scholarly journals Type-based analysis of logarithmic amortised complexity

2021 ◽  
pp. 1-33
Author(s):  
Martin Hofmann ◽  
Lorenz Leutgeb ◽  
David Obwaller ◽  
Georg Moser ◽  
Florian Zuleger
Keyword(s):  
Data Structures ◽  
Type System ◽  
Automated Analysis ◽  
Logarithmic Potential ◽  
Potential Functions ◽  
Resource Analysis

Abstract We introduce a novel amortised resource analysis couched in a type-and-effect system. Our analysis is formulated in terms of the physicist’s method of amortised analysis and is potentialbased. The type system makes use of logarithmic potential functions and is the first such system to exhibit logarithmic amortised complexity. With our approach, we target the automated analysis of self-adjusting data structures, like splay trees, which so far have only manually been analysed in the literature. In particular, we have implemented a semi-automated prototype, which successfully analyses the zig-zig case of splaying, once the type annotations are fixed.

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

2021 ◽  
pp. 99-122
Author(s):  
Lorenz Leutgeb ◽  
Georg Moser ◽  
Florian Zuleger
Keyword(s):  
Potential Function ◽  
Data Structures ◽  
Recent Progress ◽  
Complexity Analysis ◽  
Target Function ◽  
Linear Constraint ◽  
Potential Functions ◽  
Resource Analysis ◽  
Constraint System ◽  
Scalable Analysis

AbstractBeing 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.

scholarly journals Construction of potential functions associated with a given energy spectrum — An inverse problem

2020 ◽  
Vol 35 (20) ◽  
pp. 2050104
Author(s):  
A. D. Alhaidari
Keyword(s):  
Energy Spectrum ◽  
Three Dimensional ◽  
Logarithmic Potential ◽  
Potential Functions ◽  
Physical Parameters ◽  
Linear Potential ◽  
Hahn Polynomial ◽  
Exactly Solvable ◽  
The One ◽  
Solvable Problems

Using a formulation of quantum mechanics based on orthogonal polynomials in the energy and physical parameters, we present a method that gives the class of potential functions for exactly solvable problems corresponding to a given energy spectrum. In this work, we study the class of problems associated with the continuous dual Hahn polynomial. These include the one-dimensional logarithmic potential and the three-dimensional Coulomb plus linear potential.

Type-based confinement

2005 ◽  
Vol 16 (1) ◽  
pp. 83-128 ◽  
Author(s):  
TIAN ZHAO ◽  
JENS PALSBERG ◽  
JAN VITEK
Keyword(s):  
Data Structures ◽  
Type System ◽  
Object Oriented ◽  
Point Of View ◽  
Generic Extension ◽  
Practical Point ◽  
Related Research ◽  
Language Based Security ◽  
Object Graphs ◽  
Object Oriented Systems

Confinement properties impose a structure on object graphs which can be used to enforce encapsulation properties. From a practical point of view, encapsulation is essential for building secure object-oriented systems as security requires that the interface between trusted and untrusted components of a system be clearly delineated and restricted to the smallest possible set of operations and data structures. This paper investigates the notion of package-level confinement and proposes a type system that enforces this notion for a call-by-value object calculus as well as a generic extension thereof. We give a proof of soundness of this type system, and establish links between this work and related research in language-based security.

scholarly journals Data Structures in Leelus Programming Language: A Research

2021 ◽  
Author(s):  
Frank Appiah
Keyword(s):  
Data Structure ◽  
Programming Language ◽  
Data Structures ◽  
Type System

This is a research on data structures using pointer declaration for array dimensional type of representation in variable declaration. In this research, I will describe type array, term array and value array drawn from both data structure and type system.

scholarly journals Exponential Automatic Amortized Resource Analysis

2020 ◽  
pp. 359-380
Author(s):  
David M. Kahn ◽  
Jan Hoffmann
Keyword(s):  
Stirling Numbers ◽  
Linear Constraint ◽  
Potential Functions ◽  
Binomial Coefficients ◽  
Combined System ◽  
Resource Analysis ◽  
Asymptotic Bounds ◽  
Exponential Bounds ◽  
Efficient Type ◽  
Cost Semantics

AbstractAutomatic amortized resource analysis (AARA) is a type-based 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.

Intensional polymorphism in type-erasure semantics

2002 ◽  
Vol 12 (6) ◽  
pp. 567-600 ◽  
Author(s):  
KARL CRARY ◽  
STEPHANIE WEIRICH ◽  
GREG MORRISETT
Keyword(s):  
Data Structures ◽  
Garbage Collection ◽  
Type System ◽  
Implementation Techniques ◽  
Run Time ◽  
Closure Conversion ◽  
Type Information ◽  
Time Type

Intensional polymorphism, the ability to dispatch to different routines based on types at run time, enables a variety of advanced implementation techniques for polymorphic languages, including tag-free garbage collection, unboxed function arguments, polymorphic marshalling and attened data structures. To date, languages that support intensional polymorphism have required a type-passing (as opposed to type-erasure) interpretation where types are constructed and passed to polymorphic functions at run time. Unfortunately, type-passing suffers from a number of drawbacks: it requires duplication of run-time constructs at the term and type levels, it prevents abstraction, and it severely complicates polymorphic closure conversion. We present a type-theoretic framework that supports intensional polymorphism, but avoids many of the disadvantages of type passing. In our approach, run-time type information is represented by ordinary terms. This avoids the duplication problem, allows us to recover abstraction, and avoids complications with closure conversion. In addition, our type system provides another improvement in expressiveness; it allows unknown types to be refined in place, thereby avoiding certain beta-expansions required by other frameworks.

scholarly journals Automatic amortized resource analysis with the Quantum physicist’s method

2021 ◽  
Vol 5 (ICFP) ◽  
pp. 1-29
Author(s):  
David M. Kahn ◽  
Jan Hoffmann
Keyword(s):  
Quantum Tunneling ◽  
State Of The Art ◽  
Type System ◽  
Type Systems ◽  
Quantum Superposition ◽  
Resource Analysis ◽  
Linear Type ◽  
New Type ◽  
Novel Method ◽  
Standard Library

We present a novel method for working with the physicist's method of amortized resource analysis, which we call the quantum physicist's method. These principles allow for more precise analyses of resources that are not monotonically consumed, like stack. This method takes its name from its two major features, worldviews and resource tunneling, which behave analogously to quantum superposition and quantum tunneling. We use the quantum physicist's method to extend the Automatic Amortized Resource Analysis (AARA) type system, enabling the derivation of resource bounds based on tree depth. In doing so, we also introduce remainder contexts, which aid bookkeeping in linear type systems. We then evaluate this new type system's performance by bounding stack use of functions in the Set module of OCaml's standard library. Compared to state-of-the-art implementations of AARA, our new system derives tighter bounds with only moderate overhead.

Ultrastructural characteristics of sporidia of Ustilago

1989 ◽  
Vol 47 ◽  
pp. 786-787
Author(s):  
Patricia N. Hackney
Keyword(s):  
Electron Microscope ◽  
Type System ◽  
Tube Formation ◽  
High Voltage Electron Microscopy ◽  
Voltage Electron ◽  
University Of Wisconsin ◽  
Transmission Electron ◽  
Ustilago Hordei ◽  
Silene Alba ◽  
The University

Ustilago hordei and Ustilago violacea are yeast-like basidiomycete pathogens ofHordeum vulgare and Silene alba respectively. The mating type system in both species of Ustilago is bipolar, with alleles, A,a, (U.hordei) and a1, a2 (U.violacea) at a single locus. Haploid sporidia maintain the asexual phase by budding, while the sexual phase is initiated by conjugation tube formation between the mating types during budding and conjugation.For observation of budding, sporidia were prepared by culturing the four types on YEG (yeast extract glucose) broth for 24 hours. After centrifugation at 5000g cells were either left unmated or mated in a1/a2,A/a combinations. The sporidia were then mixed 1:1 with 4% agar and the resulting 1mm cubes fixed in 8% gluteraldehyde and post fixed in osmium tetroxide. After dehydration and embedding cubes were thin sectioned with a LKB ultratome and photographed in a Zeiss 9s transmission electron microscope or in an AE1 electron microscope of MK11 1MEV at the High Voltage Electron Microscopy Center of the University of Wisconsin-Madison.

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