# Computer Science for Continuous Data: Algorithmic Foundations of Numerics

## Motivation

Digital Computers naturally process discrete data, such as bits or integers or strings or graphs. From bits to advanced data structures, from a first semi-conducting transistor to billions in wafer-scale integration, from individual Boolean connectives to entire CPU circuits, from kB to TB memories, from 10^{2} to 10^{9} instructions per second, from assembly code to high-level programming languages:

the success story of digital computing arguably is due to (1) hierarchical layers of abstraction and (2) the ultimate reliability of each layer for the next one to build on ― for processing discrete data.

Continuous data on the other hand commonly arises in Physics, Engineering and Science: *natura non facit saltus*.
Such data mathematically corresponds to real numbers, smooth functions, bounded operators, or compact subsets of some abstract metric space.

The rise of digital (over analog) computers led a stagnation in the realm of continuous (=*non*-discretized) data processing:
35 years after introduction and hardware standardization of IEEE 754 floating point numbers, mainstream Numerics is arguably still dominated by this forcible discretization of continuous data ― in spite of violating associative and distributive laws, breaking symmetries, introducing and propagating rounding errors, in addition to an involved (and incomplete) axiomatization including NaNs and denormalized numbers.

Figure 1: Floating Point Number Line, from https://courses.engr.illinois.edu/cs357/fa2019/assets/images/figs/floatingpoints.png. See also https://www.jasss.org/9/4/4.html#2.1

Deviations between mathematical structures and their hardware counterparts are common also in the discrete realm,
such as the “integer” wraparound 255+1=0 occurring in bytes that led to the *Nuclear Gandhi* programming bug.

Similarly, deviations between exact and approximate continuous data underlie infamous failures such as the Ariane 501 flight V88 or the Sleipner-A oil platform.

Nowadays high-level programming languages (such as Java or Python) provide a user data type (called for example `BigInt`

or `mpz_t`

) that fully agrees with mathematical integers, simulated in software using a variable number of hardware bytes.
This additional layer of abstraction provides the reliability for advanced discrete data types (such as weighted or labelled graphs) to build on, as mentioned above.

## Mission

We develop Computer Science for continuous data, to catch up with the discrete case: from foundations via practical implementation to commercial applications.

In fact some object-oriented software libraries, such as iRRAM or Core III or realLib or Ariadne or Aern, have long been providing general (=including all transcendental) real numbers as exact encapsulated user data type.
Technically they employ finite but variable precision approximations: much like `BigInt`

, but with the added challenge of choosing said precision automatically and adaptively sufficient for the user program to appear as indistinguishable from exact reals. This requires a new (namely partial) semantics for real comparison: formalizing the folklore to “avoid” testing for equality, in terms of Kleene's ternary logic.

- Sewon Park in his PhD Thesis has extended that semantics to composite expressions, and further to command sequences aka programs, whose correctness can then be symbolically verified using an extension of Floyd-Hoare Logic;

see also the preprint arXiv:1608.05787.

- Thus reliably building on single real numbers leads to higher (but finite) dimensional data types, such as vectors or matrices.

Sewon Park has designed and analyzed and implemented a reliable variant of Gaussian Elimination, in particular regarding pivot search. Seokbin Lee has designed and analyzed and implemented a reliable variant of the Grassmannian, i.e., the orthomodular lattice of subspaces of some fixed finite-dimensional Euclidean or unitary vector space.

- Infinite sequences of real numbers arise as elements of ℓ
^{p}spaces; and as coefficients to analytic function germs.

Holger Thies has implemented analytic functions for reliably solving ODEs and PDEs.

See the references below for this and more continuous data types on GitHub.

Like discrete data, processing continuous data on a digital computer eventually boils down to processing bits: finite sequences of bits in the discrete case, in the continuous case infinite sequences, approximated via finite initial segments. Coding theory of discrete data is well-established since Claude Shannon’s famous work. Encoding real numbers as infinite sequences of bits is non-trivial: the binary expansion for example renders addition uncomputable.

- Donghyun Lim in his MSc Thesis has investigated encoding more advanced (such as function) spaces;

see also the preprint arXiv:2002.04005.

- Ivan Koswara and Gleb Pogudin and Svetlana Selivanova have related the bit-complexity intrinsic to approximate solutions of linear partial differential equations to discrete complexity classes #P and PSPACE.

## References

- C++ classes for continuous data types: RealComputation, Holger Thies

- Workshop on Computational Knot Theory organized by Alexander Stoimenow

- Vision paper (in German) about Artificial Intelligence, written jointly with Andrea Reichenberger