Neural Networks and the Satisfiability Problem A
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Efficient Algorithms with Asymmetric Read and Write Costs
Efficient Algorithms with Asymmetric Read and Write Costs Guy E. Blelloch1, Jeremy T. Fineman2, Phillip B. Gibbons1, Yan Gu1, and Julian Shun3 1 Carnegie Mellon University 2 Georgetown University 3 University of California, Berkeley Abstract In several emerging technologies for computer memory (main memory), the cost of reading is significantly cheaper than the cost of writing. Such asymmetry in memory costs poses a fun- damentally different model from the RAM for algorithm design. In this paper we study lower and upper bounds for various problems under such asymmetric read and write costs. We con- sider both the case in which all but O(1) memory has asymmetric cost, and the case of a small cache of symmetric memory. We model both cases using the (M, ω)-ARAM, in which there is a small (symmetric) memory of size M and a large unbounded (asymmetric) memory, both random access, and where reading from the large memory has unit cost, but writing has cost ω 1. For FFT and sorting networks we show a lower bound cost of Ω(ωn logωM n), which indicates that it is not possible to achieve asymptotic improvements with cheaper reads when ω is bounded by a polynomial in M. Moreover, there is an asymptotic gap (of min(ω, log n)/ log(ωM)) between the cost of sorting networks and comparison sorting in the model. This contrasts with the RAM, and most other models, in which the asymptotic costs are the same. We also show a lower bound for computations on an n × n diamond DAG of Ω(ωn2/M) cost, which indicates no asymptotic improvement is achievable with fast reads. -
Great Ideas in Computing
Great Ideas in Computing University of Toronto CSC196 Winter/Spring 2019 Week 6: October 19-23 (2020) 1 / 17 Announcements I added one final question to Assignment 2. It concerns search engines. The question may need a little clarification. I have also clarified question 1 where I have defined (in a non standard way) the meaning of a strict binary search tree which is what I had in mind. Please answer the question for a strict binary search tree. If you answered the quiz question for a non strict binary search tree withn a proper explanation you will get full credit. Quick poll: how many students feel that Q1 was a fair quiz? A1 has now been graded by Marta. I will scan over the assignments and hope to release the grades later today. If you plan to make a regrading request, you have up to one week to submit your request. You must specify clearly why you feel that a question may not have been graded fairly. In general, students did well which is what I expected. 2 / 17 Agenda for the week We will continue to discuss search engines. We ended on what is slide 10 (in Week 5) on Friday and we will continue with where we left off. I was surprised that in our poll, most students felt that the people advocating the \AI view" of search \won the debate" whereas today I will try to argue that the people (e.g., Salton and others) advocating the \combinatorial, algebraic, statistical view" won the debate as to current search engines. -
Efficient Algorithms with Asymmetric Read and Write Costs
Efficient Algorithms with Asymmetric Read and Write Costs Guy E. Blelloch1, Jeremy T. Fineman2, Phillip B. Gibbons1, Yan Gu1, and Julian Shun3 1 Carnegie Mellon University 2 Georgetown University 3 University of California, Berkeley Abstract In several emerging technologies for computer memory (main memory), the cost of reading is significantly cheaper than the cost of writing. Such asymmetry in memory costs poses a fun- damentally different model from the RAM for algorithm design. In this paper we study lower and upper bounds for various problems under such asymmetric read and write costs. We con- sider both the case in which all but O(1) memory has asymmetric cost, and the case of a small cache of symmetric memory. We model both cases using the (M, ω)-ARAM, in which there is a small (symmetric) memory of size M and a large unbounded (asymmetric) memory, both random access, and where reading from the large memory has unit cost, but writing has cost ω 1. For FFT and sorting networks we show a lower bound cost of Ω(ωn logωM n), which indicates that it is not possible to achieve asymptotic improvements with cheaper reads when ω is bounded by a polynomial in M. Moreover, there is an asymptotic gap (of min(ω, log n)/ log(ωM)) between the cost of sorting networks and comparison sorting in the model. This contrasts with the RAM, and most other models, in which the asymptotic costs are the same. We also show a lower bound for computations on an n × n diamond DAG of Ω(ωn2/M) cost, which indicates no asymptotic improvement is achievable with fast reads. -
Program Synthesis with Grammars and Semantics in Genetic Programming
Program Synthesis with Grammars and Semantics in Genetic Programming Stefan Forstenlechner UCD student number: 14204817 The thesis is submitted to University College Dublin in fulfilment of the requirements for the degree of DOCTOR OF PHILOSOPHY School of Business Head of School: Prof. Anthony Brabazon Research Supervisors: Prof. Michael O’Neill Dr. Miguel Nicolau External Examiner: Dr. John R. Woodward January 2019 Contents Contents i Abstract vii Statement of Original Authorship viii Acknowledgements ix List of Figures xi List of Tables xiv List of Algorithms xvii List of Abbreviations xviii Publications Arising xx I Introduction and Literature Review 1 1 Introduction 2 1.1 Aim of Thesis . .3 1.2 Research Questions . .4 1.3 Contributions . .7 1.3.1 Technical contributions . .8 1.4 Limitations . .8 1.5 Thesis Outline . .9 i CONTENTS 2 Related Work 12 2.1 Evolutionary Computation . 12 2.2 Genetic Programming . 14 2.2.1 Representation . 15 2.2.2 Initialization . 16 2.2.3 Fitness . 17 2.2.4 Selection . 17 2.2.5 Crossover . 19 2.2.6 Mutation . 19 2.2.7 GP Summary . 20 2.2.8 Grammars . 20 2.3 Program Synthesis . 23 2.3.1 Program Synthesis in Genetic Programming . 26 2.3.2 General Program Synthesis Benchmark Suite . 30 2.4 Semantics . 32 2.4.1 Semantic operators . 34 2.4.2 Geometric Semantic GP . 37 2.5 Conclusion . 38 II Experimental Research 40 3 General Grammar Design 41 3.1 Grammars for G3P . 41 3.2 Sorting Network . 42 3.3 Structure in Grammars . 43 3.4 Sorting Network Grammar Design . -
P Versus NP Richard M. Karp
P Versus NP Computational complexity theory is the branch of theoretical computer science concerned with the fundamental limits on the efficiency of automatic computation. It focuses on problems that appear to Richard M. Karp require a very large number of computation steps for their solution. The inputs and outputs to a problem are sequences of symbols drawn from a finite alphabet; there is no limit on the length of the input, and the fundamental question about a problem is the rate of growth of the number of required computation steps as a function of the length of the input. Some problems seem to require a very rapidly growing number of steps. One such problem is the inde- pendent set problem: given a graph, consisting of points called vertices and lines called edges connect- ing pairs of vertices, a set of vertices is called independent if no two vertices in the set are connected by a line. Given a graph and a positive integer n, the problem is to decide whether the graph contains an independent set of size n. Every known algorithm to solve the independent set problem encounters a combinatorial explosion, in which the number of required computation steps grows exponentially as a function of the size of the graph. On the other hand, the problem of deciding whether a given set of vertices is an independent set in a given graph is solvable by inspection. There are many such dichotomies, in which it is hard to decide whether a given type of structure exists within an input object (the existence problem), but it is easy to decide whether a given structure is of the required type (the verification problem). -
Learning Functional Programs with Function Invention and Reuse
Learning functional programs with function invention and reuse Andrei Diaconu University of Oxford arXiv:2011.08881v1 [cs.PL] 17 Nov 2020 Honour School of Computer Science Trinity 2020 Abstract Inductive programming (IP) is a field whose main goal is synthe- sising programs that respect a set of examples, given some form of background knowledge. This paper is concerned with a subfield of IP, inductive functional programming (IFP). We explore the idea of generating modular functional programs, and how those allow for function reuse, with the aim to reduce the size of the programs. We introduce two algorithms that attempt to solve the problem and explore type based pruning techniques in the context of modular programs. By experimenting with the imple- mentation of one of those algorithms, we show reuse is important (if not crucial) for a variety of problems and distinguished two broad classes of programs that will generally benefit from func- tion reuse. Contents 1 Introduction5 1.1 Inductive programming . .5 1.2 Motivation . .6 1.3 Contributions . .7 1.4 Structure of the report . .8 2 Background and related work 10 2.1 Background on IP . 10 2.2 Related work . 12 2.2.1 Metagol . 12 2.2.2 Magic Haskeller . 13 2.2.3 λ2 ............................. 13 2.3 Invention and reuse . 13 3 Problem description 15 3.1 Abstract description of the problem . 15 3.2 Invention and reuse . 19 4 Algorithms for the program synthesis problem 21 4.1 Preliminaries . 22 4.1.1 Target language and the type system . 22 4.1.2 Combinatorial search . -
The P Versus Np Problem
THE P VERSUS NP PROBLEM STEPHEN COOK 1. Statement of the Problem The P versus NP problem is to determine whether every language accepted by some nondeterministic algorithm in polynomial time is also accepted by some (deterministic) algorithm in polynomial time. To define the problem precisely it is necessary to give a formal model of a computer. The standard computer model in computability theory is the Turing machine, introduced by Alan Turing in 1936 [37]. Although the model was introduced before physical computers were built, it nevertheless continues to be accepted as the proper computer model for the purpose of defining the notion of computable function. Informally the class P is the class of decision problems solvable by some algorithm within a number of steps bounded by some fixed polynomial in the length of the input. Turing was not concerned with the efficiency of his machines, rather his concern was whether they can simulate arbitrary algorithms given sufficient time. It turns out, however, Turing machines can generally simulate more efficient computer models (for example, machines equipped with many tapes or an unbounded random access memory) by at most squaring or cubing the computation time. Thus P is a robust class and has equivalent definitions over a large class of computer models. Here we follow standard practice and define the class P in terms of Turing machines. Formally the elements of the class P are languages. Let Σ be a finite alphabet (that is, a finite nonempty set) with at least two elements, and let Σ∗ be the set of finite strings over Σ. -
Program Synthesis with Types
University of Pennsylvania ScholarlyCommons Publicly Accessible Penn Dissertations 2015 Program Synthesis With Types Peter-Michael Santos Osera University of Pennsylvania, [email protected] Follow this and additional works at: https://repository.upenn.edu/edissertations Part of the Computer Sciences Commons Recommended Citation Osera, Peter-Michael Santos, "Program Synthesis With Types" (2015). Publicly Accessible Penn Dissertations. 1926. https://repository.upenn.edu/edissertations/1926 This paper is posted at ScholarlyCommons. https://repository.upenn.edu/edissertations/1926 For more information, please contact [email protected]. Program Synthesis With Types Abstract Program synthesis, the automatic generation of programs from specification, promises to fundamentally change the way that we build software. By using synthesis tools, we can greatly speed up the time it takes to build complex software artifacts as well as construct programs that are automatically correct by virtue of the synthesis process. Studied since the 70s, researchers have applied techniques from many different sub-fields of computer science ot solve the program synthesis problem in a variety of domains and contexts. However, one domain that has been less explored than others is the domain of typed, functional programs. This is unfortunate because programs in richly-typed languages like OCaml and Haskell are known for ``writing themselves'' once the programmer gets the types correct. In light of this observation, can we use type theory to build more expressive and efficient type-directed synthesis systems for this domain of programs? This dissertation answers this question in the affirmative by building novel type- theoretic foundations for program synthesis. By using type theory as the basis of study for program synthesis, we are able to build core synthesis calculi for typed, functional programs, analyze the calculi's meta-theoretic properties, and extend these calculi to handle increasingly richer types and language features. -
A Memorable Trip Abhisekh Sankaran Research Scholar, IIT Bombay
A Memorable Trip Abhisekh Sankaran Research Scholar, IIT Bombay It was my first trip to the US. It had not yet sunk in that I had been chosen by ACM India as one of two Ph.D. students from India to attend the big ACM Turing Centenary Celebration in San Francisco until I saw the familiar face of Stephen Cook enter a room in the hotel a short distance from mine; later, Moshe Vardi recognized me from his trip to IITB during FSTTCS, 2011. I recognized Nitin Saurabh from IMSc Chennai, the other student chosen by ACM-India; 11 ACM SIG©s had sponsored students and there were about 75 from all over the world. Registration started at 8am on 15th June, along with breakfast. Collecting my ©Student Scholar© badge and stuffing in some food, I entered a large hall with several hundred seats, a brightly lit podium with a large screen in the middle flanked by two others. The program began with a video giving a brief biography of Alan Turing from his boyhood to the dynamic young man who was to change the world forever. There were inaugural speeches by John White, CEO of ACM, and Vint Cerf, the 2004 Turing Award winner and incoming ACM President. The MC for the event, Paul Saffo, took over and the panel discussions and speeches commenced. A live Twitter feed made it possible for people in the audience and elsewhere to post questions/comments which were actually taken up in the discussions. Of the many sessions that took place in the next two days, I will describe three that I found most interesting. -
Program Synthesis
Program Synthesis Program Synthesis Sumit Gulwani Microsoft Research [email protected] Oleksandr Polozov University of Washington [email protected] Rishabh Singh Microsoft Research [email protected] Boston — Delft Foundations and Trends® in Programming Languages Published, sold and distributed by: now Publishers Inc. PO Box 1024 Hanover, MA 02339 United States Tel. +1-781-985-4510 www.nowpublishers.com [email protected] Outside North America: now Publishers Inc. PO Box 179 2600 AD Delft The Netherlands Tel. +31-6-51115274 The preferred citation for this publication is S. Gulwani, O. Polozov and R. Singh. Program Synthesis. Foundations and Trends® in Programming Languages, vol. 4, no. 1-2, pp. 1–119, 2017. ® This Foundations and Trends issue was typeset in LATEX using a class file designed by Neal Parikh. Printed on acid-free paper. ISBN: 978-1-68083-292-1 © 2017 S. Gulwani, O. Polozov and R. Singh All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, mechanical, photocopying, recording or otherwise, without prior written permission of the publishers. Photocopying. In the USA: This journal is registered at the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923. Authorization to photocopy items for in- ternal or personal use, or the internal or personal use of specific clients, is granted by now Publishers Inc for users registered with the Copyright Clearance Center (CCC). The ‘services’ for users can be found on the internet at: www.copyright.com For those organizations that have been granted a photocopy license, a separate system of payment has been arranged. -
Template-Based Program Verification and Program Synthesis
Software Tools for Technology Transfer manuscript No. (will be inserted by the editor) Template-based Program Verification and Program Synthesis Saurabh Srivastava? and Sumit Gulwani?? and Jeffrey S. Foster??? University of California, Berkeley and Microsoft Research, Redmond and University of Maryland, College Park Received: date / Revised version: date Abstract. Program verification is the task of automat- the subsequent step of writing the program in C, C++, ically generating proofs for a program's compliance with Java etc. There is additional effort in ensuring that all a given specification. Program synthesis is the task of corner cases are covered, which is a tiresome and man- automatically generating a program that meets a given ual process. The dilligent programmer will also verify specification. Both program verification and program syn- that the final program is not only correct but also me- thesis can be viewed as search problems, for proofs and chanically verifiable, which usually requires them to add programs, respectively. annotations illustrating the correctness. For these search problems, we present approaches This makes programming a subtle task, and the ef- based on user-provided insights in the form of templates. fort involved in writing correct programs is so high that Templates are hints about the syntactic forms of the in- in practice we accept that programs can only be approx- variants and programs, and help guide the search for imately correct. While developers do realize the pres- solutions. We show how to reduce the template-based ence of corner cases, the difficulty inherent in enumer- search problem to satisfiability solving, which permits ating and reasoning about them, or using a verification the use of off-the-shelf solvers to efficiently explore the framework, ensures that developers typically leave most search space. -
Alan Mathison Turing and the Turing Award Winners
Alan Turing and the Turing Award Winners A Short Journey Through the History of Computer TítuloScience do capítulo Luis Lamb, 22 June 2012 Slides by Luis C. Lamb Alan Mathison Turing A.M. Turing, 1951 Turing by Stephen Kettle, 2007 by Slides by Luis C. Lamb Assumptions • I assume knowlege of Computing as a Science. • I shall not talk about computing before Turing: Leibniz, Babbage, Boole, Gödel... • I shall not detail theorems or algorithms. • I shall apologize for omissions at the end of this presentation. • Comprehensive information about Turing can be found at http://www.mathcomp.leeds.ac.uk/turing2012/ • The full version of this talk is available upon request. Slides by Luis C. Lamb Alan Mathison Turing § Born 23 June 1912: 2 Warrington Crescent, Maida Vale, London W9 Google maps Slides by Luis C. Lamb Alan Mathison Turing: short biography • 1922: Attends Hazlehurst Preparatory School • ’26: Sherborne School Dorset • ’31: King’s College Cambridge, Maths (graduates in ‘34). • ’35: Elected to Fellowship of King’s College Cambridge • ’36: Publishes “On Computable Numbers, with an Application to the Entscheindungsproblem”, Journal of the London Math. Soc. • ’38: PhD Princeton (viva on 21 June) : “Systems of Logic Based on Ordinals”, supervised by Alonzo Church. • Letter to Philipp Hall: “I hope Hitler will not have invaded England before I come back.” • ’39 Joins Bletchley Park: designs the “Bombe”. • ’40: First Bombes are fully operational • ’41: Breaks the German Naval Enigma. • ’42-44: Several contibutions to war effort on codebreaking; secure speech devices; computing. • ’45: Automatic Computing Engine (ACE) Computer. Slides by Luis C.