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Automatic Generation of Polynomial Loop Invariants: Algebraic Foundations (2004)
| Venue: | In International Symposium on Symbolic and Algebraic Computation 2004 (ISSAC04 |
| Citations: | 34 - 6 self |
BibTeX
@INPROCEEDINGS{Rodríguez-carbonell04automaticgeneration,
author = {Enric Rodríguez-carbonell and Deepak Kapur},
title = {Automatic Generation of Polynomial Loop Invariants: Algebraic Foundations},
booktitle = {In International Symposium on Symbolic and Algebraic Computation 2004 (ISSAC04},
year = {2004},
pages = {266--273},
publisher = {ACM Press}
}
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Abstract
This paper presents the algebraic foundation for an approach for generating polynomial loop invariants in imperative programs. It is first shown that the set of polynomials serving as loop invariants has the algebraic structure of an ideal. Using this connection, a procedure for finding loop invariants is given in terms of operations on ideals, for which Gröbner basis constructions can be employed. Most importantly, it is proved that if the assignment statements in a loop are solvable (in particular, affine) mappings with positive eigenvalues, then the procedure terminates in at most 2m + 1 iterations, where m is the number of variables in the loop. The proof is done by showing that the irreducible subvarieties of the variety associated with the polynomial ideal approximating the invariant polynomial ideal of the loop either stay the same or increase their dimension in every iteration. This yields a correct and complete algorithm for inferring conjunctions of polynomial equations as invariants. The method has been implemented in Maple using the Groebner package. The implementation has been used to automatically discover nontrivial invariants for several examples to illustrate the power of the techniques.
Keyphrases
polynomial loop invariant algebraic foundation automatic generation loop invariant polynomial ideal polynomial equation assignment statement gr bner basis construction imperative program algebraic structure complete algorithm groebner package positive eigenvalue irreducible subvarieties several example invariant polynomial ideal nontrivial invariant
