Isomorphic Data Type Transformations

Abstract

In stepwise derivations of programs from specifications, data type refinements are common. Many data type refinements involve isomorphic mappings between the more abstract and more concrete data representations. Examples include refinement of finite sets to duplicate-free ordered lists or to bit vectors, adding record components that are functions of the other fields to avoid expensive recomputation, etc. This paper describes the APT (Automated Program Transformations) tools to carry out isomorphic data type refinements in the ACL2 theorem prover and gives examples of their use. Because of the inherent symmetry of isomorphisms, these tools are also useful to verify existing programs, by turning more concrete data representations into more abstract ones to ease verification. Typically, a data type will have relatively few interface functions that access the internals of the type. Once versions of these interface functions have been derived that work on the isomorphic type, higher-level functions can be derived simply by substituting the old functions for the new ones. We have implemented the APT transformations isodata to generate the former, and propagate-iso for generating the latter functions as well as theorems about the generated functions from the theorems about the original functions. Propagate-iso also handles cases where the type is a component of a more complex one such as a list of the type or a record that has a field of the type: the isomorphism on the component type is automatically lifted to an isomorphism on the more complex type. As with all APT transformations, isodata and propagate-iso generate proofs of the relationship of the transformed functions to the originals. Background, Motivation, and ContributionIn stepwise program refinement [10,24], an implementation is derived from a specification via a sequence of intermediate specifications. A derivation is a sequence s 0 ↝ s 1 ↝ ... ↝ s n c ↦ p, where: s 0 is a requirements specification in some specification language; s 1 ,...,s n are intermediate specifications in the same language; ↝ is a refinement relation (a preorder, i.e. reflexive and transitive); and p is an implementation in some programming language, obtained from s n via a code generator c