Automatic Formal Verification of Liveness for Pipelined Processors with Multicycle Functional Units

Presented is a highly automatic approach for proving bounded liveness of pipelined processors with multicycle functional units, without the need for the user to set up an inductive argument. Multicycle functional units are abstracted with a placeholder that is suitable for proving both safety and liveness. Abstracting the branch targets and directions with arbitrary terms and formulas, respectively, that are associated with each instruction, made the branch targets and directions independent of the data operands. The observation that the term variables abstracting branch targets of newly fetched instructions can be considered to be in the same equivalence class, allowed the use of a dedicated fresh term variable for all such branch targets and the abstraction of the Instruction Memory with a generator of arbitrary values. To further improve the scaling, the multicycle ALU was abstracted with a placeholder without feedback loops. Also, the equality comparison between the terms written to the PC and the dedicated fresh term variable for branch targets of new instructions was implemented as part of the circuit, thus avoiding the need to apply the abstraction function along the specification side of the commutative diagram for liveness. This approach resulted in 4 orders of magnitude speedup for a 5-stage pipelined DLX processor with a 32-cycle ALU, compared to a previous method for indirect proof of bounded liveness, and scaled for a 5-stage pipelined DLX with a 2048-cycle ALU.