Analysing the complexity of functional programs: higher-order meets first-order

Avanzini, Martin; Lago, Ugo Dal; Moser, Georg

doi:10.1145/2858949.2784753

Cited by 21 publications

(35 citation statements)

References 58 publications

(88 reference statements)

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“…To define the RPO order ≺ we explicit the use of pattern variables of the form V using a special symbol var and consequently these variables are now of the form var(id) with id the name of the variable. The quasi-precedence used for this RPO order > is then defined as follows: +, ⊥, f n , var ≺ \ for all f n ∈ F n with statuses \ : lexicographic (permutation [2, 1]), var : lexicographic( [1]), + : lexicographic( [1,2]), ⊥ : lexicographic, f n : lexicographic( [1, . .…”

Section: Resultsmentioning

confidence: 99%

“…There are a lot of works [9,13,1,8] targeting the analysis of functional languages essentially in terms of termination and complexity, and usually they involve some encoding of the match construction. These encodings are generally deep and take into account the evaluation strategy of the targeted language leading to powerful analyzing tools.…”

Section: Related Workmentioning

confidence: 99%

“…containing no \). For the RPO order ≺ we complete the quasi-precedence used in Lemma 3.1 to take into account aliases: +, ⊥, f n , var, @ ≺ \ for all f n ∈ F n with the same statuses as before and with @ : lexicographic( [1,2]). It is easy to check that for each rule l ⇒ r in R @ \ we have l > r. For the shape of the normal forms w.r.t.…”

Section: Proposition 34 (Anti-pattern Semantics Preservation) For Amentioning

confidence: 99%

“…Generally, well-established techniques and (automatic) tools used in the plain case cannot be applied directly in the general case. There have been several works in the context of functional programming like, for example [13,9,8,1] to cite only a few, but they are essentially focused on powerful techniques for analyzing the termination and complexity of functional programs with ordered matching statements. We are interested here in a transformation approach which can be used as an add-on for well-established analyzing techniques and tools but also as a generic compiler for ordered TRS involving anti-patterns which could be easily integrated in any language providing rewrite rules, or at least pattern matching primitives.…”

Section: Introductionmentioning

confidence: 99%

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Generic Encodings of Constructor Rewriting Systems

Cirstea

Moreau

2019

Proceedings of the 21st International Symposium on Principles and Practice of Declarative Programming

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Rewriting is a formalism widely used in computer science and mathematical logic. The classical formalism has been extended, in the context of functional languages, with an order over the rules and, in the context of rewrite based languages, with the negation over patterns. We propose in this paper a concise and clear algorithm computing the difference over patterns which can be used to define generic encodings of constructor term rewriting systems with negation and order into classical term rewriting systems. As a direct consequence, established methods used for term rewriting systems can be applied to analyze properties of the extended systems. The approach can also be seen as a generic compiler which targets any language providing basic pattern matching primitives. The formalism provides also a new method for deciding if a set of patterns subsumes a given pattern and thus, for checking the presence of useless patterns or the completeness of a set of patterns.[ paint (car(x, suv)) red, paint (car(electric, x)) blue, paint (car(diesel, y)) red, paint(car(x, y)) white, paint(x) red ] for the assignment of an imaginary eco-label: all electric cars but the SUVs (which are red) are blue, diesel cars are red and the remaining cars are white; all the other vehicles are red.Patterns express positive conditions and we have used the term car (electric, x) to specify electric cars of any style. Negation is nevertheless intrinsic to human thinking and most of the time when searching• it can be used to decide whether a pattern is subsumed by a given set of patterns and thus, to check the presence of useless patterns or the completeness of a set of patterns.

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Section: Resultsmentioning

confidence: 99%

Section: Related Workmentioning

confidence: 99%

Section: Proposition 34 (Anti-pattern Semantics Preservation) For Amentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Generic Encodings of Constructor Rewriting Systems

Cirstea

Moreau

2019

Proceedings of the 21st International Symposium on Principles and Practice of Declarative Programming

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“…Techniques and tools for automatic and semi-automatic resource analysis have been extensively studied. The applied methods range from deriving and analyzing recurrence relations [55,1,16,2,12,36,10,37], to abstract interpretation and static analysis [18,7,49,39], to type systems [11,56,53], to proof assistants and program logics [4,9,8,48,19,45,42], to term rewriting [6,5,47]. Many techniques focus on upper bounds on the worst-case bounds, but average-case bounds [15,35,43,54] and lower-bounds have also been studied [3,17,44].…”

Section: Introductionmentioning

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

Leutgeb

Moser

Zuleger

2021

Computer Aided Verification

Self Cite

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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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Analysing the complexity of functional programs: higher-order meets first-order

Cited by 21 publications

References 58 publications

Generic Encodings of Constructor Rewriting Systems

Generic Encodings of Constructor Rewriting Systems

Exponential Automatic Amortized Resource Analysis

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

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