Type-based analysis of logarithmic amortised complexity

Hofmann, Martin; Leutgeb, Lorenz; Obwaller, David; Moser, Georg; Zuleger, Florian

doi:10.1017/s0960129521000232

Cited by 7 publications

(28 citation statements)

References 53 publications

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“…These data structures have so far only (semi-)manually been analysed in the literature. Our analysis is based on a novel type-and-effect system, which constitutes a generalisation of the type system studied in [13,17] to the non-deterministic and probabilistic setting, as well as an extension of the type system introduced in [34] to sublinear bounds and non-determinism. We provide a prototype implementation that is able to fully automatically analyse the case studies mentioned above.…”

Section: Introductionmentioning

confidence: 99%

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ATLAS: Automated Amortised Complexity Analysis of Self-adjusting Data Structures

Leutgeb

Moser

Zuleger

2021

Computer Aided Verification

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Being able to argue about the performance of self-adjusting data structures such as splay trees has been a main objective, when Sleator and Tarjan introduced the notion of amortised complexity.Analysing these data structures requires sophisticated potential functions, which typically contain logarithmic expressions. Possibly for these reasons, and despite the recent progress in automated resource analysis, they have so far eluded automation. In this paper, we report on the first fully-automated amortised complexity analysis of self-adjusting data structures. Following earlier work, our analysis is based on potential function templates with unknown coefficients.We make the following contributions: 1) We encode the search for concrete potential function coefficients as an optimisation problem over a suitable constraint system. Our target function steers the search towards coefficients that minimise the inferred amortised complexity. 2) Automation is achieved by using a linear constraint system in conjunction with suitable lemmata schemes that encapsulate the required non-linear facts about the logarithm. We discuss our choices that achieve a scalable analysis. 3) We present our tool $$\mathsf {ATLAS}$$ ATLAS and report on experimental results for splay trees, splay heaps and pairing heaps. We completely automatically infer complexity estimates that match previous results (obtained by sophisticated pen-and-paper proofs), and in some cases even infer better complexity estimates than previously published.

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Section: Introductionmentioning

confidence: 99%

“…the cost assigned to non-terminating computations. (iii) Based on [13,17], we develop a novel type-and-effect system that strictly generalises the prior approaches from the literature. (iv) We state two soundness theorems (see Section 5.3) based on two different-but strongly related-typing rules of ticking.…”

Section: Introductionmentioning

confidence: 99%

ATLAS: Automated Amortised Complexity Analysis of Self-adjusting Data Structures

Leutgeb

Moser

Zuleger

2021

Computer Aided Verification

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“…This may not be insurmountable, as the demotion rule showed here, but new ideas are needed overall. Logarithmic potential has been explored in [32], though the approach there departs from the automatable AARA framework of linear constraint solving.…”

Section: Exponentials Polynomials and Logarithmsmentioning

confidence: 99%

Exponential Automatic Amortized Resource Analysis

Kahn

Hoffmann

2020

Lecture Notes in Computer Science

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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.

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“…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].…”

Section: Related Workmentioning

confidence: 99%

Raising Expectations: Automating Expected Cost Analysis with Types

Wang

Kahn

Hoffmann

2020

Preprint

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This article presents a type-based analysis for deriving upper bounds on the expected execution cost of probabilistic programs. The analysis is naturally compositional, parametric in the cost model, and supports higherorder functions and inductive data types. The derived bounds are multivariate polynomials that are functions of data structures. Bound inference is enabled by local type rules that reduce type inference to linear constraint solving. The type system is based on the potential method of amortized analysis and extends automatic amortized resource analysis (AARA) for deterministic programs. A main innovation is that bounds can contain symbolic probabilities, which may appear in data structures and function arguments. Another contribution is a novel soundness proof that establishes the correctness of the derived bounds with respect to a distribution-based operational cost semantics that also includes nontrivial diverging behavior. For cost models like time, derived bounds imply termination with probability one. To highlight the novel ideas, the presentation focuses on linear potential and a core language. However, the analysis is implemented as an extension of Resource Aware ML and supports polynomial bounds and user defined data structures. The effectiveness of the technique is evaluated by analyzing the sample complexity of discrete distributions and with a novel average-case estimation for deterministic programs that combines expected cost analysis with statistical methods.

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Type-based analysis of logarithmic amortised complexity

Cited by 7 publications

References 53 publications

ATLAS: Automated Amortised Complexity Analysis of Self-adjusting Data Structures

ATLAS: Automated Amortised Complexity Analysis of Self-adjusting Data Structures

Exponential Automatic Amortized Resource Analysis

Raising Expectations: Automating Expected Cost Analysis with Types

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