Sufficiency-based Filtering of Invariants for Sequential Equivalence Checking

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Virginia Tech

Verification, as opposed to Testing and Post-Silicon Validation, is a critical step for Integrated Circuits (IC) Design, answering the question "Are we designing the right function?" before the chips are manufactured. One of the core areas of Verification is Equivalence Checking (EC), which is a special yet independent case of Model Checking (MC). Equivalence Checking aims to prove that two circuits, when fed with the same inputs, produce the exact same outputs. There are broadly two ways to conduct Equivalence Checking, simulation and Formal Equivalence Checking. Simulation requires one to try out different input combinations and observe if the two circuits produce the same output. Obviously, since it is not possible to enumerate all combinations of different inputs, completeness cannot be guaranteed. On the other hand, Formal Equivalence Checking can achieve 100% confidence. As the number of gates and in particular, the number of flip-flops, in circuits has grown tremendously during the recent years, the problem of Formal Equivalence Checking has become much harder â A recent evaluation of a general-case Formal Equivalence Checking engine [1] shows that about 15% of industrial designs cannot be verified after a typical sequential synthesis flow. As a result, a lot of attention on Formal Equivalence Checking has been drawn both academically and industrially.

For years Combinational Equivalence Checking(CEC) has been the pervasive framework for Formal Equivalence Checking(FEC) in the industry. However, due to the limitation of being able to verify circuits only with 1:1 flip-flop pairing, a pure CEC-based methodology requires a full regression of the verification process, meaning that performing sequential optimizations like retiming or FSM re-encoding becomes somewhat of a bottleneck in the design cycle [2]. Therefore, a more powerful framework — Sequential Equivalence Checking (SEC) — has been gradually adopted in industry.

In this thesis, we target on Sequential Equivalence Checking by finding efficient yet powerful group of relationships (invariants) among the signals of the two circuits being compared. In order to achieve a high success rate on some of the extremely hard-to-verify circuits, we are interested in both two-node and multi-node (up to 4 nodes) invariants. Also we are interested in invariants among both flip-flops and internal signals. For large circuits, there can be too many potential invariants requiring much time to prove. However, we observed that a large portion of them may not even contribute to equivalence checking. Moreover, equivalence checking can be significantly helped if there exists a method to check if a subset of potential invariants would be sufficient (e.g., whether two-nodes are enough or multi-nodes are also needed) prior to the verification step. Therefore, we propose two sufficiency-based approaches to identify useful invariants out of the initial potential invariants for SEC. Experimental results show that our approach can either demonstrate insufficiency of the invariants or select a small portion of them to successfully prove the equivalence property.

Our approaches are quite case-independent and flexible. They can be applied on circuits with different synthesis techniques and combined with other techniques.

Invariant filtering, Assume and Verify, Boolean Satisfiability(SAT), Sequential Equivalence Checking(SEC)