Institute of Informatics
Acta Cybernetica
Past Issues
Volume 15, Number 3, 2002
An arithmetic theory of consistency enforcement
# An arithmetic theory of consistency enforcement

**Sebastian Link and Klaus-Dieter Schewe**

### Abstract (in LaTeX format)

Consistency enforcement starts from a given program spe\-ci\-fication $S$ and a static invariant $\mathcal{I}$ and aims to replace $S$ by a slightly modified program specification $S_{\mathcal{I}}$ that is provably consistent with respect to $\mathcal{I}$. One formalization which suggests itself is to define $S_{\mathcal{I}}$ as the greatest consistent specialization of $S$ with respect to $\mathcal{I}$, where specialization is a partial order on semantic equivalence classes of program specifications.
In this paper we present such a theory on the basis of arithmetic logic. We show that with mild technical restrictions and mild restrictions concerning recursive program specifications it is possible to obtain the greatest consistent specialization gradually and independently from the order of given invariants as well as by replacing basic commands by their respective greatest consistent specialization. Furthermore, this approach allows to discuss computability and decidability aspects for the first time.

### Full text

Available electronic editions: PDF.

### DOI

DOI is not available for this article.

### BibTeX entry
`
@article{Link:2002:ActaCybernetica,`

author = {Sebastian Link and Klaus-Dieter Schewe},

title = {An arithmetic theory of consistency enforcement},

journal = {Acta Cybernetica},

year = {2002},

volume = {15},

pages = {379--416},

number = {3},

abstract = {Consistency enforcement starts from a given program spe\-ci\-fication $S$ and a static invariant $\mathcal{I}$ and aims to replace $S$ by a slightly modified program specification $S_{\mathcal{I}}$ that is provably consistent with respect to $\mathcal{I}$. One formalization which suggests itself is to define $S_{\mathcal{I}}$ as the greatest consistent specialization of $S$ with respect to $\mathcal{I}$, where specialization is a partial order on semantic equivalence classes of program specifications.

In this paper we present such a theory on the basis of arithmetic logic. We show that with mild technical restrictions and mild restrictions concerning recursive program specifications it is possible to obtain the greatest consistent specialization gradually and independently from the order of given invariants as well as by replacing basic commands by their respective greatest consistent specialization. Furthermore, this approach allows to discuss computability and decidability aspects for the first time.}

}