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Symbolic quick error detection (SQED) is a formal pre-silicon verification technique targeted at processor designs. It leverages bounded model checking (BMC) to check a design for counterexamples to a self-consistency property: given the instruction set architecture (ISA) of the design, executing an instruction sequence twice on the same inputs must always produce the same outputs. Self-consistency is a universal, implementation-independent property. Consequently, in contrast to traditional verification approaches that use implementation-specific assertions (often generated manually), SQED does not require a full formal design specification or manually-written properties. Case studies have shown that SQED is effective for commercial designs and that SQED substantially improves design productivity. However, until now there has been no formal characterization of its bug-finding capabilities. We aim to close this gap by laying a formal foundation for SQED. We use a transition-system processor model and define the notion of a bug using an abstract specification relation. We prove the soundness of SQED, i.e., that any bug reported by SQED is in fact a real bug in the processor. Importantly, this result holds regardless of what the actual specification relation is. We next describe conditions under which SQED is complete, that is, what kinds of bugs it is guaranteed to find. We show that for a large class of bugs, SQED can always find a trace exhibiting the bug. Ultimately, we prove full completeness of a variant of SQED that uses specialized state reset instructions. Our results enable a rigorous understanding of SQED and its bug-finding capabilities and give insights on how to optimize implementations of SQED in practice.