158 research outputs found
Formalization of a normalization theorem in simplicial topology
In this paper we present a complete formalization of the Normalization
Theorem, a result in Algebraic Simplicial Topology stating that there exists a
homotopy equivalence between the chain complex of a simplicial set, and a smaller
chain complex for the same simplicial set, called the normalized chain complex.
Even if the Normalization Theorem is usually stated as a higher-order result (with
a Category Theory flavor) we manage to give a first-order proof of it. To this aim
it is instrumental the introduction of an algebraic data structure called simplicial
polynomial. As a demonstration of the validity of our techniques we developed a
formal proof in the ACL2 theorem prover.Ministerio de Ciencia e Innovación MTM2009-13842European Commission FP7 STREP project ForMath n. 24384
Applying ACL2 to the Formalization of Algebraic Topology: Simplicial Polynomials
In this paper we present a complete formalization, using the
ACL2 theorem prover, of the Normalization Theorem, a result in Algebraic
Simplicial Topology stating that there exists a homotopy equivalence
between the chain complex of a simplicial set, and a smaller chain
complex for the same simplicial set, called the normalized chain complex.
The interest of this work stems from three sources. First, the normalization
theorem is the basis for some design decisions in the Kenzo computer
algebra system, a program for computing in Algebraic Topology.
Second, our proof of the theorem is new and shows the correctness of
some formulas found experimentally, giving explicit expressions for the
above-mentioned homotopy equivalence. And third, it demonstrates that
the ACL2 theorem prover can be effectively used to formalize mathematics,
even in areas where higher-order tools could be thought to be more
appropriate.Ministerio de Ciencia e Innovación MTM2009-13842European Commission FP7 STREP project ForMath n. 24384
Verifying the bridge between simplicial topology and algebra: the Eilenberg–Zilber algorithm
The Eilenberg–Zilber algorithm is one of the central components of the computer algebra system called Kenzo, devoted to
computing in Algebraic Topology. In this article we report on a complete formal proof of the underlying Eilenberg–Zilber
theorem, using the ACL2 theorem prover. As our formalization is executable, we are able to compare the results of the
certified programme with those of Kenzo on some universal examples. Since the results coincide, the reliability of Kenzo is
reinforced. This is a new step in our long-term project towards certified programming for Algebraic Topology.Ministerio de Ciencia e Innovación MTM2009-13842European Union’s 7th Framework Programme [243847] (ForMath)
Certified Symbolic Manipulation: Bivariate Simplicial Polynomials
Certified symbolic manipulation is an emerging new field
where programs are accompanied by certificates that, suitably interpreted, ensure the correctness of the algorithms. In
this paper, we focus on algebraic algorithms implemented in
the proof assistant ACL2, which allows us to verify correctness in the same programming environment. The case study
is that of bivariate simplicial polynomials, a data structure
used to help the proof of properties in Simplicial Topology.
Simplicial polynomials can be computationally interpreted in
two ways. As symbolic expressions, they can be handled
algorithmically, increasing the automation in ACL2 proofs.
As representations of functional operators, they help proving
properties of categorical morphisms. As an application of this
second view, we present the definition in ACL2 of some
morphisms involved in the Eilenberg-Zilber reduction, a central part of the Kenzo computer algebra system. We have
proved the ACL2 implementations are correct and tested
that they get the same results as Kenzo does.Ministerio de Ciencia e Innovación MTM2009-13842Unión Europea nr. 243847 (ForMath
Proving and Computing: Applying Automated Reasoning to the Verification of Symbolic Computation Systems
The application of automated reasoning to the formal verification
of symbolic computation systems is motivated by the need of
ensuring the correctness of the results computed by the system, beyond
the classical approach of testing. Formal verification of properties of the
implemented algorithms require not only to formalize the properties of
the algorithm, but also of the underlying (usually rich) mathematical
theory.
We show how we can use ACL2, a first-order interactive theorem
prover, to reason about properties of algorithms that are typically implemented
as part of symbolic computation systems. We emphasize two
aspects. First, how we can override the apparent lack of expressiveness we
have using a first-order approach (at least compared to higher-order logics).
Second, how we can execute the algorithms (efficiently, if possible)
in the same setting where we formally reason about their correctness.
Three examples of formal verification of symbolic computation algorithms
are presented to illustrate the main issues one has to face in this
task: a Gr¨obner basis algorithm, a first-order unification algorithm based
on directed acyclic graphs, and the Eilenberg-Zilber algorithm, one of
the central components of a symbolic computation system in algebraic
topology
On the stability of persistent entropy and new summary functions for Topological Data Analysis
Persistent entropy of persistence barcodes, which is based on the Shannon entropy, has
been recently defined and successfully applied to different scenarios: characterization of the
idiotypic immune network, detection of the transition between the preictal and ictal states in
EEG signals, or the classification problem of real long-length noisy signals of DC electrical
motors, to name a few. In this paper, we study properties of persistent entropy and prove its
stability under small perturbations in the given input data. From this concept, we define three
summary functions and show how to use them to detect patterns and topological features
Karoubi's relative Chern character, the rigid syntomic regulator, and the Bloch-Kato exponential map
We construct a variant of Karoubi's relative Chern character for smooth,
separated schemes over the ring of integers in a p-adic field and prove a
comparison with the rigid syntomic regulator. For smooth projective schemes we
further relate the relative Chern character to the etale p-adic regulator via
the Bloch-Kato exponential map. This reproves a result of Huber and Kings for
the spectrum of the ring of integers and generalizes it to all smooth
projective schemes as above.Comment: v1:33 pages; v2:major revision (28 pages); v3:minor changes; v4:minor
changes following suggestions by a refere
The notion of dimension in geometry and algebra
This talk reviews some mathematical and physical ideas related to the notion
of dimension. After a brief historical introduction, various modern
constructions from fractal geometry, noncommutative geometry, and theoretical
physics are invoked and compared.Comment: 29 pages, a revie
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