Titles and Abstracts
Peter Aczel
Two possible principles for constructive set theory
In my talk I will formulate the two principles, discuss their
significance for CST (Constructive Set Theory) and state some results
concerning them.
The first principle, sigmaCP expresses a strong completeness property
for intuitionistic propositional logic that implies the negations of
LPO, WLPO, MP, SEP, etc, ..., but is nevertheless relatively
consistent with CZF.
The second principle, RRS (Relation Reflection Scheme) is a
consequence of RDC (Relative Dependent Choices Scheme) but appears not
to be a choice principle, although it can be used where previously
RDC has seemed to be needed.
Thorsten Altenkirch
Informative and noninformative existentials in Type Theory
Andrej Bauer
Continuity begets continuity (joint work with Alex Simpson)
Ulrich Berger
Strong normalisation
via domaintheoretic computability predicates
It is wellknown that intersection types on the one hand characterise
the strongly normalisable lambda terms and on the other hand give rise
to a syntactically defined domain model for the lambda calculus.
Recently, Coquand and Spiwack have combined and extended these facts
to give a domaintheoretic criteria for strong normalisability for the
lambda calculus with recursively defined constants. In the talk I
will show that their normalisation proof can also be carried out in an
abstract axiomatic setting where the computability predicates are
indexed by the elements of a (not syntactically presented) domain. The
more abstract setting simplifies the proof considerably without
sacrificing predicativity or constructivity.
Marian Baroni
Archimedean spaces
Archimedean ordered vector spaces are examined constructively. In
the classical theory, if e is an order unit of an Archimedean
space, then the Minkowski functional of the order interval
[e,e] is a norm with respect to which [e,e] is the closed
unit ball. Ishihara's theorems on the constructive existence of
Minkowski functionals can be used to prove similar results
constructively.
Douglas Bridges
Four seasons in constructive mathematics
Looking back over nearly 40 years as an aficionado of constructive
mathematics, I will wax sentimental about a few results that brought
me particular pleasure. The results will be drawn from approximation
theory, complex function theory, integration theory, and mathematical
economics.
Jesper Carlström
A constructive version of Birkhoff's theorem
The classical version of Birkhoff's theorem says that a class of
algebras (in universal algebra) is closed under subalgebras, homomorphic
images and products if and only if it is axiomatizable by equations. I will
discuss a constructive version, with 'closed under inductive limits' added as
a fourth condition. To get a predicative version, we also add the requirement
that the class be 'setbased' in a sense typical for predicativity. Finally, I
would like to comment on in which sense the proof lets us 'compute' a set of
axioms for a class of
algebras.
Francesco Ciraulo
A constructive semantical treatment of LJnondeducibility
Formal topologies have been used to give a constructive semantical
characterization of deducibility in intuitionistic predicate calculus
(Sambin, Coquand). I will show that the recently introduced notion of a
binary positivity predicate (Sambin) is a natural tool in order to give a
constructive semantic for nondeducibility (and for some related notions).
Thierry Coquand
Maximal and minimal prime ideals in constructive algebra
Using some ideas from pointfree topology, one can interpret
constructively the prime spectrum of a ring, and so analyse
constructively proofs that use a prime ideal in a generic way. We
present some analysis of proofs in commutative algebra that use a
maximal or minimal prime ideal. One is a lemma used by Suslin in his
solution of Serre's problem (following Yengui). The other examples are
TraversoSwan's theorem on seminormal rings, and a proof of Zariski
Main Theorem due to Peskine. We present some remarks on the algorithms
corresponding to these arguments, compared to algorithms corresponding
to proofs using prime ideals generically.
Giovanni Curi
On some peculiar aspects of the category of formal spaces
In contrast with what happens in the topostheoretic context, in
constructive settings the category of locales or formal spaces is
not locally small. A particular example shows that, as a second
important difference, not every locale can be presented by a site
in the sense of Johnstone (equivalently, not every formal space can
be inductively generated). I will present results aimed at
deepening the comprehension of these peculiarities, and at
evaluating, in each case, the extent of the divergence.
In particular, I will show that no Boolean formal space (=formal
space whose associated frame is a Boolean algebra) can be
inductively generated. This provides us with a `topologically'
relevant example of a noninductively generated formal space for any
given formal space, namely its least dense subformal space. It will
also be shown how one can use similar arguments for proving, in
topoi as in type theory, that Boolean formal spaces cannot be
compact or open, and (as already known) cannot have a point. Similar
results hold true for De Morgan (i.e. extremally disconnected)
locales (with some intriguing nuances, though). It can also be shown
that products of noninductively generated formal spaces can
sometimes be constructed (e.g. the selfproduct of the
DedekindMacNeille formal topology), thus answering a problem left
open in T.Coquand et al., "Inductively generated formal
topologies".
In the second part of the talk I will briefly survey some
results concerning the size of homsets in the category of formal
spaces.
Fredrik Dahlgren
Effective Distribution Theory via Domain Theory
The theory of distributions is central to the modern theory of
ordinary and partial differential equations. Thus, to be able to show that
solution operators to certain wellknown differential equations are
effective, we need an effective theory of distributions. We will use the
theory of effective domains and domain representability to introduce a
notion of effectivity on test functions, and on distributions. We will
show that many of the standard operations on distributions are effective
in this setting.
Hannes Diener
Generalising compactness
We will present recent work on extending the notion of
compactness into a more general setting than the uniform/metric space
one.
Martin Escardó
Closed sublocales in constructive topology.
I'll examine notions of closed subspace in various flavours
of constructive topology, and relate it to other topological notions.
Peter Hancock
Representations of continuous functions on final coalgebras
Brouwer's bar theorem contains a representation of continuous
functions from Baire space to a discrete space by elements of
an inductively defined datatype. This can be generalised to a
representation of continuous functions
between final coalgebras for a certain class of functors, by means
of a datatype in the definition of which least fixed points are nested within
greatest fixed
points.
Robin Havea
On firmness of the state space and positive elements of a Banach
algebra
Following Bridges and Havea, this is an investigation of positive
elements in a Banach algebra. Under the firmness of the state space of a
Banach algebra, it is shown that even powers of positive Hermitian elements
are in fact
positive.
Hajime Ishihara
The uniform boundedness theorem and a boundedness principle
In Bishop's constructive mathematics, one can prove the contraposition of
the classical uniform boundedness theorem.
We show that the original classical uniform boundedness theorem is
equivalent to the boundedness principle, called BDN, which was introduced
as an equivalent of a continuity principle: every sequentially continuous
mapping from a (complete) separable metric space to a metric space is
pointwise continuous, and hence is provable both in intuitionistic
mathematics
and in constructive recursive mathematics.
Henri Lombardi
What is a real closed field?
Angus Macintyre
Issues of Constructivity in HardCore Number Theory
I will survey the numbertheoretic and logical problems concerning
the effective content of Finiteness Theorems in
arithmetic.
Milly Maietti
Towards a minimal twolevel foundation for constructive
mathematics (Joint work with Giovanni Sambin)
We propose a proofsasprograms foundation equipped with a
level given by an intensional type theory, and a level given by an
extensional theory, that is minimal (at the appropriate level) among
relevant existing foundations for constructive mathematics.
Erik Palmgren
Locally compact metric spaces as formal topologies
We show that the category of locally compact metric spaces may be
embedded in the category of formal topologies via a full
and
faithful functor. The proof is constructive in the sense of Bishop's
constructive mathematics BISH. This makes it possible to directly use
results from BISH in formal topology. A crucial construction is
Steve Vickers' notion of a localic completion of a metric space.
Michael Rathjen
On solving A=A>A
Giovanni Sambin
Basic picture as invariance
Helmut Schwichtenberg
A direct proof of the equivalence between Brouwer's fan theorem and
König's lemma with a uniqueness hypothesis
From results of Ishihara it is known that the weak (that is, binary)
form of König's lemma (WKL) implies Brouwer's fan theorem (Fan).
Moreover, Berger and Ishihara [MLQ 2005] have shown that a weakened
form WKL! of WKL, where as an additional hypothesis it is required
that in an effective sense infinite paths are unique, is equivalent
to Fan. The proof that WKL! implies Fan is done explicitely. The
other direction (Fan implies WKL!) is far less directly proved; the
emphasis is rather to provide a fair number of equivalents to Fan,
and to do the proofs economically by giving a circle of
implications. Here we give a direct construction. Moreover, we go
one step further and formalize the equivalence proof (in the Minlog
proof assistant). Since the statements of both Fan and WKL! have
computational content, we can automatically extract terms from the
two proofs. It turns out that these terms express in a rather
perspicuous way the informal constructions.
Dieter Spreen
Effectivity and effective continuity of multifunctions
Multifunctions are used with great success in analysis and logic. In
the talk different effectivity notions for such functions are discussed as
well as effective version of various continuity notion proposed in the
literature. A general result on the effective continuity of effective
multifunctions is presented and some important special cases are
considered.
The results extend those presented on a similar workshop in Venice some years
ago.
Thomas Streicher
Independence Results for BISH
Ruben van den Brink
Characterizing finite subsets of N and compact subsets of N
by logical schemes
Wim Veldman
Perhaps: the subtlety of intuitionistic mathematics
Steve Vickers
The Vietoris powerlocale and compact intervals of the localic reals
(This is part of an investigation into localic real analysis with Martin
Escardo.)
In classical real analysis, some basic results such as the intermediate value
theorem can be derived from the fact that closed intervals [a,b] are compact
and
connected.
The proofs are thoroughly classical, but I shall describe how similar ideas
can be used in a constructive localic approach to the reals. The techniques
used appear also to be suitable for translation to inductively generated
formal
topology.
The fundamental tool is the Vietoris powerlocale V(X), whose points are
certain sublocales of X. It is analogous to hyperspaces or powerdomains. A
clopen sublocale V+(X) comprises the nonempty points of V(X), I shall also
describe a sublocale Vc(X), comprising the connected points of
V(X).
In the case of the localic real line R, the points of V(R) correspond
classically to the compact subspaces of the reals. These can also be
described localically as metric
completions.
Some classical constructions can now be described localically.
* There are locale maps inf, sup: V+(R) > R calculating infimum and supremum
of points of
V+(R).
* Let LEQ be the sublocale of RxR comprising those pairs (x,y) for which x <=
y. Then there is a locale map [,]: LEQ > Vc(R) mapping each point (a,b) to
the closed interval [a,b]. (Hence [a,b] is compact and
connected.)
The intermediate value theorem can now be stated as follows. Let f: R > R be
a locale map, and let a <= b be such that f(a) <= 0 <= f(b). Then
inf(V+(f)([a,b])) = 0. (f is f composed with the absolute value map.
Remember also that the powerlocale is functorial, so V+( f): V+(R) >
V+(R).)
Similarly, Rolle's theorem. Let f: R > R have continuous first derivative
f'. (This can be expressed localically using Caratheodory's characterization.
We have some g: RxR > R such
that
f(y)  f(x) = (y  x) * g(x,y)
and then f'(x) = g(x,x).)
If a < b and f(a) = f(b) then inf(V+(f')([a,b])) = 0.
