On procedures as open subroutines. I

Langmaack, Hans

doi:10.1007/bf00289503

Cited by 31 publications

(3 citation statements)

References 4 publications

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“…The origin of Apt's obserbjation is the argument of Theorem 2 in [lo] which shows that HLo(A) is incompiete in case A is Presburger Arithmetic; and we again find it in use in [9], there applied to finite structures A which interpret much richer programming languages than WV with undecidable halting problems for IAl 2 2. Actually, the termination of programs over finite interpretations was considered, independently of Hoare's logic, in Langmaack [14,15]; for example the observation applies there to sharpen one of Clarke's incompleteness results to include structures A with IA I= 1, and it a;speai ;. often in Langmaack and Olderog's extensive studies of Hoare-like logics for the higher progralm languages, see [16f and the references there cited.…”

Section: If the Ha!kyg Problem For Programs In Wp Is Undecidable On Amentioning

confidence: 99%

Some natural structures which fail to possess a sound and decidable hoare-like logic for their while-programs

Bergstra¹,

Tucker²

1982

Theoretical Computer Science

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&A b(p)s(q}} wherein A t= {p)S{q} means henever p is true of ad initid stat4 for S then either S terminates in a state for whi th q is true or S diverges. With the same level of gererality, one can define tile standard Hoare logic HLo(A) for-pa/"g, on A as the set of all triples (p)S{q} generated by Hoare's proof rules for-?V'9' and including the usually undecidable first-order thctory Th(A) of A as axioms. For any set&Se program semantics 9, I-&,(A) is sound in the sense that HLo(A)c PC(A). In [IO], Cook showed that if I, is expressive for-Wg over A then HI&A) is compMe in the sense that HLO(A) = PC(A), Of more interest to us, on this occasion, is another theorem of [IO]: if A is Presburger arithmetic then HLo(A) is not complete; and also Wand's more detailed analysis of incompleteness [25] wherein he gave a simple, although artificial, structure A and al.1 asserted program {p}S{q} such that { p)S{q) E PC(A) butr { p}S{q} & HLo(A). After settling on a weak criterion for a set of asserted programs to qualify as a Hoare logil.:, we will build up some general theory from which one can read off more extreme examples of incompleteness: Theorem, Let A be Presburger arithmetic, the field of real algebraic numbers, or the field of algebraic numbers. Then A is a computable @ebraic structure with decidable @rsf-order theory Th(A) such that (I) each sound Howe Iogic HL(A) 3 HLo(A) is r.&. but not recursive; (2) PC(A) is co-r.e. but not recursive; in fact, PCI'A? is a complete co-r.e. set. Therefure, A has no sound and complete Hoare logk for its while-programs ; and, in particular, A fails to possess even a sound, if incomplete, Hoare logic which is recursive.

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Section: If the Ha!kyg Problem For Programs In Wp Is Undecidable On Amentioning

confidence: 99%

Some natural structures which fail to possess a sound and decidable hoare-like logic for their while-programs

Bergstra¹,

Tucker²

1982

Theoretical Computer Science

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“…Es entsteht ein formal ~iquivalentes Programm [2]. Eine Prozedur mit globalen Parametern wird gemiil3 Lemma 4.1 auf Schachtelungstiefe 1 gebracht; es bleibt eine ,,unwesentliche" Prozedur auf Schachtelungstiefe n stehen.…”

Section: Beweisunclassified

Zur Elimination von Prozedurschachtelungen

Lippe

Simon

1979

Computing

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On the Elimination of Procedure Nestings. In ALGOL-like programming languages with procedures there are several properties of programs concerning procedures that may be used to improve resp. optimise the performance of the runtime system. In these considerations problems very soon arise, because in general one has to discuss programs with an arbitrary nesting of procedure declarations. This paper shows that an arbitrary ALGOL 60-program can be transformed into an ALGOL 60-program which has a maximal procedure nesting 2, and both programs define the same input-output function. In order to prove this property a new notion for equivalence of ALGOL-like programs with procedures is introduced.Zur Elimination von Prozedurschachtelungen. In ALGOL-~ihnlichen Programmiersprachen mit Prozeduren gibt es im Zusammenhang mit diesen Prozeduren eiue Reihe yon Programmeigenschaften, die zu einer Verbesserung des Laufzeitsystems benutzt werden k6nnen. Bei solchen 13berlegungen treten jedoch h~iufig Probleme auf, da man Programme mit beliebigen Prozedurschachtelungen betrachten muB. Es wird nun gezeigt, dab sich beliebige ALGOL-60-Programme in funktional aquivalente Programme mit einer maximalen Prozedurschachtelungstiefe 2 transforrnieren lassen. Als HilfsmitteI wird hierzu ein neuer Aquivalenzbegriff fiir ALGOL-~ihnliche Programme mit Prozeduren eingefiihrt.

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“…Earlier, Kruseman Aretz [5] has used Algol 60's name parameter mechanism instead of formal procedures. Actually, arbitrary Turing machines and similar devices can be simulated by procedures and recursion only (and without other data types like integer and boolean), as shown by Langmaack [6,7,8] and Fokkinga [3]. (Of course, for anyone familiar with the Lambda calculus this comes as no surprise, but what is important here is that procedures as result are not needed in these programs.…”

Section: Introductionmentioning

confidence: 99%