Abstract
We present here an application of abstract single threaded objects (abstract stobjs) in the ACL2 theorem prover, to define a formally verified algorithm that given a matrix with elements in the ring of integers, computes an equivalent matrix in column echelon form. Abstract stobjs allow us to define a sound logical interface between matrices defined as lists of lists, convenient for reasoning but inefficient, and matrices represented as unidimensional stobjs arrays, which implement accesses and (destructive) updates in constant time. Also, by means of the abstract stobjs mechanism, we use a more convenient logical representation of the transformation matrix, as a sequence of elemental transformations. Although we describe here a particular normalization algorithm, we think this approach could be useful to obtain formally verified and efficient executable implementations of a number of matrix normal form algorithms.
Supported by Ministerio de Ciencia e Innovación, projects TIN2013-41086-P and MTM2014-54151-P.
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Notes
- 1.
Some algorithms for computing matrix normal forms, like the Smith normal form, need to compute two transformation matrices, but similar ideas would apply in that case.
- 2.
Of course, other normal forms algorithms needs different elementary transformations, and possible more than one. But again, the same ideas described here could be applied in such cases.
- 3.
We used ACL2 Version 7.2 compiled with SBCL 1.2.16.
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Lambán, L., Martín-Mateos, F.J., Rubio, J., Ruiz-Reina, JL. (2017). Using Abstract Stobjs in ACL2 to Compute Matrix Normal Forms. In: Ayala-Rincón, M., Muñoz, C.A. (eds) Interactive Theorem Proving. ITP 2017. Lecture Notes in Computer Science(), vol 10499. Springer, Cham. https://doi.org/10.1007/978-3-319-66107-0_23
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