RZ: a Tool for Bringing Constructive and Computable Mathematics Closer to Programming Practice

Abstract

Realizability theory is not just a fundamental tool in logic and computability. It also has direct application to the design and implementation of programs, since it can produce code interfaces for the data structure corresponding to a mathematical theory. Our tool, called RZ, serves as a bridge between the worlds of constructive mathematics and programming. By using the realizability interpretation of constructive mathematics, RZ translates specifications in constructive logic into annotated interface code in Objective Caml. The system supports a rich input language allowing descriptions of complex mathematical structures. RZ does not extract code from proofs, but allows any implementation method, from handwritten code to code extracted from proofs by other tools.

Minlog - A Tool for Program Extraction Supporting Algebras and Coalgebras

“…Also, Minlog seems to be the only system implementing the Dialectica Interpretation and program extraction from classical proofs. We also mention RZ [BS07], a tool that computes the realizability interpretation of a mathematical statement (but does not extract programs from proofs).…”

Implementing Real Numbers With RZ

“…In order to keep mathematics and programming close to each other, we replaced the customary Type 2 representations with representations in a programming language, in our case Objective Caml [11], but other languages could be used. Then we used RZ [5], a tool written in a related project by Chris Stone and Andrej Bauer, to automatically translate the constructive theory of reals to a formal program specification. Finally, we implemented the specification in Objective Caml.…”

“…We refer to [4] for resources on RZ, and to [3,2] for background on how realizability theory is used to connect constructive and computable mathematics. In this section we explain enough to make the rest of the paper comprehensible.…”

“…RZ uses the realizability interpretation of constructive logic and dependent type theory in the category of pers to compute specifications from mathematical theories, as is explained in [5].…”

“…Any general-purpose programming language with sufficiently well-defined semantics could be used instead. The technical requirement is that P forms a typed partial combinatory algebra[12] 4. The type τ A is not a set but just a formal expression in P, which is why we distinguish between τ A and its set of values [[τ A ]].…”