172 research outputs found
Bar recursion in classical realisability : dependent choice and continuum hypothesis
This paper is about the bar recursion operator in the context of classical
realizability. After the pioneering work of Berardi, Bezem & Coquand [1], T.
Streicher has shown [10], by means of their bar recursion operator, that the
realizability models of ZF, obtained from usual models of -calculus
(Scott domains, coherent spaces, . . .), satisfy the axiom of dependent choice.
We give a proof of this result, using the tools of classical realizability.
Moreover, we show that these realizability models satisfy the well ordering of
and the continuum hypothesis These formulas are therefore realized
by closed -terms. This allows to obtain programs from proofs of
arithmetical formulas using all these axioms.Comment: 11 page
On the structure of classical realizability models of ZF
The technique of "classical realizability" is an extension of the method of
"forcing"; it permits to extend the Curry-Howard correspondence between proofs
and programs, to Zermelo-Fraenkel set theory and to build new models of ZF,
called "realizability models". The structure of these models is, in general,
much more complicated than that of the particular case of "forcing models". We
show here that the class of constructible sets of any realizability model is an
elementary extension of the constructibles of the ground model (a trivial fact
in the case of forcing, since these classes are identical). It follows that
Shoenfield absoluteness theorem applies to realizability models.Comment: 17 page
Realizability algebras: a program to well order R
The theory of classical realizability is a framework in which we can develop
the proof-program correspondence. Using this framework, we show how to
transform into programs the proofs in classical analysis with dependent choice
and the existence of a well ordering of the real line. The principal tools are:
The notion of realizability algebra, which is a three-sorted variant of the
well known combinatory algebra of Curry. An adaptation of the method of forcing
used in set theory to prove consistency results. Here, it is used in another
way, to obtain programs associated with a well ordering of R and the existence
of a non trivial ultrafilter on N
Valid formulas, games and network protocols
We describe a remarkable relation between the notion of valid formula of
predicate logic and the specification of network protocols. We give several
examples such as the acknowledgement of one packet or of a sequence of packets.
We show how to specify the composition of protocols.Comment: 12 page
Realizability algebras II : new models of ZF + DC
Using the proof-program (Curry-Howard) correspondence, we give a new method
to obtain models of ZF and relative consistency results in set theory. We show
the relative consistency of ZF + DC + there exists a sequence of subsets of R
the cardinals of which are strictly decreasing + other similar properties of R.
These results seem not to have been previously obtained by forcing.Comment: 28
A program for the full axiom of choice
The theory of classical realizability is a framework for the Curry-Howard
correspondence which enables to associate a program with each proof in
Zermelo-Fraenkel set theory. But, almost all the applications of mathematics in
physics, probability, statistics, etc. use Analysis i.e. the axiom of dependent
choice (DC) or even the (full) axiom of choice (AC). It is therefore important
to find explicit programs for these axioms. Various solutions have been found
for DC, for instance the lambda-term called "bar recursion" or the instruction
"quote" of LISP. We present here the first program for AC.Comment: 22 pages - The paper has been formatted for publication in Log. Meth.
Comp. S
A note about Grothendieck's constant
A remark on the proof that the Grothendieck constant satisfies
A call-by-name lambda-calculus machine
International audienceA simple lazy machine which runs programs written in lambda-calculus. It was introduced by the author in 1985. It has been, since, used and implemented by several authors but remained unpublished. It will appear soon in a special issue of Higher Order and Symbolic Computation
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