# Probabilistic abstraction for model checking: an approach based on property testing

### From LRDE

- Authors
- Sophie Laplante, Richard Lassaigne, Frédéric Magniez, Sylvain Peyronnet, Michel de Rougemont
- Journal
- ACM Transactions on Computational Logic
- Type
- article
- Projects
- APMC
- Date
- 2005-11-21

## Abstract

The goal of model checking is to verify the correctness of a given program, on all its inputs. The main obstacle, in many cases, is the intractably large size of the program's transition system. Property testing is a randomized method to verify whether some fixed property holds on individual inputs, by looking at a small random part of that input. We join the strengths of both approaches by introducing a new notion of probabilistic abstraction, and by extending the framework of model checking to include the use of these abstractions. Our abstractions map transition systems associated with large graphs to small transition systems associated with small random subgraphs. This reduces the original transition system to a family of small, even constant-size, transition systems. We prove that with high probability, “sufficiently” incorrect programs will be rejected (**Failed to parse (unknown function "\eps"): {\displaystyle \eps}**
-robustness). We also prove that under a certain condition (exactness), correct programs will never be rejected (soundness). Our work applies to programs for graph properties such as bipartiteness, **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k}**
-colorabilityor any **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \exists\forall}**
first order graph properties. Our main contribution is to show how to apply the ideas of property testing to syntactic programs for such properties. We give a concrete example of an abstraction for a program for bipartiteness. Finally, we show that the relaxation of the test alone does not yield transition systems small enough to use the standard model checking method. More specifically, we prove, using methods from communication complexity, that the OBDD size remains exponential for approximate bipartiteness.

## Bibtex (lrde.bib)

@Article{ laplante.07.tocl, author = {Sophie Laplante and Richard Lassaigne and Fr\'ed\'eric Magniez and Sylvain Peyronnet and Michel de Rougemont}, title = {Probabilistic abstraction for model checking: an approach based on property testing}, journal = {ACM Transactions on Computational Logic}, year = 2007, month = aug, volume = 8, number = 4, abstract = {The goal of model checking is to verify the correctness of a given program, on all its inputs. The main obstacle, in many cases, is the intractably large size of the program's transition system. Property testing is a randomized method to verify whether some fixed property holds on individual inputs, by looking at a small random part of that input. We join the strengths of both approaches by introducing a new notion of probabilistic abstraction, and by extending the framework of model checking to include the use of these abstractions. Our abstractions map transition systems associated with large graphs to small transition systems associated with small random subgraphs. This reduces the original transition system to a family of small, even constant-size, transition systems. We prove that with high probability, ``sufficiently'' incorrect programs will be rejected ($\eps$-robustness). We also prove that under a certain condition (exactness), correct programs will never be rejected (soundness). Our work applies to programs for graph properties such as bipartiteness, $k$-colorability, or any $\exists\forall$ first order graph properties. Our main contribution is to show how to apply the ideas of property testing to syntactic programs for such properties. We give a concrete example of an abstraction for a program for bipartiteness. Finally, we show that the relaxation of the test alone does not yield transition systems small enough to use the standard model checking method. More specifically, we prove, using methods from communication complexity, that the OBDD size remains exponential for approximate bipartiteness.} }