164 research outputs found
Equality between Functionals in the Presence of Coproducts
AbstractWe consider the lambda calculus obtained from the simply typed calculus by adding products, coproducts, and a terminal type. We prove the following theorem: The equations provable in this calculus are precisely those true in any set-theoretic model with an infinite base type
The exp-log normal form of types
Lambda calculi with algebraic data types lie at the core of functional
programming languages and proof assistants, but conceal at least two
fundamental theoretical problems already in the presence of the simplest
non-trivial data type, the sum type. First, we do not know of an explicit and
implemented algorithm for deciding the beta-eta-equality of terms---and this in
spite of the first decidability results proven two decades ago. Second, it is
not clear how to decide when two types are essentially the same, i.e.
isomorphic, in spite of the meta-theoretic results on decidability of the
isomorphism.
In this paper, we present the exp-log normal form of types---derived from the
representation of exponential polynomials via the unary exponential and
logarithmic functions---that any type built from arrows, products, and sums,
can be isomorphically mapped to. The type normal form can be used as a simple
heuristic for deciding type isomorphism, thanks to the fact that it is a
systematic application of the high-school identities.
We then show that the type normal form allows to reduce the standard beta-eta
equational theory of the lambda calculus to a specialized version of itself,
while preserving the completeness of equality on terms. We end by describing an
alternative representation of normal terms of the lambda calculus with sums,
together with a Coq-implemented converter into/from our new term calculus. The
difference with the only other previously implemented heuristic for deciding
interesting instances of eta-equality by Balat, Di Cosmo, and Fiore, is that we
exploit the type information of terms substantially and this often allows us to
obtain a canonical representation of terms without performing sophisticated
term analyses
Perspectives for proof unwinding by programming languages techniques
In this chapter, we propose some future directions of work, potentially
beneficial to Mathematics and its foundations, based on the recent import of
methodology from the theory of programming languages into proof theory. This
scientific essay, written for the audience of proof theorists as well as the
working mathematician, is not a survey of the field, but rather a personal view
of the author who hopes that it may inspire future and fellow researchers
Twisted Hopf symmetries of canonical noncommutative spacetimes and the no-pure-boost principle
We study the twisted-Hopf-algebra symmetries of observer-independent
canonical spacetime noncommutativity, for which the commutators of the
spacetime coordinates take the form [x^{mu},x^{nu}]=i theta^{mu nu} with
observer-independent (and coordinate-independent) theta^{mu nu}. We find that
it is necessary to introduce nontrivial commutators between transformation
parameters and spacetime coordinates, and that the form of these commutators
implies that all symmetry transformations must include a translation component.
We show that with our noncommutative transformation parameters the Noether
analysis of the symmetries is straightforward, and we compare our
canonical-noncommutativity results with the structure of the conserved charges
and the "no-pure-boost" requirement derived in a previous study of
kappa-Minkowski noncommutativity. We also verify that, while at intermediate
stages of the analysis we do find terms that depend on the ordering convention
adopted in setting up the Weyl map, the final result for the conserved charges
is reassuringly independent of the choice of Weyl map and (the corresponding
choice of) star product.Comment: 12 page
Gamma spaces and information
We investigate the role of Segal’s Gamma-spaces in the context of classical and quantum information, based on categories of finite probabilities with stochastic maps and density matrices with quantum channels. The information loss functional extends to the setting of probabilistic Gamma-spaces considered here. The Segal construction of connective spectra from Gamma-spaces can be used in this setting to obtain spectra associated to certain categories of gapped systems
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