K. R. Apt. Ten years of Hoare's Logic, a Survey---Part 1. ACM TOPLAS, 3:431--483, 1981.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... 1 Introduction The treatment of higher order procedures (procedures can be passed as parameters) in Hoare like axiomatic systems has been a major research topic since the problem of sound and relatively complete calculi for simple procedures was solved in a satisfactory way, Coo78] Old81] [Apt81]. All known proof systems for higher order procedures extend the basic formalism introduced by Hoare, Hoa69] in a significant way. Olderog, Old84] and Damm and Josko, DJ83] extend the language for pre and postconditions while the form of the partial correctness assertions is maintained. In ....

K. R. Apt. Ten years of Hoare's Logic, a Survey---Part 1. ACM TOPLAS, 3:431--483, 1981.

....is expressible in Io by assignment r : r(F : r 0 :F ) to individual elds. The notation e is used in rules (record ) 12 Re use of variable identi ers is a standard feature of block structured languages, but for formal reasoning they are rather a bother due to the need for renamings [5]. With the use of modules to control scope, re declaration ceases to be much of an issue; moreover, if B p : com( is derivable without constraint on re declarations, then so is B p 0 : com( where p 0 is some renaming of p such that p 0 has no re declarations. 13 R : pr( x : T ) ....

K. R. Apt. Ten years of Hoare's logic, a survey, part I. ACM Trans. Prog. Lang. Syst., 3(4):431{ 483, 1981.

....V post 0 . Our central concern will be to nd necessary conditions for (pre 0 ; post 0 ) pre; post) The reason should become clear as we continue to review classical proof systems. 2 To deal with procedure calls, the rule of consequence is no longer adequate for a complete proof system [2]. For example, to show correctness of a recursive procedure named P with body S some form of induction is needed: calls of P in S are assumed to meet the speci cation of S. More accurately, a sound recursion rule allows as an assumption the speci cation augmented with a predicate used to ensure ....

....nor instantiation license that conclusion. The speci cations (pre0; post0) and (pre1; post1) can be satis ed by many simple programs without procedures, and of course those programs also satisfy the weaker speci cations. By completeness of Hoare logic for the simple imperative language [2], those triples are all provable. This is unsatisactory however. Say a proof system is adaptation complete [11] if the following holds: If f pre g S f post g and (pre 0 ; post 0 ) pre; post) then the proof of f pre g S f post g can be completed to a proof of f pre 0 g S f post 0 g . ....

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K. Apt. Ten years of Hoare's logic, a survey, part I. ACM Trans. Prog. Lang. Syst., 3(4):431-483, 1981.

....based on a signature that contains the basic sorts, function symbols and relation symbols with their associated type where a type is a list of sorts. The statements of the language consist of the usual while programs. The first order logic is based on the same signature. Hoare logic (Hoare 1969, Apt 1981, Cousot 1990) is obtained by mixing programs and first order formulae. The first order language is intended to be used for describing the properties of a set of variables (the store ) The intuitive reading of the Hoare triple fOEgSf g is: if the store initially has property OE, then the store ....

Apt, K.R. (1981), Ten years of Hoare's logic. ACM Trans. on Programming Languages and Systems, 3:431--483.

....Adaptation Rules David A. Naumann Stevens Institute of Technology, Hoboken, NJ 07030 USA 0 Introduction For reasoning about total correctness of while programs, the rules proposed by Hoare [10] have stood the test of time. But for procedure calls, a number of di erent rules have appeared (e. g, [11,9,2,1,5,12]) There appears to be no consensus on the right rule, and some proposals even turn out to be unsound. The results reported in this note were found in an attempt to derive an adaptation rule rather than pulling it from a magician s hat using tools from re nement calculus. This sheds new light ....

....Hoare proposed one rule for each program construct, along with a rule of consequence that lets us deduce f pre 0 g S f post 0 g from f preg S f post g provided the implications pre 0 ) pre and post ) post 0 are valid. These rules are relatively complete for programs without procedures [2]. For a procedure, modular reasoning requires veri cation with respect to a single speci cation that can be adapted to various calling contexts. Rules for such adaptation are subject to a number of complications to do with parameter passing and aliasing; for recursive procedures some form of ....

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K. Apt. Ten years of Hoare's logic, a survey, part I. ACM Trans. Prog. Lang. Syst., 3(4):431-483, 1981.

....F; s j= OE 9 = F j= s OE: x Gamma Gamma Gamma Gamma s 0 : 7 Floyd Hoare Logic over Frames A partial correctness assertion has syntax fOEgP f g where OE; are assertions, and P is a (conditional) process term. A general overview of correctness assertions and their logic is given in Apt 1981. See Ponse 1991, and Groote and Ponse 1994 for a process algebraic approach. The interpretation of correctness assertions is defined as usual: F j= fOEgP f g if 8s 2 jF j; F; s j= fOEgP f g F; s j= fOEgP f g if 8s 0 2 jF j F; s j= OE; and F j= s P Gamma s 0 9 = F; s 0 j= ....

Apt, K.R. 1981. Ten Years of Hoare's Logic, a Survey, Part I. ACM Transactions on Programming Languages and Systems 3(4):431--483.

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