Computable decision making on the reals and other spaces via partiality and nondeterminism

TL;DR

This paper introduces a programming semantics framework using constructive topology with nondeterminism and partiality, enabling computable decision making on connected spaces like the real numbers, and implements these ideas in the Marshall language.

Contribution

It develops a novel semantics based on constructive topology that allows decision making on real spaces using nondeterminism and partiality, and introduces pattern matching on spaces.

Findings

Nondeterminism or partiality enables decision making on connected spaces.

Pattern matching on spaces generalizes functional programming patterns.

Implementation in the Marshall language demonstrates practical applicability.

Abstract

Though many safety-critical software systems use floating point to represent real-world input and output, programmers usually have idealized versions in mind that compute with real numbers. Significant deviations from the ideal can cause errors and jeopardize safety. Some programming systems implement exact real arithmetic, which resolves this matter but complicates others, such as decision making. In these systems, it is impossible to compute (total and deterministic) discrete decisions based on connected spaces such as $\mathbb{R}$. We present programming-language semantics based on constructive topology with variants allowing nondeterminism and/or partiality. Either nondeterminism or partiality suffices to allow computable decision making on connected spaces such as $\mathbb{R}$. We then introduce pattern matching on spaces, a language construct for creating programs on spaces, generalizing pattern matching in functional programming, where patterns need not represent decidable predicates and also may overlap or be inexhaustive, giving rise to nondeterminism or partiality, respectively. Nondeterminism and/or partiality also yield formal logics for constructing approximate decision procedures. We implemented these constructs in the Marshall language for exact real arithmetic.