Complexity verification using guided theorem enumeration

Abstract

Determining if a given program satisfies a given bound on the amount of resources that it may use is a fundamental problem with critical practical applications. Conventional automatic verifiers for safety properties cannot be applied to address this problem directly because such verifiers target properties expressed in decidable theories; however, many practical bounds are expressed in nonlinear theories, which are undecidable. In this work, we introduce an automatic verification algorithm, CAMPY, that determines if a given program P satisfies a given resource bound B, which may be expressed using polynomial, exponential, and logarithmic terms. The key technical contribution behind our verifier is an interpolating theorem prover for non-linear theories that lazily learns a sufficiently accurate approximation of non-linear theories by selectively grounding theorems of the nonlinear theory that are relevant to proving that P satisfies B. To evaluate CAMPY, we implemented it to target Java Virtual Machine bytecode. We applied CAMPY to verify that over 20 solutions submitted for programming problems hosted on popular online coding platforms satisfy or do not satisfy expected complexity bounds.