Automating the Diagram Method to Prove Correctness of Program Transformations

Abstract

We report on the automation of a technique to prove the correctness of program transformations in higher-order program calculi which may permit recursive let-bindings as they occur in functional programming languages. A program transformation is correct if it preserves the observational semantics of programs. In our LRSX Tool the so-called diagram method is automated by combining unification, matching, and reasoning on alpha-renamings on the higher-order meta-language, and automating induction proofs via an encoding into termination problems of term rewrite systems. We explain the techniques, we illustrate the usage of the tool, and we report on experiments. * This research is supported by the Deutsche Forschungsgemeinschaft (DFG) under grant SA2908/3-1 1 available from http://goethe.link/LRSXTOOL61 Illustration of the Diagram Method -ExamplesWe illustrate the concept of observational semantics, correctness of program transformations, and the diagram method (and its automation) using a quite simple example. In Fig. 2 we define a program calculus Simple. The syntax of Simple-expressions consists of two constants, ⊥ to represent a failing computation, and ⊤ to represent success, a unary operator ¬ for negation, and a binary operator ∧ which computes the conjunction of ⊤ and ⊥, i.e. evaluation of e 1 ∧ e 2 results in ⊤ iff e 1 and e 2 both evaluate to ⊤ and otherwise the evaluation ends with ⊥. The reduction strategy which evaluates the ∧-operator from left to right is defined by using evaluation contexts A (defined in Fig. 2 where [·] denotes the context hole). The standard reduction sr − → is the union of the rules (sr, bot), (sr,top), (sr, neg1), and (sr, neg2).

A Probabilistic Call-by-Need Lambda-Calculus -- Extended Version

“…2 are correct. We use the so-called diagram method [21][22][23]28] to prove their correctness. Given a transformation →, the method uses Proposition 4.4 to show correctness: for a step S, → (note that we work with the closure of → w.r.t.…”

“…In earlier work (e.g. [23,28]), diagrams were computed manually, but nowadays they are computed automatically by a tool, that unifies the left-hand sides of the standard reductions with left-(forking) and also right-hand sides (commuting) of the transformation, and then searches to show joinability (the unification algorithm is described in [27], the tool is described in [22]). If not stated otherwise, the diagrams in this paper were computed by this tool.…”

“…The latter is a preparation for further correctness proofs of program transformations, that are used as local optimizations in compilers: here we apply a syntactic method to show correctness. This so-called diagram method [21][22][23]28] computes all overlaps between transformation steps and reductions of the operational semantics, and joins these overlaps, such that the results are complete sets of diagrams. The diagrams are then used to inductively show the (expected) convergence equivalence.…”

“…The diagrams are then used to inductively show the (expected) convergence equivalence. The diagram computation and the inductive construction are semi-automated by techniques and tools, from unification and term rewriting, that were developed in previous work [21,22,27] for deterministic and non-deterministic calculi. We show that they are transferable and still very useful for the probabilistic setting.…”