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Efficient Craig interpolation for linear Diophantine (dis)equations and linear modular equations

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Abstract

The use of Craig interpolants has enabled the development of powerful hardware and software model checking techniques. Efficient algorithms are known for computing interpolants in rational and real linear arithmetic. We focus on subsets of integer linear arithmetic. Our main results are polynomial time algorithms for obtaining interpolants for conjunctions of linear Diophantine equations, linear modular equations (linear congruences), and linear Diophantine disequations. We also present an interpolation result for conjunctions of mixed integer linear equations. We show the utility of the proposed interpolation algorithms for discovering modular/divisibility predicates in a counterexample guided abstraction refinement (CEGAR) framework. This has enabled verification of simple programs that cannot be checked using existing CEGAR based model checkers.

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Authors and Affiliations

  1. Synopsys, 2025 NW Cornelius Pass Rd, Hillsboro, OR, 97124, USA

    Himanshu Jain

  2. Computer Science Department, Carnegie Mellon University, 5000 Forbes Avenue, Pittsburgh, PA, 15213, USA

    Edmund M. Clarke

  3. Department of Computer Science, Technion—Israel Institute of Technology, Haifa, 32000, Israel

    Orna Grumberg

Authors
  1. Himanshu Jain
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  2. Edmund M. Clarke
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  3. Orna Grumberg
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Corresponding author

Correspondence to Himanshu Jain.

Additional information

This paper is an extended version of [14]. This research was sponsored by the Gigascale Systems Research Center (GSRC), Semiconductor Research Corporation (SRC), the National Science Foundation (NSF), the Office of Naval Research (ONR), the Naval Research Laboratory (NRL), the Defense Advanced Research Projects Agency (DARPA), the Army Research Office (ARO), and the General Motors Collaborative Research Lab at CMU. The views and conclusions contained in this document are those of the author and should not be interpreted as representing the official policies, either expressed or implied, of GSRC, SRC, NSF, ONR, NRL, DARPA, ARO, GM, or the U.S. government.

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Jain, H., Clarke, E.M. & Grumberg, O. Efficient Craig interpolation for linear Diophantine (dis)equations and linear modular equations. Form Methods Syst Des 35, 6–39 (2009). https://doi.org/10.1007/s10703-009-0069-x

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