Correctness of Programs with Function Procedures

The correctness of programs with programmer-declared functions is investigated. We use the framework of the typed lambda calculus with explicit declaration of (possibly recursive) functions. Its expressions occur in the statements of a simple language with assignment, composition and conditionals. A denotational and an operational semantics for this language are provided, and their equivalence is proved. Next, a proof system for partial correctness is presented, and its soundness is shown. Completeness is then established for the case that only call-by-value is allowed. Allowing call-by-name as well, completeness is shown only for the case that the type structure is restricted, and at the cost of extending the language of the proof system. The completeness problem for the general case remains open. In the technical considerations, an important role is played by a reduction system which essentially allows us to reduce expression evaluation to systematic execution of auxiliary assignments. Termination of this reduction system is shown using Tait's computability technique. Complete proofs will appear in the full version of the paper.

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