Automatic abstraction for congruences

  • Authors:

  • Andy King;Harald Søndergaard

  • Affiliations:

  • Portcullis Computer Security Limited, Pinner, UK;The University of Melbourne, Victoria, Australia

  • Venue:
  • VMCAI'10 Proceedings of the 11th international conference on Verification, Model Checking, and Abstract Interpretation
  • Year:
  • 2010

Quantified Score

Hi-index 0.00

Visualization

Abstract

One approach to verifying bit-twiddling algorithms is to derive invariants between the bits that constitute the variables of a program. Such invariants can often be described with systems of congruences where in each equation ${c} \cdot {x} = d \mod m$, m is a power of two, c is a vector of integer coefficients, and x is a vector of propositional variables (bits). Because of the low-level nature of these invariants and the large number of bits that are involved, it is important that the transfer functions can be derived automatically. We address this problem, showing how an analysis for bit-level congruence relationships can be decoupled into two parts: (1) a SAT-based abstraction (compilation) step which can be automated, and (2) an interpretation step that requires no SAT-solving. We exploit triangular matrix forms to derive transfer functions efficiently, even in the presence of large numbers of bits. Finally we propose program transformations that improve the analysis results.