Exponential Automatic Amortized Resource Analysis

Abstract

Automatic amortized resource analysis (AARA) is a typebased technique for inferring concrete (non-asymptotic) bounds on a program's resource usage. Existing work on AARA has focused on bounds that are polynomial in the sizes of the inputs. This paper presents and extension of AARA to exponential bounds that preserves the benefits of the technique, such as compositionality and efficient type inference based on linear constraint solving. A key idea is the use of the Stirling numbers of the second kind as the basis of potential functions, which play the same role as the binomial coefficients in polynomial AARA. To formalize the similarities with the existing analyses, the paper presents a general methodology for AARA that is instantiated to the polynomial version, the exponential version, and a combined system with potential functions that are formed by products of Stirling numbers and binomial coefficients. The soundness of exponential AARA is proved with respect to an operational cost semantics and the analysis of representative example programs demonstrates the effectiveness of the new analysis.

Synthesis with Asymptotic Resource Bounds

“…Another set of methods to generate resource bounds are type-based [9,10,14,19]. As we discussed throughout the paper, the complexities generated by these methods are concrete functions and not expressed with big-O notation, although [19] is sometimes able to pattern match a case of the Master Theorem.…”

“…These type systems differ from ours in a few ways. The AARA line of research [9,10,14] is able to assign amortized complexity to programs, but is not able to generate logarithmic bounds. [19] is also able to perform amortized analysis; however, the technique is not fully automated, and instead requires the user to provide type annotations on terms, which are then checked by the type system.…”

Raising expectations: automating expected cost analysis with types

“…To focus on the changes that probabilistic choice induces on the type system, we describe its action here in linear AARA, where all potential functions are linear in terms of list sizes. In other work, potential functions have been expanded to cover polynomials [Hoffmann et al 2011; Hoffmann and Hofmann 2010b] and exponentials [Kahn and Hoffmann 2020], but this extension to AARA is orthogonal to probabilistic choice. Indeed, we have carried over the implementation and soundness of probabilistic AARA to support multivariate-polynomial potential functions and user-defined datatypes without problem, which we use to perform analyses in ğ6 and beyond.…”

“…Our work is based on AARA, which was initially introduced [Hofmann and Jost 2003] to automatically derive linear heap-space bounds for first-order functional programs. AARA has been extended to polynomial bounds [Hoffmann et al 2011;Hoffmann and Hofmann 2010b;Hofmann and Moser 2015], exponential bounds [Kahn and Hoffmann 2020], logarithmic bounds [Hofmann and Moser 2018], higher-order functions Jost et al 2010], user-defined datatypes Jost et al 2009], and separation logic [Atkey 2010]. The technique has also been generalized to imperative arithmetic programs [Carbonneaux et al 2017[Carbonneaux et al , 2015, as well as integrated into formal proof assistants [Charguéraud and Pottier 2015;Nipkow 2015].…”

Type-based analysis of logarithmic amortised complexity

“…Most research has focused on the inference of polynomial bounds on the worst-case cost of the program under analysis. A few papers also target the inference of exponential and logarithmic bounds (Albert et al, 2008;Avanzini et al, 2011;Chatterjee et al, 2017;Kahn and Hoffmann, 2020;Wang et al, 2017;Winkler and Moser, 2020). Some of the cited approaches are able to conduct an automated amortised analysis in the sense of Sleator and Tarjan: The work on type-based cost analysis by Martin Hofmann and his collaborators (Hoffmann, 2011;Hoffmann et al, 2012aHoffmann et al, , 2017Hofmann and Jost, 2003;Moser, 2014, 2015;Hoffmann and Shao, 2015a,b;Jost et al, 2009Jost et al, , 2010Jost et al, , 2017), which we discuss in more detail below, directly employs potential functions as proposed in Sleator and Tarjan (1985); Tarjan (1985).…”