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

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

Automatic Static Cost Analysis for Parallel Programs

Abstract. Static analysis of the evaluation cost of programs is an extensively studied problem that has many important applications. However, most automatic methods for static cost analysis are limited to sequential evaluation while programs are increasingly evaluated on modern multicore and multiprocessor hardware. This article introduces the first automatic analysis for deriving bounds on the worst-case evaluation cost of parallel first-order functional programs. The analysis is performed by a novel type system for amortized resource analysis. The main innovation is a technique that separates the reasoning about sizes of data structures and evaluation cost within the same framework. The cost semantics of parallel programs is based on call-by-value evaluation and the standard cost measures work and depth. A soundness proof of the type system establishes the correctness of the derived cost bounds with respect to the cost semantics. The derived bounds are multivariate resource polynomials which depend on the sizes of the arguments of a function. Type inference can be reduced to linear programming and is fully automatic. A prototype implementation of the analysis system has been developed to experimentally evaluate the effectiveness of the approach. The experiments show that the analysis infers bounds for realistic example programs such as quick sort for lists of lists, matrix multiplication, and an implementation of sets with lists. The derived bounds are often asymptotically tight and the constant factors are close to the optimal ones

Automatic Static Cost Analysis for Parallel Programs

Abstract Static analysis of the evaluation cost of programs is an extensively studied problem that has many important applications. However, most automatic methods for static cost analysis are limited to sequential evaluation while programs are increasingly evaluated on modern multicore and multiprocessor hardware. This article introduces the first automatic analysis for deriving bounds on the worst-case evaluation cost of parallel first-order functional programs. The analysis is performed by a novel type system for amortized resource analysis. The main innovation is a technique that separates the reasoning about sizes of data structures and evaluation cost within the same framework. The cost semantics of parallel programs is based on call-by-value evaluation and the standard cost measures work and depth. A soundness proof of the type system establishes the correctness of the derived cost bounds with respect to the cost semantics. The derived bounds are multivariate resource polynomials which depend on the sizes of the arguments of a function. Type inference can be reduced to linear programming and is fully automatic. A prototype implementation of the analysis system has been developed to experimentally evaluate the effectiveness of the approach. The experiments show that the analysis infers bounds for realistic example programs such as quick sort for lists of lists, matrix multiplication, and an implementation of sets with lists. The derived bounds are often asymptotically tight and the constant factors are close to the optimal ones

Automatic Static Cost Analysis for Parallel Programs

Abstract Static analysis of the evaluation cost of programs is an extensively studied problem that has many important applications. However, most automatic methods for static cost analysis are limited to sequential evaluation while programs are increasingly evaluated on modern multicore and multiprocessor hardware. This article introduces the first automatic analysis for deriving bounds on the worst-case evaluation cost of parallel first-order functional programs. The analysis is performed by a novel type system for amortized resource analysis. The main innovation is a technique that separates the reasoning about sizes of data structures and evaluation cost within the same framework. The cost semantics of parallel programs is based on call-by-value evaluation and the standard cost measures work and depth. A soundness proof of the type system establishes the correctness of the derived cost bounds with respect to the cost semantics. The derived bounds are multivariate resource polynomials which depend on the sizes of the arguments of a function. Type inference can be reduced to linear programming and is fully automatic. A prototype implementation of the analysis system has been developed to experimentally evaluate the effectiveness of the approach. The experiments show that the analysis infers bounds for realistic example programs such as quick sort for lists of lists, matrix multiplication, and an implementation of sets with lists. The derived bounds are often asymptotically tight and the constant factors are close to the optimal ones

Automated Amortised Analysis

Steffen Jost researched a novel static program analysis that automatically infers formally guaranteed upper bounds on the use of compositional quantitative resources. The technique is based on the manual amortised complexity analysis. Inference is achieved through a type system annotated with linear constraints. Any solution to the collected constraints yields the coefficients of a formula, that expresses an upper bound on the resource consumption of a program through the sizes of its various inputs. The main result is the formal soundness proof of the proposed analysis for a functional language. The strictly evaluated language features higher-order types, full mutual recursion, nested data types, suspension of evaluation, and can deal with aliased data. The presentation focuses on heap space bounds. Extensions allowing the inference of bounds on stack space usage and worst-case execution time are demonstrated for several realistic program examples. These bounds were inferred by the created generic implementation of the technique. The implementation is highly efficient, and solves even large examples within seconds.Steffen Jost stellt eine neuartige statische Programmanalyse vor, welche vollautomatisch Schranken an den Verbrauch quantitativer Ressourcen berechnet. Die Grundidee basiert auf der Technik der Amortisierten Komplexitätsanalyse, deren nicht-triviale Automatisierung durch ein erweitertes Typsystem erreicht wird. Das Typsystem berechnet als Nebenprodukt ein lineares Gleichungssystem, dessen Lösungen Koeffizienten für lineare Formeln liefern. Diese Formeln stellen garantierte obere Schranken an den Speicher- oder Zeitverbrauch des analysierten Programms dar, in Abhängigkeit von den verschiedenen Eingabegrößen des Programms. Die Relevanz der einzelnen Eingabegrößen auf den Ressourcenverbrauch wird so deutlich beziffert. Die formale Korrektheit der Analyse wird für eine funktionale Programmiersprache bewiesen. Die strikte Sprache erlaubt: Typen höherer Ordnung, volle Rekursion, verschachtelte Datentypen, explizites Aufschieben der Auswertung und Aliasing. Die formale Beschreibung der Analyse befasst sich primär mit dem Verbrauch von dynamischen Speicherplatz. Für eine Reihe von realistischen Programmbeispielen wird demonstriert, dass die angefertigte generische Implementation auch gute Schranken an den Verbrauch von Stapelspeicher und der maximalen Ausführungszeit ermitteln kann. Die Analyse ist sehr effizient implementierbar, und behandelt auch größere Beispielprogramme vollständig in wenigen Sekunden