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Isabelle/HOL Theories of Algebras for Iteration, Infinite Executions and Correctness of Sequential Computations
This technical report contains Isabelle/HOL theories that describe algebras for iteration, infinite executions and correctness of sequential computations. The results are explained in a separate document. The following files are available:
Technical Report
Isabelle/HOL theory file
Hilbert spaces built on a similarity and on dynamical renormalization
We develop a Hilbert space framework for a number of general multi-scale
problems from dynamics. The aim is to identify a spectral theory for a class of
systems based on iterations of a non-invertible endomorphism.
We are motivated by the more familiar approach to wavelet theory which starts
with the two-to-one endomorphism in the one-torus \bt, a
wavelet filter, and an associated transfer operator. This leads to a scaling
function and a corresponding closed subspace in the Hilbert space
L^2(\br). Using the dyadic scaling on the line \br, one has a nested family
of closed subspaces , n \in \bz, with trivial intersection, and with
dense union in L^2(\br). More generally, we achieve the same outcome, but in
different Hilbert spaces, for a class of non-linear problems. In fact, we see
that the geometry of scales of subspaces in Hilbert space is ubiquitous in the
analysis of multiscale problems, e.g., martingales, complex iteration dynamical
systems, graph-iterated function systems of affine type, and subshifts in
symbolic dynamics. We develop a general framework for these examples which
starts with a fixed endomorphism (i.e., generalizing ) in a
compact metric space . It is assumed that is onto, and
finite-to-one.Comment: v3, minor addition
Guard Your Daggers and Traces: On The Equational Properties of Guarded (Co-)recursion
Motivated by the recent interest in models of guarded (co-)recursion we study
its equational properties. We formulate axioms for guarded fixpoint operators
generalizing the axioms of iteration theories of Bloom and Esik. Models of
these axioms include both standard (e.g., cpo-based) models of iteration
theories and models of guarded recursion such as complete metric spaces or the
topos of trees studied by Birkedal et al. We show that the standard result on
the satisfaction of all Conway axioms by a unique dagger operation generalizes
to the guarded setting. We also introduce the notion of guarded trace operator
on a category, and we prove that guarded trace and guarded fixpoint operators
are in one-to-one correspondence. Our results are intended as first steps
leading to the description of classifying theories for guarded recursion and
hence completeness results involving our axioms of guarded fixpoint operators
in future work.Comment: In Proceedings FICS 2013, arXiv:1308.589
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