Email Gateway (Aug 22 2022 at 12:42): Email Gateway (Aug 22 2022 at 12:42):From: Gerwin Klein <Gerwin.Klein@nicta.com.au>
AFP 2016 is now available from http://afp.sf.net
All entries work with Isabelle2016, previous releases remain available, and the following new entries have become available from the front page:
List Update
by Maximilian Haslbeck and Tobias Nipkow
These theories formalize the quantitative analysis of a number of classical algorithms for the list update problem: 2-competitiveness of move-to-front, the lower bound of 2 for the competitive- ness of deterministic list update algorithms and 1.6-competitiveness of the randomized COMB algorithm, the best randomized list update algorithm known to date.
An informal description is found in an accompanying report. The material is based on the first two chapters of Online Computation and Competitive Analysis by Borodin and El-Yaniv.
Polynomial Interpolation
by René Thiemann and Akihisa Yamada
We formalized three algorithms for polynomial interpolation over arbitrary fields: Lagrange's explicit expression, the recursive algorithm of Neville and Aitken, and the Newton interpolation in combination with an efficient implementation of divided differences. Variants of these algorithms for integer polynomials are also available, where sometimes the interpolation can fail; e.g., there is no linear integer polynomial p such that p(0) = 0 and p(2) = 1. Moreover, for the Newton interpolation for integer polynomials, we proved that all intermediate results that are computed during the algorithm must be integers. This admits an early failure detection in the implementation. Finally, we proved the uniqueness of polynomial interpolation.
The development also contains improved code equations to speed up the division of integers in target languages.
Polynomial Factorization
by René Thiemann and Akihisa Yamada
Based on existing libraries for polynomial interpolation and matrices, we formalized several factorization algorithms for polynomials, including Kronecker's algorithm for integer polynomials, Yun's square-free factorization algorithm for field polynomials, and Berlekamp's algorithm for polynomials over finite fields. By combining the last one with Hensel's lifting, we derive an efficient factorization algorithm for the integer polynomials, which is then lifted for rational polynomials by mechanizing Gauss' lemma. Finally, we assembled a combined factorization algorithm for rational polynomials, which combines all the mentioned algorithms and additionally uses the explicit formula for roots of quadratic polynomials and a rational root test.
As side products, we developed division algorithms for polynomials over integral domains, as well as primality-testing and prime-factorization algorithms for integers.
Ergodic Theory
by Sebastien Gouezel
Ergodic theory is the branch of mathematics that studies the behaviour of measure preserving transformations, in finite or infinite measure. It interacts both with probability theory (mainly through measure theory) and with geometry as a lot of interesting examples are from geometric origin. We implement the first definitions and theorems of ergodic theory, including notably Poicaré recurrence theorem for finite measure preserving systems (together with the notion of conservativity in general), induced maps, Kac's theorem, Birkhoff theorem (arguably the most important theorem in ergodic theory), and variations around it such as conservativity of the corresponding skew product, or Atkinson lemma.
The Tortoise and Hare Algorithm
by Peter Gammie
We formalize the Tortoise and Hare cycle-finding algorithm ascribed to Floyd by Knuth, and an improved version due to Brent.
Planarity Certificates
by Lars Noschinski
This development provides a formalization of planarity based on combinatorial maps and proves that Kuratowski's theorem implies combinatorial planarity. Moreover, it contains verified implementations of programs checking certificates for planarity (i.e., a combinatorial map) or non-planarity (i.e., a Kuratowski subgraph).
Enjoy!
Gerwin
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Last updated: Nov 21 2024 at 12:39 UTC