1,373 research outputs found
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
Bifinite Chu Spaces
This paper studies colimits of sequences of finite Chu spaces and their
ramifications. Besides generic Chu spaces, we consider extensional and
biextensional variants. In the corresponding categories we first characterize
the monics and then the existence (or the lack thereof) of the desired
colimits. In each case, we provide a characterization of the finite objects in
terms of monomorphisms/injections. Bifinite Chu spaces are then expressed with
respect to the monics of generic Chu spaces, and universal, homogeneous Chu
spaces are shown to exist in this category. Unanticipated results driving this
development include the fact that while for generic Chu spaces monics consist
of an injective first and a surjective second component, in the extensional and
biextensional cases the surjectivity requirement can be dropped. Furthermore,
the desired colimits are only guaranteed to exist in the extensional case.
Finally, not all finite Chu spaces (considered set-theoretically) are finite
objects in their categories. This study opens up opportunities for further
investigations into recursively defined Chu spaces, as well as constructive
models of linear logic
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
Computational contents of classical logic and extensional choice
We present here a logical system mini PML which is an extension of HOL with the Curry-Howard correspondence allowing both classical logic and the extensional axiom of choice for natural numbers and higher-order functionals on natural numbers
On the strength of proof-irrelevant type theories
We present a type theory with some proof-irrelevance built into the
conversion rule. We argue that this feature is useful when type theory is used
as the logical formalism underlying a theorem prover. We also show a close
relation with the subset types of the theory of PVS. We show that in these
theories, because of the additional extentionality, the axiom of choice implies
the decidability of equality, that is, almost classical logic. Finally we
describe a simple set-theoretic semantics.Comment: 20 pages, Logical Methods in Computer Science, Long version of IJCAR
2006 pape
Functions out of Higher Truncations
In homotopy type theory, the truncation operator ||-||n (for a number n > -2)
is often useful if one does not care about the higher structure of a type and
wants to avoid coherence problems. However, its elimination principle only
allows to eliminate into n-types, which makes it hard to construct functions
||A||n -> B if B is not an n-type. This makes it desirable to derive more
powerful elimination theorems. We show a first general result: If B is an
(n+1)-type, then functions ||A||n -> B correspond exactly to functions A -> B
which are constant on all (n+1)-st loop spaces. We give one "elementary" proof
and one proof that uses a higher inductive type, both of which require some
effort. As a sample application of our result, we show that we can construct
"set-based" representations of 1-types, as long as they have "braided" loop
spaces. The main result with one of its proofs and the application have been
formalised in Agda.Comment: 15 pages; to appear at CSL'1
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