Runtime Complexity Bounds Using Squeezers

Abstract

Determining upper bounds on the time complexity of a program is a fundamental problem with a variety of applications, such as performance debugging, resource certification, and compile-time optimizations. Automated techniques for cost analysis excel at bounding the resource complexity of programs that use integer values and linear arithmetic. Unfortunately, they fall short when the complexity depends more intricately on the evolution of data during execution. In such cases, state-of-the-art analyzers have shown to produce loose bounds, or even no bound at all. We propose a novel technique that generalizes the common notion of recurrence relations based on ranking functions. Existing methods usually unfold one loop iteration and examine the resulting arithmetic relations between variables. These relations assist in establishing a recurrence that bounds the number of loop iterations. We propose a different approach, where we derive recurrences by comparing whole traces with whole traces of a lower rank, avoiding the need to analyze the complexity of intermediate states. We offer a set of global properties, defined with respect to whole traces, that facilitate such a comparison and show that these properties can be checked efficiently using a handful of local conditions. To this end, we adapt state squeezers , an induction mechanism previously used for verifying safety properties. We demonstrate that this technique encompasses the reasoning power of bounded unfolding, and more. We present some seemingly innocuous, yet intricate, examples that previous tools based on cost relations and control flow analysis fail to solve, and that our squeezer-powered approach succeeds.