Proof Methods for Corecursive Programs

Recursion is a well-known and powerful programming technique, with a wide variety of applications. The dual technique of corecursion is less well-known, but is increasingly proving to be just as useful. This article is a tutorial on the four main methods for proving properties of corecursive programs: fixpoint induction, the approximation (or take) lemma, coinduction, and fusion.

Proof methods for structured corecursive programs

Corecursive programs produce values of greatest fixpoint types, in contrast to recursive programs, which consume values of least fixpoint types. There are a number of widely used methods for proving properties of corecursive programs, including fixpoint induction, the take lemma, and coinduction. However, these methods are all rather low level, in that they do not exploit the common structure that is often present in corecursive definitions. We argue for a more structured approach to proving properties of corecursive programs. In particular, we show that by writing corecursive programs using a simple operator that encapsulates a common pattern of corecursive definition, we can then use high-level algebraic properties of this operator to conduct proofs in a purely calculational style that avoids the use of inductive or coinductive methods.

The Use of Proof Planning Critics to Diagnose Errors in the Base Cases of Recursive Programs

This paper reports the use of proof planning to diagnose errors in program code. In particular it looks at the errors that arise in the base cases of recursive programs produced by undergraduates. It describes two classes of error that arise in this situation. The use of test cases would catch these errors but would fail to distinguish between them. The system adapts proof critics, commonly used to patch faulty proofs, to diagnose such errors and distinguish between the two classes. It has been implemented in Lambda-clam, a proof planning system, and applied successfully to a small set of examples.

Run-time profiling of Algol 68-R programs using DIDYMUS and SCAMP

Information concerning the run-time behaviour of programs ("program profiling") can be of the greatest assistance in improving program efficiency. Two software devices have been developed for use on ICL 1900 Series machines to provide such information. DIDYMUS is probabilistic in approach and uses multi- tasking facilities to sample the instruction addresses used by a program at run time. It will work regardless of the source language of the program and matches the detected addresses against a loader map to produce a histogram. SCAMP is restricted to profiling Algol 68-R programs, but provides deterministic information concerning those language constructs that are monitored. Procedure calls to appropriate counting routines are inserted into the source text in a pre-pass prior to compilation. The profile information is printed out at the end of the program run. It has been found that these two approaches complement each other very effectively.