76 research outputs found
Normalization of IZF with Replacement
ZF is a well investigated impredicative constructive version of
Zermelo-Fraenkel set theory. Using set terms, we axiomatize IZF with
Replacement, which we call \izfr, along with its intensional counterpart
\iizfr. We define a typed lambda calculus \li corresponding to proofs in
\iizfr according to the Curry-Howard isomorphism principle. Using realizability
for \iizfr, we show weak normalization of \li. We use normalization to prove
the disjunction, numerical existence and term existence properties. An inner
extensional model is used to show these properties, along with the set
existence property, for full, extensional \izfr
A Normalizing Intuitionistic Set Theory with Inaccessible Sets
We propose a set theory strong enough to interpret powerful type theories
underlying proof assistants such as LEGO and also possibly Coq, which at the
same time enables program extraction from its constructive proofs. For this
purpose, we axiomatize an impredicative constructive version of
Zermelo-Fraenkel set theory IZF with Replacement and -many
inaccessibles, which we call \izfio. Our axiomatization utilizes set terms, an
inductive definition of inaccessible sets and the mutually recursive nature of
equality and membership relations. It allows us to define a weakly-normalizing
typed lambda calculus corresponding to proofs in \izfio according to the
Curry-Howard isomorphism principle. We use realizability to prove the
normalization theorem, which provides a basis for program extraction
capability.Comment: To be published in Logical Methods in Computer Scienc
Extracting Programs from Constructive HOL Proofs via IZF Set-Theoretic<br> Semantics
Church's Higher Order Logic is a basis for influential proof assistants --
HOL and PVS. Church's logic has a simple set-theoretic semantics, making it
trustworthy and extensible. We factor HOL into a constructive core plus axioms
of excluded middle and choice. We similarly factor standard set theory, ZFC,
into a constructive core, IZF, and axioms of excluded middle and choice. Then
we provide the standard set-theoretic semantics in such a way that the
constructive core of HOL is mapped into IZF. We use the disjunction, numerical
existence and term existence properties of IZF to provide a program extraction
capability from proofs in the constructive core.
We can implement the disjunction and numerical existence properties in two
different ways: one using Rathjen's realizability for IZF and the other using a
new direct weak normalization result for IZF by Moczydlowski. The latter can
also be used for the term existence property.Comment: 17 page
Relational Parametricity for Computational Effects
According to Strachey, a polymorphic program is parametric if it applies a
uniform algorithm independently of the type instantiations at which it is
applied. The notion of relational parametricity, introduced by Reynolds, is one
possible mathematical formulation of this idea. Relational parametricity
provides a powerful tool for establishing data abstraction properties, proving
equivalences of datatypes, and establishing equalities of programs. Such
properties have been well studied in a pure functional setting. Many programs,
however, exhibit computational effects, and are not accounted for by the
standard theory of relational parametricity. In this paper, we develop a
foundational framework for extending the notion of relational parametricity to
programming languages with effects.Comment: 31 pages, appears in Logical Methods in Computer Scienc
Aerospace Medicine and Biology: A continuing bibliography with indexes (supplement 291)
This bibliography lists 131 reports, articles and other documents introduced into the NASA scientific and technical information system in November 1986
Lower bound of the parabolic Hilbert commutator
Funding Information: T. Oikari was supported by the Academy of Finland project numbers 306901 and 314829 , by the Finnish Centre of Excellence in Analysis and Dynamics Research project No. 307333 , by the three-year research grant of the University of Helsinki No. 75160010 and by the Jenny and Antti Wihuri Foundation grant No. 00200253 . Publisher Copyright: © 2022 The Author(s)Answering a key point left open in the recent work of Bongers, Guo, Li and Wick [2], we provide the lower bound ‖b‖BMOγ(R2)≲‖[b,Hγ]‖Lp(R2)→Lp(R2), where Hγ is the parabolic Hilbert transform.Peer reviewe
An analysis of the constructive content of Henkin's proof of G\"odel's completeness theorem
G{\"o}del's completeness theorem for classical first-order logic is one of
the most basic theorems of logic. Central to any foundational course in logic,
it connects the notion of valid formula to the notion of provable formula.We
survey a few standard formulations and proofs of the completeness theorem
before focusing on the formal description of a slight modification of Henkin's
proof within intuitionistic second-order arithmetic.It is standard in the
context of the completeness of intuitionistic logic with respect to various
semantics such as Kripke or Beth semantics to follow the Curry-Howard
correspondence and to interpret the proofs of completeness as programs which
turn proofs of validity for these semantics into proofs of derivability.We
apply this approach to Henkin's proof to phrase it as a program which
transforms any proof of validity with respect to Tarski semantics into a proof
of derivability.By doing so, we hope to shed an effective light on the relation
between Tarski semantics and syntax: proofs of validity are syntactic objects
with which we can compute.Comment: R{\'e}dig{\'e} en 4 {\'e}tapes: 2013, 2016, 2022, 202
Martin-L\"of \`a la Coq
We present an extensive mechanization of the meta-theory of Martin-L\"of Type
Theory (MLTT) in the Coq proof assistant. Our development builds on
pre-existing work in Agda to show not only the decidability of conversion, but
also the decidability of type checking, using an approach guided by
bidirectional type checking. From our proof of decidability, we obtain a
certified and executable type checker for a full-fledged version of MLTT with
support for , , , and identity types, and one
universe. Furthermore, our development does not rely on impredicativity,
induction-recursion or any axiom beyond MLTT with a schema for indexed
inductive types and a handful of predicative universes, narrowing the gap
between the object theory and the meta-theory to a mere difference in
universes. Finally, we explain our formalization choices, geared towards a
modular development relying on Coq's features, e.g. meta-programming facilities
provided by tactics and universe polymorphism
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