By William D. Stansfield
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In Proc. of PLDI ’01, 2001. Also in SIGPLAN Notices 36(5) (May 2001). Verifying Programs with Dynamic 1-Selector-Linked Structures 29 16. S. W. Reps, and R. Wilhelm. Parametric Shape Analysis via 3-Valued Logic. TOPLAS, 24(3), 2002. 17. T. Touili. Widening Techniques for Regular Model Checking. ENTCS, 50, 2001. 18. A. Venet. Automatic Analysis of Pointer Aliasing for Untyped Programs. Science of Computer Programming, 35(2), 1999. 19. P. Wolper and B. Boigelot. Verifying Systems with Infinite but Regular State Spaces.
When checking just its basic consistency property, we have completely abstracted away the data values stored in the list and made all the conditional jumps fully nondeterministic. To check that the procedure really sorts, we used a technique inspired by [15]. We considered the values of the list elements to be abstracted to being either greater or less than or equal than their successors. The abstracted data values were represented by two special letters (gt and lte) associated with every list item.
Lemma 3. Let R be a serialisable extrapolation system and let Ri1 Ri2 . . Rin be a total ordering of the rules of R which is compatible with ≺. Then, R∗ = R∗in ◦ R∗in−1 · · · ◦ R∗i1 . Proof. Follows from Lemma 2: closing by some Ri j never creates new rewriting contexts for any of the Ri with < j. From the two lemmas 1 and 3 we deduce the following fact: Theorem 1. For every serialisable extrapolation system R and for every regular language L, the set R∗ (L) is regular and effectively constructible.