Abstract
Abstraction is one of the most important approaches for reducing the number of states in formal verification. An important abstraction technique is the usage of three-valued logic, extensible to bit-vectors. The best abstract bit-vector results for movement and logical operations can be computed quickly. However, for widely-used arithmetic operations, efficient algorithms for computation of the best possible output have not been known up to now.
In this paper, we present new efficient polynomial-time algorithms for abstract addition and multiplication with three-valued bit-vector inputs. These algorithms produce the best possible three-valued bit-vector output and remain fast even with 32-bit inputs.
To obtain the algorithms, we devise a novel modular extreme-finding technique via reformulation of the problem using pseudo-Boolean modular inequalities. Using the introduced technique, we construct an algorithm for abstract addition that computes its result in linear time, as well as a worst-case quadratic-time algorithm for abstract multiplication. Finally, we experimentally evaluate the performance of the algorithms, confirming their practical efficiency.
This work was supported by the Czech Technical University (CTU) grant No. SGS20/211/OHK3/3T/18 and institutional financing of the Institute of Computer Science (RVO:67985807).
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Onderka, J., Ratschan, S. (2022). Fast Three-Valued Abstract Bit-Vector Arithmetic. In: Finkbeiner, B., Wies, T. (eds) Verification, Model Checking, and Abstract Interpretation. VMCAI 2022. Lecture Notes in Computer Science(), vol 13182. Springer, Cham. https://doi.org/10.1007/978-3-030-94583-1_12
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