Arithmetic circuits are the fundamental building blocks of circuitry with applications including digital signal processing, cryptography processors, and multimedia. Integer multiplier circuits with high bit width of the operands dominate the extensive circuitry area of new-generation technologies. Traditionally, various multiplication algorithms are available to generate multiplier circuits considering area, delay, and power. Custom optimization is performed to reduce the circuit size, which increases the probability of logical bugs in the design. In the past over thirty years, prominent formal verification techniques such as Satisfiability (SAT) checking, Binary Decision Diagram (BDD), and Symbolic Computer Algebra (SCA) made massive progress in analyzing the correctness of the circuits. In this paper, we study the best state-of-the-art techniques from each method available in the academic domain and perform a comparative analysis to verify integer multiplier circuits with different architectures after logic optimization. Although the complexity of BDDs is constantly exponential with the input size of the circuit, and BDDs can be constructed only up to 18 bits, the method is robust to verify a variety of multiplier structures. Algebraic backward rewriting based on Symbolic Computer Algebra (SCA) facilitates the formal verification of high-bit-width multiplier circuits. Conventional approaches that leverage hierarchical structural information are constrained to algebraic-friendly multipliers, wherein adder sub-circuits are preserved in their canonical form, an assumption often invalidated post logic synthesis and optimization. In contrast, advanced algebraic techniques that operate directly on flattened net-lists demonstrate scalability and robustness in verifying large multiplier designs. Formal analysis with straightforward SAT techniques does not work well for comparing two structural non-similar circuits, which is often the case after applying logic optimization. If the degree of similarity is not excessively low, SAT-Sweeping can effectively reduce structural non-similarity, and SAT techniques can verify multipliers up to 512 bits. However, the verification of complex circuits, characterized by their non-algebraic-friendly nature, near-zero similarity to reference circuits, and larger input sizes, remains an open challenge.
扫码关注我们
求助内容:
应助结果提醒方式:
