Specified blocks

Eric C. R. Hehner
[doi] [ISBN] [Google Scholar] [DBLP] [Citeseer]
Read: 21 February 2020

Verified Software: Theories, Tools, Experiments: First IFIP TC 2/WG 2.3 Conference, VSTTE 2005, Zurich, Switzerland, October 10-13, 2005, Revised Selected Papers and Discussions
Springer Berlin Heidelberg
Berlin, Heidelberg
Pages 384-391
Topic(s): tools verification
Note(s): loop invariant
Papers: tuerk:vstte:2010, schwerhoff:ecoop:2015

Program verification based on assertions is fundamentally wrong. Like Ptolemy’s planetary theory of cycles within cycles, they are overly complex and we need something as simple as Galileo’s theory of elliptical orbits. This paper forcefully argues that they should be replaced with “specified blocks” that form a specification of an entire block of code. That is, we should describe what a block of code is intended to do instead of what is true at strategic points in the code.

A specified block consists of a block of code plus a predicate that relates the new and old values of any variables. For example, a block that increments every element of an array might have the specification

(forall j: 0..n . L'[j] = L[j]+1)

while the obvious loop that implements this specification would have the specification

(forall j: 0,..i . L'[j] = L[j]) &&
(forall j: i,..n . L'[j] = L[j]+1)

Compare this with the loop invariant that looks like this

(forall j: 0,..i . L'[j] = L[j]+1) &&
(forall j: i,..n . L'[j] = L[j]) &&
0 ≤ i ≤ n

The loop invariant is different from the loop specification in three ways:

  • The loop invariant describes what has been done so far whereas the loop specification describes what is still to be done. You can see this in the first line where the loop specification says that the loop does not change some initial segment of the loop while the loop invariant says that, so far, the initial segment has been incremented.

  • Likewise, the second line of the loop specification says that the loop is going to increment the tail of the array while the loop invariant says that the loop has not yet been changed.

  • Finally, the loop invariant describes a single iteration of the loop and so it has to say something about the range of i while the loop specification describes all remaining iterations of the loop and does not need to constrain i.

For loops, the difference of describing what is still to be done enables local reasoning about loops because the focus is on the action of the loop, not the code that comes before it. This was independently rediscovered by Tuerk two years later in the context of separation logic where it eliminates the need to invent predicates for describing partial data structures. This was later used as the motivation for supporting lightweight magic wands.

The paper contains four examples to illustrate how specified blocks are easier to work with: incrementing arrays, binary search, exponentiation and product of power series. The exponentiation example involves gotos: contradicting Dijkstra by saying “Apparently, unstructured goto’s pose no more verification problem than structured loops.”

This approach to specification has been implemented in HOL.

Loop invariant