Type-2 effectivity in abstract state machines for algorithms with exact real arithmetic

Abstract

• Integration of exact real arithmetic into Abstract State Machines (ASMs) based on type-2 theory of effectivity (TTE). • Representation of real numbers by rapidly converging Cauchy sequences. • ASM background structure with many operations on reals such as binary comparisons, exponentials, trigonometric functions. • Stream computations with concurrent ASMs. • ASM specification of algorithms over the reals illustrated by the well known Newton algorithm. Computations with real numbers are decisive for all scientific and technical applications, in particular cyber-physical systems, and precision in the results is essential for quality and safety. The type-2 theory of effectivity (TTE) is a well established theory of computability on infinite strings, which can be used to represent real numbers by rapidly converging Cauchy sequences, on top of which standard operations such as addition, multiplication, division, exponentials, trigonometric functions, etc. can be defined. In this paper we develop an extension of Abstract State Machines (ASMs) handling streams in an incremental way in accordance with TTE. This enables defining a data type Real as part of the background structure and based on this exact computation with real numbers. Output can be generated at any degree of precision by exploring only sufficiently long prefixes of the representing Cauchy sequences. We then outline an ASM development process that starts from a specification of an algorithm using Real , proceeds by making the rational elements of the Cauchy sequences explicit in a refinement, reconsiders this refined specification as a concurrent ASM, where each agent cares about computation up to some precision, and finally, based on a detailed analysis of the required precision, prunes the concurrent ASM down to an ASM that guarantees a sufficient level of precision in the outputs. In doing so, backward precision propagation replaces the common forward error propagation that is common for implementations exploiting floating point arithmetic.