Automated Test Generation for Debugging Multiple Bugs in Arithmetic Circuits

Optimized and custom arithmetic circuits are widely used in embedded systems such as multimedia applications, cryptography systems, signal processing and console games. Debugging of arithmetic circuits is a challenge due to increasing complexity coupled with non-standard implementations. Existing algebraic rewriting techniques produce a remainder to indicate the presence of a potential bug. However, bug localization remains a major bottleneck. Simulation-based validation using random or constrained-random tests are not effective for complex arithmetic circuits due to bit-blasting. In this paper, we present an automated test generation and bug localization technique for debugging arithmetic circuits. This paper makes four important contributions. We propose an automated approach for generating directed tests by suitable assignments of input variables to make the remainder non-zero. The generated tests are guaranteed to activate bugs. We also propose an automatic bug fixing technique by utilizing the patterns of the remainder terms as well as by analyzing the regions activated by the generated tests to detect and correct the error(s). We also propose an efficient debugging algorithm that can handle multiple dependent as well as independent bugs. Finally, our proposed framework, consisting of directed test generation, bug localization and bug correction, is fully automated. In other words, our framework is capable of producing a corrected implementation of arithmetic circuits without any manual intervention. Our experimental results demonstrate that the proposed approach can be used for automated debugging of large and complex arithmetic circuits.