Abstraction discovery and refinement for model checking by symbolic trajectory evaluation
Keywords: model checking
symbolic trajectory evaluation
abstraction
formal verification
hardware verification

This dissertation documents two contributions to automating the formal verification of hardware – particularly memory-intensive circuits – by Symbolic Trajectory Evaluation (STE), a model checking technique based on symbolic simulation over abstract sets of states. The contributions focus on improvements to the use of BDD-based STE, which uses binary decision diagrams internally.
We introduce a solution to one of the major hurdles in using STE: finding suitable abstractions. Our work has produced the first known algorithm that addresses this problem by automatically discovering good, non-trivial abstractions. These abstractions are computed from the specification, and essentially encode partial input combinations sufficient for determining the specification’s output value. They can then be used to verify whether the hardware model meets its specification using a technique based on and significantly extending previous work by Melham and Jones [2]. Moreover, we prove that our algorithm delivers correct results by construction. We demonstrate that the abstractions received by our algorithm can greatly reduce verification costs with three example hardware designs, typical of the kind of problems faced by the semiconductor design industry.
We further propose a refinement method for abstraction schemes when over- abstraction occurs, i.e., when the abstraction hides too much information of the original design to determine whether it meets its specification. The refinement algorithm we present is based on previous work by Chockler et al. [3], which selects refinement candidates by approximating which abstracted input is likely the biggest cause of the abstraction being unsuitable. We extend this work substantially, concentrating on three aspects. First, we suggest how the approach can also work for much more general abstraction schemes. This enables refining any abstraction allowed in STE, rather than just a subset. Second, Chockler et al. describe how to refine an abstraction once a refinement candidate has been identified. We present three additional variants of refining the abstraction. Third, the refinement at its core depends on evaluating circuit logic gates. The previous work offered solutions for NOT- and AND-gates. We propose a general approach to evaluating arbitrary logic gates, which improves the selection process of refinement candidates. We show the effectiveness of our work by automatically refining an abstraction for a content-addressable memory that exhibits over-abstraction, and by evaluating some common logic gates.
These two contributions can be used independently to help automate the hard- ware verification by STE, but they also complement each other. To show this, we combine both algorithms to create a fully automatic abstraction discovery and refinement loop. The only inputs required are the hardware design and the specification, which the design should meet. While only small circuits could be verified completely automatically, it clearly shows that our two contributions allow the construction of a verification framework that does not require any user interaction.