60 research outputs found
On choice rules in dependent type theory
In a dependent type theory satisfying the propositions as
types correspondence together with the proofs-as-programs paradigm,
the validity of the unique choice rule or even more of the choice rule says
that the extraction of a computable witness from an existential statement
under hypothesis can be performed within the same theory.
Here we show that the unique choice rule, and hence the choice rule,
are not valid both in Coquand\u2019s Calculus of Constructions with indexed
sum types, list types and binary disjoint sums and in its predicative
version implemented in the intensional level of the Minimalist Founda-
tion. This means that in these theories the extraction of computational
witnesses from existential statements must be performed in a more ex-
pressive proofs-as-programs theory
A predicative variant of a realizability tripos for the Minimalist Foundation.
open2noHere we present a predicative variant of a realizability tripos validating
the intensional level of the Minimalist Foundation extended with Formal Church
thesis.the file attached contains the whole number of the journal including the mentioned pubblicationopenMaietti, Maria Emilia; Maschio, SamueleMaietti, MARIA EMILIA; Maschio, Samuel
Elementary quotient completion
We extend the notion of exact completion on a weakly lex category to
elementary doctrines. We show how any such doctrine admits an elementary
quotient completion, which freely adds effective quotients and extensional
equality. We note that the elementary quotient completion can be obtained as
the composite of two free constructions: one adds effective quotients, and the
other forces extensionality of maps. We also prove that each construction
preserves comprehensions
An extensional Kleene realizability semantics for the Minimalist Foundation
We build a Kleene realizability semantics for the two-level Minimalist
Foundation MF, ideated by Maietti and Sambin in 2005 and completed by Maietti
in 2009. Thanks to this semantics we prove that both levels of MF are
consistent with the (Extended) formal Church Thesis CT. MF consists of two
levels, an intensional one, called mTT and an extensional one, called emTT,
based on versions of Martin-L\"of's type theory. Thanks to the link between the
two levels, it is enough to build a semantics for the intensional level to get
one also for the extensional level. Hence here we just build a realizability
semantics for the intensional level mTT. Such a semantics is a modification of
the realizability semantics in Beeson 1985 for extensional first order
Martin-L\"of's type theory with one universe. So it is formalised in Feferman's
classical arithmetic theory of inductive definitions. It is called extensional
Kleene realizability semantics since it validates extensional equality of
type-theoretic functions extFun, as in Beeson 1985. The main modification we
perform on Beeson's semantics is to interpret propositions, which are defined
primitively in MF, in a proof-irrelevant way. As a consequence, we gain the
validity of CT. Recalling that extFun+ CT+ AC are inconsistent over arithmetics
with finite types, we conclude that our semantics does not validate the full
Axiom of Choice AC. On the contrary, Beeson's semantics does validate AC, being
this a theorem of Martin-L\"of's theory, but it does not validate CT. The
semantics we present here appears to be the best Kleene realizability semantics
for the extensional level emTT of MF. Indeed Beeson's semantics is not an
option for emTT since the full AC added to it entails the excluded middle
Quotient completion for the foundation of constructive mathematics
We apply some tools developed in categorical logic to give an abstract
description of constructions used to formalize constructive mathematics in
foundations based on intensional type theory. The key concept we employ is that
of a Lawvere hyperdoctrine for which we describe a notion of quotient
completion. That notion includes the exact completion on a category with weak
finite limits as an instance as well as examples from type theory that fall
apart from this.Comment: 32 page
Joyal's arithmetic universes via type theory
Abstract Andre Joyal constructed arithmetic universes to provide a categorical proof of Godel incompleteness results. He built them in three stages: he first took a Skolem theory, then the category of its predicates and finally he made the exact completion out of the latter. Here, we prove that the construction of an initial arithmetic universe is equivalent to that of an initial list-arithmetic pretopos and also of an initial arithmetic pretopos. The initial list-arithmetic pretopos is built out of its internal language formulated as a dependent typed calculus in the style of Martin-Lof's extensional type theory. Analogously, we prove that the second stage of Joyal's construction is equivalent to taking an initial arithmetic lextensive category or an initial regular locos. We conclude by proposing the notion of list-arithmetic pretopos as the general definition of arithmetic universe. We are motivated from the fact in any list-arithmetic pretopos we can show the existence of free internal categories and diagrams as in any of Joyal's arithmetic universes
Constructive version of Boolean algebra
The notion of overlap algebra introduced by G. Sambin provides a constructive
version of complete Boolean algebra. Here we first show some properties
concerning overlap algebras: we prove that the notion of overlap morphism
corresponds classically to that of map preserving arbitrary joins; we provide a
description of atomic set-based overlap algebras in the language of formal
topology, thus giving a predicative characterization of discrete locales; we
show that the power-collection of a set is the free overlap algebra
join-generated from the set. Then, we generalize the concept of overlap algebra
and overlap morphism in various ways to provide constructive versions of the
category of Boolean algebras with maps preserving arbitrary existing joins.Comment: 22 page
A realizability semantics for inductive formal topologies, Church's Thesis and Axiom of Choice
We present a Kleene realizability semantics for the intensional level of the
Minimalist Foundation, for short mtt, extended with inductively generated
formal topologies, Church's thesis and axiom of choice. This semantics is an
extension of the one used to show consistency of the intensional level of the
Minimalist Foundation with the axiom of choice and formal Church's thesis in
previous work. A main novelty here is that such a semantics is formalized in a
constructive theory represented by Aczel's constructive set theory CZF extended
with the regular extension axiom
A characterization of regular and exact completions of pure existential completions
The notion of existential completion in the context of Lawvere's doctrines
was introduced by the second author in his PhD thesis, and it turned out to be
a restriction to faithful fibrations of Peter Hofstra's construction used to
characterize Dialectica fibrations. The notions of regular and exact
completions of elementary and existential doctrines were brought up in recent
works by the first author with F. Pasquali and P. Rosolini, inspired by those
done by M. Hyland, P. Johnstone and A. Pitts on triposes. Here, we provide a
characterization of the regular and exact completions of (pure) existential
completions of elementary doctrines by showing that these amount to the
and -completions,
respectively, of the category of predicates of their generating elementary
doctrines. This characterization generalizes a previous result obtained by the
first author with F. Pasquali and P. Rosolini on doctrines equipped with
Hilbert's -operators. Relevant examples of applications of our
characterization, quite different from those involving doctrines with Hilbert's
-operators, include the regular syntactic category of the regular
fragments of first-order logic (and his effectivization) as well as the
construction of Joyal's Arithmetic Universes
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