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Proof Rules for Recursive Procedures Hesselink, Wim H View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by University of Groningen University of Groningen Proof Rules for Recursive Procedures Hesselink, Wim H. Published in: Formal Aspects of Computing IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite from it. Please check the document version below. Document Version Publisher's PDF, also known as Version of record Publication date: 1993 Link to publication in University of Groningen/UMCG research database Citation for published version (APA): Hesselink, W. H. (1993). Proof Rules for Recursive Procedures. Formal Aspects of Computing, 5. Copyright Other than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of the author(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons). Take-down policy If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim. Downloaded from the University of Groningen/UMCG research database (Pure): http://www.rug.nl/research/portal. For technical reasons the number of authors shown on this cover page is limited to 10 maximum. Download date: 12-11-2019 Formal Aspects of Computing (1993) 5:554-570 @ 1993 BCS Formal Aspects of Computing Proof Rules for Recursive Procedures Wirn H. Hesselink Rijksuniversiteit Groningen, Department of Computing Science, Groningen, The Netherlands Keywords: Proof rule; Recursive procedure; Specification; Weakest precondition; Well-founded relation Abstract. Four proof rules for recursive procedures in a Pascal-like language are presented. The main rule deals with total correctness and is based on results of Gries and Martin. The rule is easier to apply than Martin's. It is introduced as an extension of a specification format for Pascal-procedures, with its associated correctness and invocation rules. It uses well-founded recursion and is proved under the postulate that a procedure is semantically equal to its body. This rule for total correctness is compared with Hoare's rule for partial correct- ness of recursive procedures, in which no well-founded relation is needed. Both rules serve to prove correctness, i.e. sufficiency of certain preconditions. There are also two rules for proving necessity of preconditions. These rules can be used to give formal proofs of nontermination and refinement. They seem to be completely new. 1. Announcement of Aims and Results In iterative programming, the proof rules for correctness are well established and have led to powerful methods for program derivation, culminating in [Kal90]. For recursive procedures in a language like Pascal, however, a generally accepted proof rule is still lacking. In [Hoa71], Hoare presented a derivation system to prove partial correctness of recursive procedures. This system has the disadvantages that it can only yield partial correctness and that it is incompatible with the full power of predicate calculus. In [Heh79] and [Mar83], methods are given to treat total correctness, but a concrete proof rule is not given. In particular, the combination of recursion and parameters remains implicit. The purpose of the present paper is to propose a proof rule for total correctness Correspondence and offprint requests to: Wire H. Hesselink, Rijksuniversiteit Groningen, Department of Computing Science, P.O. Box 800, 9700 AV Groningen, The Netherlands. email: [email protected] Proof Rules for Recursive Procedures 555 of recursive procedures with value and reference parameters in a language like Pascal. The specifications are given in terms of preconditions and postconditions. The proof rule is given in three versions of increasing abstractness. The most abstract version is compared with a version of Hoare's Rule that is compatible with predicate calculus. A second purpose of this paper is to present two proof rules complementary to the rules for total and partial correctness. These new rules serve to prove necessity of preconditions instead of sufficiency. 2. Introduction The main role of the procedure mechanism in programming is that it allows abstraction and thus serves to separate the invocation of a command from its implementation. This separation is accomplished by a full specification. The user of a procedure relies on the specification and the implementer has only the task to fulfil it. The user has an invocation rule to derive properties of the invocation from the specification. The implementer has a correctness rule to prove that a body can serve as an implementation. Recursion arises when the implementers take the opportunity to use the procedure(s) they are implementing. In that case, the correctness rule requires the invocation rule as an induction hypothesis. The resulting problem of circularity has two solutions. The circularity can be ignored or it can be broken by means of well-founded relations. In [Hoa71], Hoare proposes the first solution. He gives a rule that guarantees only partial correctness: if execution terminates, a postcondition is guaranteed to be established. Hoare's rule requires a concept of derivability. It can be formalized in so-called Hoare logic or in dynamic logic, but it has no obvious rendering in predicate calculus. In [Mar83], Martin opts for the second solution. He gives a method to prove total correctness, i.e. partial correctness as well as termination. The method is not formalized to a concrete proof rule, and it has the disadvantage that it requires predicate calculus on two different state spaces: the space of the call and the space of the declaration. The main result of the present paper is a proof rule for total correctness more explicit than Martin's method. The rule enables complete and rather concise correctness proofs by annotation. It has the same power as Martin's method, though -at first sight- it may seem to be weaker. The rule uses predicates on only one state space, the space of the call. In exchange for this advantage, we get the burden that there are more specification values (logical variables) needed to relate preconditions and postconditions. In fact, following [Gri81] p. 151, we require that the postcondition does not contain the value parameters, so that specification values are needed instead. In section 3, we present a format for specified declaration of Pascal procedures. This format is accompanied by a correctness rule for the implementer and an invocation rule for the user of the procedure. The proof rule for total correctness of recursive procedures is given in three versions: (10), (17) and (19). Section 4 contains rule (10), in which natural numbers are sufficient to prove termination. Section 5 contains the stronger rule (17), which captures well-founded recursion. Section 6 contains rule (19), an abstraction in which parametrization, specification values, invariant predicates and mutual recursion are unified. Subsequently, rule (19) is proved. 556 W.H. Hesselink Hoare's rule for partial correctness of recursive procedures is presented as rule (25) in section 7. Instead of a variant function, it uses quantification over semantic functions w that generalize the weakest precondition functions wp and wlp. This quantification serves to eliminate the need for a separate logic or derivation system, as in [Hoa71]. The new setting of Hoare's rule is used in section 8 to present a new rule, (27), for proving that the weakest precondition of a procedure implies a given predicate. This necessity rule can serve in formal proofs of nontermination and incorrectness. In section 9, we give an example where rule (27) is used to prove the correctness of a program transformation. The three rules (19), (25) and (27) suggest the existence of a fourth rule. Indeed, in section 10, we present necessity rule (31) for partial correctness, which uses a variant function in a way similar to the rule for total correctness. We use Hoare triples to denote total correctness, so that (0) {P} S {Q} =- [P ~ wp.S.Q] for predicates P and Q and a command S. The truth of either side of (0) expresses that P is a precondition that guarantees termination of command S in any state in which Q holds. All results in this paper are sound under (unbounded) nondeterminacy. 3. Declaration and Invocation We begin with the declaration format, which is inspired by [Gri81] and [Mar83] but is more closely tuned to Pascal, in that we consider only global variables, value parameters and reference parameters. For simplicity, we assume that the procedure has one value parameter x and one reference parameter y and that it refers to two global variables u and v. The types of x, y, u, v are left out because they don't concern us here. We assume that precondition P and postcondition Q use one specification value i ~ C. In this way we arrive at the specified declaration: (1) proe h(x ; var y) {glou, v!; ailicC :: preP, post Q}. The meaning of the specification is defined by Correctness Rule. An implementation body.h of procedure h is correct if it satisfies the conditions: (2) all global variables used in body.h and in predicates P and Q are listed after the key word glo; global variables that are threatened to be modified (in the sense of [JEW85]) are indicated by "!", (3) the value parameters (x) do not occur in postcondition Q, (4) for all specification values i ~ C, we have {P} body.h {Q}. Of course, variable modalities that do not occur can be omitted from the specification. The list after key word glo is needed to exclude aliasing upon invocation of h. Condition (3) may seem unnecessarily restrictive. There are three reasons for imposing it. Firstly, it encourages specifications with simple Proof Rules for Recursive Procedures 557 postconditions (J.E.
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