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5644 results in Programming Languages and Applied Logic

Page 72 of 283

  • 2 - Coalgebras of Polynomial Functors

  • Bart Jacobs, Radboud Universiteit Nijmegen
  • Book:
    Introduction to Coalgebra
    Published online:
    22 December 2016
    Print publication:
    27 October 2016, pp 33-113

  • Summary

    The previous chapter has introduced several examples of coalgebras and has illustrated basic coalgebraic notions such as behaviour and invariance (for those examples). This chapter will go deeper into the study of the area of coalgebra, introducing some basic notions, definitions and terminology. It will first discuss several fundamental set theoretic constructions, such as products, coproducts, exponents and powersets in a suitably abstract (categorical) language. These constructs are used to define a collection of elementary functors, the so-called polynomial functors. As will be shown in Section 2.2, this class of functors is rich enough to capture many examples of interesting coalgebras, including deterministic and non-deterministic automata. One of the attractive features of polynomial functors is that almost all of them have a final coalgebra – except when the (non-finite) powerset occurs. The unique map into a final coalgebra will appear as behaviour morphism, mapping a state to its behaviour. The two last sections of this chapter, Sections 2.4 and 2.5, provide additional background information, namely on algebras (as duals of coalgebras) and on adjunctions. The latter form a fundamental categorical notion describing backand- forth translations that occur throughout mathematics.

    Constructions on Sets

    This section describes familiar constructions on sets, such as products, coproducts (disjoint unions), exponents and powersets. It does so in order to fix notation and to show that these operations are functorial, i.e. give rise to functors. This latter aspect is maybe not so familiar. Functoriality is essential for properly developing the theory of coalgebras; see Definition 1.4.5.

    These basic constructions on sets are instances of more general constructions in categories. We shall give a perspective on these categorical formulations, but we do not overemphasise this point. Readers without much familiarity with the theory of categories may then still follow the development, and readers who are quite comfortable with categories will recognise this wider perspective anyway.

    Products

    We recall that for two arbitrary sets X, Y the product X × Y is the set of pairs

    There are then obvious projection functions π1 : X×Y → X and π2 : X×Y → Y by π1(x, y) = x and π2(x, y) = y.

  • 4 - Logic, Lifting and Finality

  • Bart Jacobs, Radboud Universiteit Nijmegen
  • Book:
    Introduction to Coalgebra
    Published online:
    22 December 2016
    Print publication:
    27 October 2016, pp 159-245

  • Summary

    The previous three chapters have introduced some basic elements of the theory of coalgebras, focusing on coalgebraic system descriptions, homomorphisms, behaviour, finality and bisimilarity. So far, only relatively simple coalgebras have been used, for inductively defined classes of polynomial functors, on the category Sets of sets and functions. This chapter will go beyond these polynomial functors and will consider other examples. But more important, it will follow a different, more systematic approach, relying not on the way functors are constructed but on the properties they satisfy – and work from there. Inevitably, this chapter will technically be more challenging, requiring more categorical maturity from the reader.

    The chapter starts with a concrete description of two new functors, namely the multiset and distribution functors, written as M and D, respectively. As we shall see, on the one hand, from an abstract point of view, they are much like powerset P, but on the other hand they capture different kinds of computation: D is used for probabilistic computation and M for resourcesensitive computation.

    Subsequently, Sections 4.2–4.5 will take a systematic look at relation lifting – used in the previous chapter to define bisimulation relations. Relation lifting will be described as a certain logical operation, which will be developed on the basis of a moderate amount of categorical logic, in terms of so-called factorisation systems. This will give rise to the notion of ‘logical bisimulation’ in Section 4.5. It is compared with several alternative formulations. For weak pullback-preserving functors on Sets these different formulations coincide. With this theory in place Section 4.6 concentrates on the existence of final coalgebras. Recall that earlier we skipped the proof of Theorem 2.3.9, claiming the existence of final coalgebras for finite Kripke polynomial functors. Here we present general existence results, for ‘bounded’ endofunctors on Sets. Finally, Section 4.7 contains another characterisation of simple polynomial functors in terms of size and preservation properties. It also contains a characterisation of more general ‘analytical’ functors, which includes for instance the multiset functor M.

    Multiset and Distribution Functors

    A set is a collection of elements. Such an element, if it occurs in the set, occurs only once. This sounds completely trivial. But one can imagine situations in which multiple occurrences of the same element can be relevant.

  • Preface

  • Bart Jacobs, Radboud Universiteit Nijmegen
  • Book:
    Introduction to Coalgebra
    Published online:
    22 December 2016
    Print publication:
    27 October 2016, pp vii-xvi

  • Summary

    Mathematics is about the formal structures underlying counting, measuring, transforming etc. It has developed fundamental notions like number systems, groups, vector spaces (see e.g. [345]) and has studied their properties. In more recent decades also ‘dynamical’ features have become a subject of research. The emergence of computers has contributed to this development. Typically, dynamics involves a ‘state of affairs’, which can possibly be observed and modified. For instance, the contents of a tape of a Turing machine contribute to its state. Such a machine may thus have many possible states and can move from one state to another. Also, the combined contents of all memory cells of a computer can be understood as the computer's state. A user can observe part of this state via the screen (or via the printer) and modify this state by typing commands. In reaction, the computer can display certain behaviour. Describing the behaviour of such a computer system is a non-trivial matter. However, formal descriptions of such complicated systems are needed if we wish to reason formally about their behaviour. Such reasoning is required for the correctness or security of these systems. It involves a specification describing the required behaviour, together with a correctness proof demonstrating that a given implementation satisfies the specification.

    Both mathematicians and computer scientists have introduced various formal structures to capture the essence of state-based dynamics, such as automata (in various forms), transition systems, Petri nets and event systems. The area of coalgebra has emerged within theoretical computer science with a unifying claim. It aims to be the mathematics of computational dynamics. It combines notions and ideas from the mathematical theory of dynamical systems and from the theory of state-based computation. The area of coalgebra is still in its infancy but promises a perspective on uniting, say, the theory of differential equations with automata and process theory and with biological and quantum computing, by providing an appropriate semantical basis with associated logic. The theory of coalgebras may be seen as one of the original contributions stemming from the area of theoretical computer science.

  • 17 - Exception Handling in C++

  • from PART-II - Object Oriented Programming
  • Kunal Pimparkhede
  • Book:
    Computer Programming with C++
    Published online:
    28 May 2018
    Print publication:
    18 October 2016, pp 969-993

  • Summary

    Overview

    Exception is an abnormal situation in the program that can terminate the program if it is not handled by the programmer. Some examples of exception are as follows:

  • 1. If you accidently perform division by zero in the program, then it is called division by zero exception

  • 2. If you access array elements beyond the specific boundaries of an array index, then it is called array index out of bounds exception. For example, if the size of array a is 5 and if you accidently try to access a[7] that does not exist, then it could give logically incorrect results, which is an exception

  • 3. If your program is required to take salary of an employee as input from the user and by mistake the user inputs a negative number, then it is called an invalid input exception

  • 4. If your program is writing to a file but the file is read-only, then there is an access rights violation in performing the output operation, which is also an example of exception

  • 5. In network communication, if you are writing your own C++ client to communicate with the server on the same network and if server is down when you are sending messages through the client program, then it is called ‘Resource Not available’ type of exception

    • There are many more specific types of exceptions that may occur in your program from case to case. What is important is to understand what exceptions are and how effectively we can handle them to ensure that the program does not terminate abruptly when exception occurs. It is the programmer's responsibility to create robust programs that do not fail abruptly even if any abnormal situation occurs.

    NOTES

    There is a difference between an exception and an error. If there is an error in the program, the program will certainly fail. For example, a semicolon missing after a statement is an example of syntax error; program going out of memory while running a large loop is an example of memory error and so on. Errors cannot be handled and hence the program will certainly fail when an error occurs. Whereas, exception is something that can be handled in the program thereby avoiding the abrupt failure of the program. Exception handlers in C++ are used to handle exceptions (not errors).

Page 72 of 283