Publications:
- F. Ciraulo - T. Kawai - S. Maschio "Factorizing
the Top-Loc adjunction through positive topologies" (submitted;
preprint version here arXiv:1812.09190).
- F. Ciraulo "σ-locales in
Formal Topology" (submitted; preprint version here arXiv:1801.09644).
- F. Ciraulo "Overlap Algebras
as Almost Discrete Locales" (submitted; preprint version
here arXiv:1601.04830).
- S. Bredariol - F.
Ciraulo "Educare al pensiero
razionale nella scuola primaria: una proposta didattica", L'Insegnamento della Matematica e
delle Scienza Integrate 42 (2019), pp. 139-158.
- O. Gaggi - F. Ciraulo - M. Casagrande. 2018.
"Eating Pizza to learn fractions".
In
International Conference on Smart Objects and Technologies for
Social Good (Goodtechs ’18), November 28–30, 2018, Bologna, Italy.
ACM, New York. DOI:10.1145/3284869.3284921
- F. Ciraulo "Geometria fra
immagini, immaginazione e... "magia"!", L'insegnamento
della matematica e delle scienze integrate 41
(2018), pp. 691-709.
- F. Ciraulo - D. Rinaldi - P. Schuster "Lindenbaum's
Lemma via Open Induction", in R. Kahle and T. Strahm
and T.Studer (eds.) "Advances in Proof Theory", Progress
in Computer Science and Applied Logic 28, Birkhäuser
Basel (2016), pp. 65-77.
We
begin the study of (not necessarily complete) partial
orders with some notion of "overlap" or "positivity". We
propose several kinds of structures which, classically,
turn out to be Boolean algebras. For instance, it is
well-known that the collection of all finite and cofinite
subsets of a given set form a Boolean algebra,
classically. Intuitionistcally, on the contrary, we
are
able to describe it as a partial order with overlap.
I show
that the regular open subsets of a formal topology form an
overlap algebra and that every (set-based) overlap algebra
can be represented in this way.
In this
paper we study what are the links between closure and
interior operators from an intuitionistic point of view.
In particular, we are interested in understanding whether
a closure operator can determine an interior operator or
not; and vice versa. Classically, the picture is very
clear: closure and interior are determined by each other
via complement. Intuitionistically, all is more complex,
as usual. We describe a way to construct the largest
interior ``compatible'' with a given closure and, dually,
the largest closure ``compatible'' with a given interior.
Here ``compatible'' stands for a suitable link between
interior and closure which is valid intuitionistically. In
contrast with the classical picture, these two
constructions are not inverse one to another. In fact,
they form a Galois connection.
We
develop a piece of constructive Topology within the
language of overlap algebras. In particular, we show how
to express the notion of regular open set and that of
regular space in such an algebraic framework. This
approach is fully intuitionistic and so we can avoid any
use of set-theoretic complement. We succeed in
characterize the link between the interior operator and
the closure operator in an intuitionistic way. The main
result of the paper is that the regular open sets of a
regular space form an overlap algebra which, in general,
is atom-less.
- F. Ciraulo "Soddisfacibilità
costruttiva", La
Matematica nella Società e nella Cultura, Rivista
dell'Unione Matematica Italiana, Serie I, Vol. I, pp. 275-278
(2008).
- F. Ciraulo - G. Sambin "Tense
logic within a constructive metatheory", Preprint n. 341,
Dipartimento di Matematica ed Applicazioni, Università degli Studi
di Palermo (2008).
My Ph.D. thesis: Constructive
Satisfiability (
.pdf)
A constructive (i.e. intuitionistic and predicative) analysis of
the logical notion of satisfiability (and also non-deducibility) for
first-order intuitionistic logic. From a semantic point of view, the
main tool is formal topology theory and, in particular, the notion of
(binary) positivity. Co-inductive methods are used in many proofs.
Conference Presentations:
- "Equazioni booleane e matematizzazione... nell'isola di
Smullyan", convegno "Educare alla razionalità", Torino, 22-23
maggio 2019.
- "Overtness and density for σ-locales" at "CCC 2018", Faro, Portugal, September 24-28,
2018.
- "Notions of Booleanization in
pointfree Topology" at "Constructive Mathematics", Haudorff
research Insitute for Mathematics, Bonn, Germany, August
6-10, 2018 (video).
- "σ-locales and Booleanization in Formal Topology" at "CCC 2017", Nancy, France, June 26-30 2017.
- "Boolean locales: a constructive analog" at the "Logic Colloquium 2013", Évora, Portugal,
July 22-27 2013.
- "The overlap relation in intuitionistic lattice theory" at
the "XXIV
Incontro di Logica", Bologna, Italy, 2-4 February 2011.
- "Satisfiability and Consistency from a constructive point of
view" at the "XXIII Incontro di Logica", Genova, Italy,
20-23 February 2008.