CERFACS CERFACS Scientific Activity Report Jan. 2010 – Dec. 2011

6 Qualitative Computing
Group member : Françoise Chatelin, CERFACS and Université Toulouse 1.
The main activity of the unique member in the Qualitative Computing group in 2010-2011 has been
to continue the exploration of the multiplicative computing land presented in the book “Qualitative
Computing : A computational journey into nonlinearity” to be published by World Scientific, Singapore
in 2012. This book, written in mathematical language, is about the domain of mathematical computation
which extends beyond modern calculus and classical analysis when numbers are not restricted to belong
to a commutative field. It describes the dynamics of complexification, resulting in an endless remorphing
of the computational landscape. Computation weaves a colourful tapestry always in a state of becoming.
In the process, some meta-principles emerge which guide the autonomous evolution of mathematical
computation. These organic principles are essential keys to analyse very large numerical simulations of
unstable phenomena.
High tech industries are in desperate need for adequate tools to assess the validity of simulations produced
by ever faster computers for ever more unstable problems. In order to meet these industrial expectations,
the applied mathematicians are facing a formidable challenge summarized by the two words 1) nonlinearity
and 2) coupling. The book proposes to explore radically new paths in the unchartered jungle of nonlinear
computation.
– 1) Use hypercomputation in quadratic algebras, rather than the computation in linear vector spaces that
is traditional since the early 20th century.
– 2) Complement the classical linear logic (based on the sequence of natural integers) with a complex logic
which expresses the potential of the complex plane for organic intelligence.
The presentation of the technical content is almost everywhere kept at an undergraduate level. The
prerequisites are classical calculus, analysis and numerical linear algebra.
The subject of Qualitative Computing covers theoretical and practical aspects of nonlinear computation.
Multiplication is the lead actor : multiplication of numbers, vectors and matrices. The theoretical aspects
which have been chosen for presentation in the book describe hypercomputation over vectors in Dickson
algebras (Chapters 2 to 6, 9 and 11), the theory of Homotopic Deviation for matrices (Chapter 7), and
Fourier analysis for complex signals (Chapter 10). As for Chapter 8, it addresses more practical aspects.
Inter alia, it clarifies why the scientific computer not only is a most efficient tool to speed-up intractable
computations in every corner of our technological society, but also has an epistemological potential which
begs to be put to good use in our attempt to decipher the organic evolution of life. Then, the final Chapter 12
concludes with organic intelligence for dicksonian numbers (i.e. vectors in nonassociative algebras A k ,k ≥
1 of dimension 2 k , recursively defined from A 0 = R) and wraps up some of the lessons in computation that
were taught in the book. A few of them follow. Numbers need not be confined to commutative fields. They
can be vectors or matrices in algebras equipped with a noncommutative multiplication that is the source of
evolution. And Nature’s computation based on electromagnetic information coming from physical light can
be paradoxical ! Moreover, the informations processed by computation in A k , k ≥ 5, cannot have a purely
electromagnetic origin.
The work of the years 2010-2011 is presented in detail in the five technical reports [ALG41, ALG46,
ALG45, ALG43, ALG42]. In order to account for the phenomenon known as Special Relativity in Physics
and that is found everywhere in Life Sciences, the classical addition of vectors can be generalized to
define a relativistic addition, whose departure from commutativity and associativity is controled by an
22 Jan. 2010 – Dec. 2011