embedding of a suitable category of topological spaces into a category of generalized point-free topologies, which is an improvement on the usual adjunction between topological spaces and locales.…”

mentioning

confidence: 99%

Formalization of Formal Topology by Means of the Interactive Theorem Prover Matita

Asperti

Maietti²,

Coen

et al. 2011

Lecture Notes in Computer Science

View full text Add to dashboard Cite

The project entitled "Formalization of Formal Topology by means of the interactive theorem prover Matita" is an official bilateral project between the Universities of Padova and Bologna, funded by the former, active ¡¡¡¡¡¡¡ .mine from march 2008 until august 2010. The project brought together and exploited the synergic collaboration of two communities of researchers, both centered around constructive type theory: ======= from March 2008 until August 2010. The project aimed to bring together and exploit the synergic collaboration of two communities of researchers, both centered around constructive type theory: ¿¿¿¿¿¿¿ .r3432 on one side the Logic Group at the University of Padova, focused on developing formal, pointfree topology within a constructive and predicative framework; on the other side, the Helm group at the University of Bologna, developing the Matita Interactive Theorem Prover [2], a young proof assistant based on the Calculus of Inductive Constructions as its logical foundation. The idea of the project was to formalize and check the new approach to formal topology being developed in Padova by means of Matita, with the aim on one side to assess the truly foundational nature of the theoretical framework (i.e. its reduction to notions so elementary to be easily understood by an automatic device), and on the other to drive the development of Matita, testing the tool on a non trivial set of mathematical results, and addressing from an original theoretical perspective some key problems of constructive interactive proving (general recursion, extensionality, quotients, . . .).The project is a rare example of a significant collaboration between mathematicians and computer scientists in handling mathematical knowledge.It is worth to emphasize that the interest in the formalization from the mathematical perspective is not in the automatic verification of the results (on which mathematicians are already largely confident with) but in the phenomenological goal to investigate the most natural way to organize a new foundational framework in a coherent set of interconnected components, their mutual relations and dependencies, our interaction with these representations, and their influence on the concrete mathematical experience (see [1]). Formalization is neither a goal nor a technique, but first and foremost a methodology.The formalization work was mainly focused on Overlap Algebras [4,3], new algebraic structures designed to ease reasoning about subsets within intuitionistic logic. The main result checked in Matita [8] is the embedding of a suitable

show abstract

Intuitionistic mereology

Maffezioli

Varzi

2019

Synthese

View full text Add to dashboard Cite

Two mereological theories are presented based on a primitive apartness relation along with binary relations of mereological excess and weak excess, respectively. It is shown that both theories are acceptable from the standpoint of constructive reasoning while remaining faithful to the spirit of classical mereology. The two theories are then compared and assessed with regard to their extensional import.

show abstract

Constructive version of Boolean algebra

Cited by 8 publications

References 11 publications

Intuitionistic overlap structures

Intuitionistic overlap structures

Formalization of Formal Topology by Means of the Interactive Theorem Prover Matita

Intuitionistic mereology

Contact Info

Product

Resources

About