Open Access

Dominik Stoffel

• This thesis has explored how structural techniques can be applied to the problem of formal verification for sequential circuits. Algorithms for formal verification which operate on non-canonical gate netlist representations of digital circuits have certain advantages over the traditional techniques based on canonical representations as BDDs. They allow to exploit problem-specific knowledge because they can take into account structural properties of the designs being analyzed. This allows us to break the problem down into sub-problems which are (hopefully) easier to be solved. However, in the past, the main application of such structural techniques was in the field of combinational equivalence checking. One reason for this is that the behaviour of a sequential system does not only depend on its inputs but also on its internal states, and no concepts had been developed to-date allowing structural methods to deal with large sets of states. An important goal of this research was therefore to develop structural, non-canonical forms of representing the reachable states of a finite state machine and to develop methods for reachability analysis based on such representations. In order to reach this goal, two steps were taken. Firstly, a framework for manipulating Boolean functions represented as gate netlists has been established. Secondly, using this framework, a structural method for FSM traversal was developed serving as the basis for an equivalence checking algorithm for sequential circuits. The framework for manipulating Boolean functions represented as multi-level combinational networks is based on a new concept of an implicant in a multi-level network and on an AND/ORtype enumeration technique which allows us to derive such implicants. This concept extends the classical notion of an implicant in two-level circuits to the multi-level case. Using this notion, arbitrary transformations in multi-level combinational networks can be performed. The multi-level network implicants can be determined from AND/OR reasoning graphs, which are associated with an AND/OR reasoning technique operating directly on the gate netlist description of a multi-level circuit. This reasoning technique has the important property that it is complete, i.e. the associated AND/OR trees contain all prime implicants of a Boolean function at an arbitrary node in a combinational circuit. In other words, AND/OR graphs constructed for a network function serve as a representation of this function. A great advantage over BDDs is that AND/OR graphs, besides representing the logic function, also represent some structural properties of the analyzed circuitry. This permits to develop heuristics that are specially tailored for certain applications such as logic optimization or verification. Another advantage which is especially useful for logic optimization is the fact that the proposed AND/OR enumeration scheme is not restricted to the use of a specific logic alphabet such as B3 = {0, 1, X}. By using Roth’s D-calculus based on B5 = {0, 1, D, D-Komplement} permissible implicants can be determined. Transformations based on permissible implicants exploit observability don’t-care conditions in logic synthesis by creating permissible functions at internal network nodes. In order to evaluate the new structural framework for manipulating Boolean functions represented as gate netlists, several experiments with implicant-based optimization of multi-level circuits were performed. The results show that implicant-based circuit transformations lead to significantly better optimization results than traditional synthesis techniques. Next, based on the proposed structural methods for Boolean function manipulation, techniques for representing and manipulating the set of states of a sequential circuit have been developed. The concept of a “stub circuit” was introduced which implicitly represents a set of state vectors as the range of a multi-output function given as a gate netlist. The stub circuit is the result of an existential quantification operation which is obtained by functional decomposition using implicant-based netlist transformations and a network cutting procedure. Using this existential quantification operation, a new structural FSM traversal algorithm was formulated which performs a fixed point iteration on the set of reachable states represented by the stub circuit. The proposed approach performs a reachability analysis of the states of a sequential circuit. It operates on gate netlists and naturally allows to incorporate structural properties of a design under consideration into the reasoning. Therefore, structural FSM traversal is an interesting alternative to traditional symbolic FSM traversal, especially in those applications of formal verification, where structural properties can be exploited. Structural FSM traversal was applied to the problem of sequential equivalence checking. Here, structural similarities between the designs to be compared can effectively reduce the complexity of the verification task. The FSM to be traversed is a special product machine called sequential miter. The special structural properties of this product machine have made it possible to formulate an approximate algorithm for structural FSM traversal, called record and play(). This algorithm uses an approximation on the reachable state set represented by the stub circuit which is very beneficial for performance. Instead of calculating the stub circuit using the exact algorithm, implicant-based transformations directly using structural design similarities are performed. These transformations, together with existential quantification implemented by the cutting procedure, lead to an over-approximation of the reachable state set. By this overapproximation, only such unreachable product states are added to the set of states represented by the stub circuit which are unreachable at the current point in time but which are nevertheless equivalent. Therefore, more product states are added to the set of reachable states sometimes leading to drastic acceleration of the traversal, i.e. the fixed point is reached in much fewer steps. The algorithm record and play() was applied to the problem of checking the equivalence of a circuit with its optimized and retimed version. Retiming is a form of sequential circuit optimization which can radically alter the state encoding of a circuit. Traditional FSM traversal techniques often fail because the BDDs needed to represent the reachable state set and the transition relation of the product machine become too large. Experiments were conducted to evaluate the performance of record and play() on a standard set of sequential benchmark circuits. The algorithm was capable of proving the equivalence of optimized and retimed circuits with their original versions, some of which (to our knowledge) have never before been verified using traditional techniques like symbolic FSM traversal. The experimental results are very promising. Future research will therefore explore how structural FSM traversal can be applied to model checking.