Types with potential: polynomial resource bounds via automatic amortized analysis

A primary feature of a computer program is its quantitative performance characteristics: the amount of resources such as time, memory, and power the program needs to perform its task. Concrete resource bounds for specific hardware have many important applications in software development but their manual determination is tedious and error-prone. This dissertation studies the problem of automatically determining concrete worst-case bounds on the quantitative resource consumption of functional programs. Traditionally, automatic resource analyses are based on recurrence relations. The difficulty of both extracting and solving recurrence relations has led to the development of type-based resource analyses that are compositional, modular, and formally verifiable. However, existing automatic analyses based on amortization or sized types can only compute bounds that are linear in the sizes of the arguments of a function. This work presents a novel type system that derives polynomial bounds from first-order functional programs. As pioneered by Hofmann and Jost for linear bounds, it relies on the potential method of amortized analysis. Types are annotated with multivariate resource polynomials, a rich class of functions that generalize non-negative linear combinations of binomial coefficients. The main theorem states that type derivations establish resource bounds that are sound with respect to the resource-consumption of programs which is formalized by a big-step operational semantics. Simple local type rules allow for an efficient inference algorithm for the type annotations which relies on linear constraint solving only. This gives rise to an analysis system that is fully automatic if a maximal degree of the bounding polynomials is given. The analysis is generic in the resource of interest and can derive bounds on time and space usage. The bounds are naturally closed under composition and eventually summarized in closed, easily understood formulas. The practicability of this automatic amortized analysis is verified with a publicly available implementation and a reproducible experimental evaluation. The experiments with a wide range of examples from functional programming show that the inference of the bounds only takes a couple of seconds in most cases. The derived heap-space and evaluation-step bounds are compared with the measured worst-case behavior of the programs. Most bounds are asymptotically tight, and the constant factors are close or even identical to the optimal ones. For the first time we are able to automatically and precisely analyze the resource consumption of involved programs such as quick sort for lists of lists, longest common subsequence via dynamic programming, and multiplication of a list of matrices with different, fitting dimensions.