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Integer Linear Programs
20 ________________________________________________________________________________________________ Integer Linear Programs Many linear programming problems require certain variables to have whole number, or integer, values. Such a requirement arises naturally when the variables represent enti- ties like packages or people that can not be fractionally divided — at least, not in a mean- ingful way for the situation being modeled. Integer variables also play a role in formulat- ing equation systems that model logical conditions, as we will show later in this chapter. In some situations, the optimization techniques described in previous chapters are suf- ficient to find an integer solution. An integer optimal solution is guaranteed for certain network linear programs, as explained in Section 15.5. Even where there is no guarantee, a linear programming solver may happen to find an integer optimal solution for the par- ticular instances of a model in which you are interested. This happened in the solution of the multicommodity transportation model (Figure 4-1) for the particular data that we specified (Figure 4-2). Even if you do not obtain an integer solution from the solver, chances are good that you’ll get a solution in which most of the variables lie at integer values. Specifically, many solvers are able to return an ‘‘extreme’’ solution in which the number of variables not lying at their bounds is at most the number of constraints. If the bounds are integral, all of the variables at their bounds will have integer values; and if the rest of the data is integral, many of the remaining variables may turn out to be integers, too. -
Lecture Notes on Types for Part II of the Computer Science Tripos
Q Lecture Notes on Types for Part II of the Computer Science Tripos Prof. Andrew M. Pitts University of Cambridge Computer Laboratory c 2016 A. M. Pitts Contents Learning Guide i 1 Introduction 1 2 ML Polymorphism 6 2.1 Mini-ML type system . 6 2.2 Examples of type inference, by hand . 14 2.3 Principal type schemes . 16 2.4 A type inference algorithm . 18 3 Polymorphic Reference Types 25 3.1 The problem . 25 3.2 Restoring type soundness . 30 4 Polymorphic Lambda Calculus 33 4.1 From type schemes to polymorphic types . 33 4.2 The Polymorphic Lambda Calculus (PLC) type system . 37 4.3 PLC type inference . 42 4.4 Datatypes in PLC . 43 4.5 Existential types . 50 5 Dependent Types 53 5.1 Dependent functions . 53 5.2 Pure Type Systems . 57 5.3 System Fw .............................................. 63 6 Propositions as Types 67 6.1 Intuitionistic logics . 67 6.2 Curry-Howard correspondence . 69 6.3 Calculus of Constructions, lC ................................... 73 6.4 Inductive types . 76 7 Further Topics 81 References 84 Learning Guide These notes and slides are designed to accompany 12 lectures on type systems for Part II of the Cambridge University Computer Science Tripos. The course builds on the techniques intro- duced in the Part IB course on Semantics of Programming Languages for specifying type systems for programming languages and reasoning about their properties. The emphasis here is on type systems for functional languages and their connection to constructive logic. We pay par- ticular attention to the notion of parametric polymorphism (also known as generics), both because it has proven useful in practice and because its theory is quite subtle. -
Open C++ Tutorial∗
Open C++ Tutorial∗ Shigeru Chiba Institute of Information Science and Electronics University of Tsukuba [email protected] Copyright c 1998 by Shigeru Chiba. All Rights Reserved. 1 Introduction OpenC++ is an extensible language based on C++. The extended features of OpenC++ are specified by a meta-level program given at compile time. For dis- tinction, regular programs written in OpenC++ are called base-level programs. If no meta-level program is given, OpenC++ is identical to regular C++. The meta-level program extends OpenC++ through the interface called the OpenC++ MOP. The OpenC++ compiler consists of three stages: preprocessor, source-to-source translator from OpenC++ to C++, and the back-end C++ com- piler. The OpenC++ MOP is an interface to control the translator at the second stage. It allows to specify how an extended feature of OpenC++ is translated into regular C++ code. An extended feature of OpenC++ is supplied as an add-on software for the compiler. The add-on software consists of not only the meta-level program but also runtime support code. The runtime support code provides classes and functions used by the base-level program translated into C++. The base-level program in OpenC++ is first translated into C++ according to the meta-level program. Then it is linked with the runtime support code to be executable code. This flow is illustrated by Figure 1. The meta-level program is also written in OpenC++ since OpenC++ is a self- reflective language. It defines new metaobjects to control source-to-source transla- tion. The metaobjects are the meta-level representation of the base-level program and they perform the translation. -
Let's Get Functional
5 LET’S GET FUNCTIONAL I’ve mentioned several times that F# is a functional language, but as you’ve learned from previous chapters you can build rich applications in F# without using any functional techniques. Does that mean that F# isn’t really a functional language? No. F# is a general-purpose, multi paradigm language that allows you to program in the style most suited to your task. It is considered a functional-first lan- guage, meaning that its constructs encourage a functional style. In other words, when developing in F# you should favor functional approaches whenever possible and switch to other styles as appropriate. In this chapter, we’ll see what functional programming really is and how functions in F# differ from those in other languages. Once we’ve estab- lished that foundation, we’ll explore several data types commonly used with functional programming and take a brief side trip into lazy evaluation. The Book of F# © 2014 by Dave Fancher What Is Functional Programming? Functional programming takes a fundamentally different approach toward developing software than object-oriented programming. While object-oriented programming is primarily concerned with managing an ever-changing system state, functional programming emphasizes immutability and the application of deterministic functions. This difference drastically changes the way you build software, because in object-oriented programming you’re mostly concerned with defining classes (or structs), whereas in functional programming your focus is on defining functions with particular emphasis on their input and output. F# is an impure functional language where data is immutable by default, though you can still define mutable data or cause other side effects in your functions. -
Run-Time Type Information and Incremental Loading in C++
Run-time Type Information and Incremental Loading in C++ Murali Vemulapati Sriram Duvvuru Amar Gupta WP #3772 September 1993 PROFIT #93-11 Productivity From Information Technology "PROFIT" Research Initiative Sloan School of Management Massachusetts Institute of Technology Cambridge, MA 02139 USA (617)253-8584 Fax: (617)258-7579 Copyright Massachusetts Institute of Technology 1993. The research described herein has been supported (in whole or in part) by the Productivity From Information Technology (PROFIT) Research Initiative at MIT. This copy is for the exclusive use of PROFIT sponsor firms. Productivity From Information Technology (PROFIT) The Productivity From Information Technology (PROFIT) Initiative was established on October 23, 1992 by MIT President Charles Vest and Provost Mark Wrighton "to study the use of information technology in both the private and public sectors and to enhance productivity in areas ranging from finance to transportation, and from manufacturing to telecommunications." At the time of its inception, PROFIT took over the Composite Information Systems Laboratory and Handwritten Character Recognition Laboratory. These two laboratories are now involved in research re- lated to context mediation and imaging respectively. LABORATORY FOR ELEC RONICS A CENTERCFORENTER COORDINATION LABORATORY ) INFORMATION S RESEARCH RESE CH E53-310, MITAL"Room Cambridge, MA 02M142-1247 Tel: (617) 253-8584 E-Mail: [email protected] Run-time Type Information and Incremental Loading in C++ September 13, 1993 Abstract We present the design and implementation strategy for an integrated programming environment which facilitates specification, implementation and execution of persistent C++ programs. Our system is implemented in E, a persistent programming language based on C++. The environment provides type identity and type persistence, i.e., each user-defined class has a unique identity and persistence across compilations. -
Kodak Color Control Patches Kodak Gray
Kodak Color Control Patches 0 - - _,...,...,. Blue Green Yellow Red White 3/Color . Black Kodak Gray Scale A 1 2 a 4 s -- - --- - -- - -- - -- -- - A Study of Compile-time Metaobject Protocol :J/1\'-(Jt.-S~..)(:$/;:t::;i':_;·I'i f-.· 7 •[:]f-. :::JJl. (:~9 ~~Jf'?E By Shigeru Chiba chiba~is.s.u-tokyo.ac.jp November 1996 A Dissertation Submitted to Department of Information Science Graduate School of Science The Un iversity of Tokyo In Partial Fulfillment of the Requirements For the Degree of Doctor of Science. Copyright @1996 by Shigeru Chiba. All Rights Reserved. - - - ---- - ~-- -- -~ - - - ii A Metaobject Protocol for Enabling Better C++ Libraries Shigeru Chiba Department of Information Science The University of Tokyo chiba~is.s.u-tokyo.ac.jp Copyrighl @ 1996 by Shi geru Chiba. All Ri ghts Reserved. iii iv Abstract C++ cannot be used to implement control/data abstractions as a library if their implementations require specialized code for each user code. This problem limits programmers to write libraries in two ways: it makes some kinds of useful abstractions that inherently require such facilities impossi ble to implement, and it makes other abstractions difficult to implement efficiently. T he OpenC++ MOP addresses this problem by providing li braries the ability to pre-process a program in a context-sensitive and non-local way. That is, libraries can instantiate specialized code depending on how the library is used and, if needed, substitute it for the original user code. The basic protocol structure of the Open C++ MOP is based on that of the CLOS MOP, but the Open C++ MOP runs metaobjects only at compile time. -
1 Simply-Typed Lambda Calculus
CS 4110 – Programming Languages and Logics Lecture #18: Simply Typed λ-calculus A type is a collection of computational entities that share some common property. For example, the type int represents all expressions that evaluate to an integer, and the type int ! int represents all functions from integers to integers. The Pascal subrange type [1..100] represents all integers between 1 and 100. You can see types as a static approximation of the dynamic behaviors of terms and programs. Type systems are a lightweight formal method for reasoning about behavior of a program. Uses of type systems include: naming and organizing useful concepts; providing information (to the compiler or programmer) about data manipulated by a program; and ensuring that the run-time behavior of programs meet certain criteria. In this lecture, we’ll consider a type system for the lambda calculus that ensures that values are used correctly; for example, that a program never tries to add an integer to a function. The resulting language (lambda calculus plus the type system) is called the simply-typed lambda calculus (STLC). 1 Simply-typed lambda calculus The syntax of the simply-typed lambda calculus is similar to that of untyped lambda calculus, with the exception of abstractions. Since abstractions define functions tht take an argument, in the simply-typed lambda calculus, we explicitly state what the type of the argument is. That is, in an abstraction λx : τ: e, the τ is the expected type of the argument. The syntax of the simply-typed lambda calculus is as follows. It includes integer literals n, addition e1 + e2, and the unit value ¹º. -
The Number Line Topic 1
Topic 1 The Number Line Lesson Tutorials Positive numbers are greater than 0. They can be written with or without a positive sign (+). +1 5 +20 10,000 Negative numbers are less than 0. They are written with a negative sign (−). − 1 − 5 − 20 − 10,000 Two numbers that are the same distance from 0 on a number line, but on opposite sides of 0, are called opposites. Integers Words Integers are the set of whole numbers and their opposites. Opposite Graph opposites When you sit across from your friend at Ź5 ź4 Ź3 Ź2 Ź1 0 1234 5 the lunch table, you negative integers positive integers sit opposite your friend. Zero is neither negative nor positive. Zero is its own opposite. EXAMPLE 1 Writing Positive and Negative Integers Write a positive or negative integer that represents the situation. a. A contestant gains 250 points on a game show. “Gains” indicates a number greater than 0. So, use a positive integer. +250, or 250 b. Gasoline freezes at 40 degrees below zero. “Below zero” indicates a number less than 0. So, use a negative integer. − 40 Write a positive or negative integer that represents the situation. 1. A hiker climbs 900 feet up a mountain. 2. You have a debt of $24. 3. A student loses 5 points for being late to class. 4. A savings account earns $10. 408 Additional Topics MMSCC6PE2_AT_01.inddSCC6PE2_AT_01.indd 408408 111/24/101/24/10 88:53:30:53:30 AAMM EXAMPLE 2 Graphing Integers Graph each integer and its opposite. Reading a. 3 Graph 3. -
Query Rewriting Using Views in a Typed Mediator
Язык СИНТЕЗ как ядро канонической информационной модели Л. А. Калиниченко ([email protected]) C. А. Ступников ([email protected]) Lecture outline Objectives of the language Frame language: semi-structured capabilities Type system of SYNTHESIS Classes and metaclasses Assertions Processes, workflows Formulae facilities of the language 2 History of the language . Preliminary draft version published in 1991 . First version published in 1993 . English version prepared in 1995 and extended in 1997 . Current version in English published in 2007 3 The SYNTHESIS is a multipurpose language The language is oriented on the following basic kinds of application: . serving as a kernel of the canonical information model; . providing facilities for unifying representation of heterogeneous information models of different kinds (of data, services, processes, ontologies); . providing sufficient modeling facilities for formalized definitions of mediators for various subject domains; . supporting mediation-based design of information systems; . providing modeling facilities for mediator and resource specifications for the mediation architecture; . serving for semantic reconciliation of the mediator and resource specifications to form compositions of resources refining mediator specifications; . serving as an interface for users during problem formulation and solving in various applications 4 Mediation orientation . Two different approaches to the problem of integrated representation of multiple information resources for an IS are distinguished: 1. moving from resources to problems (resource driven approach) 2. moving from an application to resources (application driven approach) The SYNTHESIS language is oriented on support of application-driven approach for mediation-based IS development. The mediator's layer is introduced to provide the users with the metainformation uniformly characterizing subject definitions in the canonical information model – providing application domain conceptual schema . -
Problem Solving with Programming Course Material
Problem Solving with Programming Course Material PROBLEM SOLVING WITH PROGRAMMING Hours/Week Marks Year Semester C L T P/D CIE SEE Total I II 2 - - 2 30 70 100 Pre-requisite Nil COURSE OUTCOMES At the end of the course, the students will develop ability to 1. Analyze and implement software development tools like algorithm, pseudo codes and programming structure. 2. Modularize the problems into small modules and then convert them into modular programs 3. Apply the pointers, memory allocation techniques and use of files for dealing with variety of problems. 4. Apply C programming to solve problems related to scientific computing. 5. Develop efficient programs for real world applications. UNIT I Pointers Basics of pointers, pointer to array, array of pointers, void pointer, pointer to pointer- example programs, pointer to string. Project: Simple C project by using pointers. UNIT II Structures Basics of structure in C, structure members, accessing structure members, nested structures, array of structures, pointers to structures - example programs, Unions- accessing union members- example programs. Project: Simple C project by using structures/unions. UNIT III Functions Functions: User-defined functions, categories of functions, parameter passing in functions: call by value, call by reference, recursive functions. passing arrays to functions, passing strings to functions, passing a structure to a function. Department of Computer Science and Engineering Problem Solving with Programming Course Material Project: Simple C project by using functions. UNIT IV File Management Data Files, opening and closing a data file, creating a data file, processing a data file, unformatted data files. Project: Simple C project by using files. -
Integersintegersintegers Chapter 6Chapter 6 6 Chapter 6Chapter Chapter 6Chapter Chapter 6
IntegersIntegersIntegers Chapter 6Chapter 6 Chapter 6 Chapter 6 Chapter 6 6.1 Introduction Sunita’s mother has 8 bananas. Sunita has to go for a picnic with her friends. She wants to carry 10 bananas with her. Can her mother give 10 bananas to her? She does not have enough, so she borrows 2 bananas from her neighbour to be returned later. After giving 10 bananas to Sunita, how many bananas are left with her mother? Can we say that she has zero bananas? She has no bananas with her, but has to return two to her neighbour. So when she gets some more bananas, say 6, she will return 2 and be left with 4 only. Ronald goes to the market to purchase a pen. He has only ` 12 with him but the pen costs ` 15. The shopkeeper writes ` 3 as due amount from him. He writes ` 3 in his diary to remember Ronald’s debit. But how would he remember whether ` 3 has to be given or has to be taken from Ronald? Can he express this debit by some colour or sign? Ruchika and Salma are playing a game using a number strip which is marked from 0 to 25 at equal intervals. To begin with, both of them placed a coloured token at the zero mark. Two coloured dice are placed in a bag and are taken out by them one by one. If the die is red in colour, the token is moved forward as per the number shown on throwing this die. If it is blue, the token is moved backward as per the number 2021-22 MATHEMATICS shown when this die is thrown. -
The Gaudi Reflection Tool Crash Course
SDLT Single File Template 1 4. Sept. 2001 Version/Issue: 2/1 European Laboratory for Particle Physics Laboratoire Européen pour la Physique des Particules CH-1211 Genève 23 - Suisse The Gaudi Reflection Tool Crash Course Document Version: 1 Document Date: 4. Sept. 2001 Document Status: Draft Document Author: Stefan Roiser Abstract 1 Introduction The Gaudi Reflection Tool is a model for the metarepresentation of C++ classes. This model has to be filled with the corresponding information to each class that shall be represented through this model. Ideally the writing of this information and filling of the model will be done during the creation of the real classes. Once the model is filled with information a pointer to the real class will be enough to enter this model and retrieve information about the real class. If there are any relations to other classes, and the information about these corresponding classes is also part of the metamodel one can jump to these classes and also retrieve information about them. So an introspection of C++ classes from an external point of view can be realised quite easily. 2 The Meta Model The struture of the Gaudi Reflection Model is divided into four parts. These four parts are represented by four C++-classes. The corepart is the class called MetaClass. Within this class everything relevant to the real object will be stored. This information includes the name, a description, the author, the list of methods, members and constructors of this class etc. This class is also the entrypoint to the MetaModel. Using the MetaClass function forName and the Template page 1 SDLT Single File Template 2 The Meta Model Version/Issue: 2/1 string of the type of the real class as an argument, a pointer to the corresponding instance of MetaClass will be returned.