Computation. Philosophical Issues. Instructor: Viola Schiaffonati. March, 26 th 2018

Transcription

1 Computation Philosophical Issues Instructor: Viola Schiaffonati March, 26 th 2018

2 Computer science: what kind of object? 2 Computer science: science/disciplines of computersor of computation? History of computers in terms of goals To realize a machine able to perform computations To establish foundations for mathematics Computer as a machine that can take, as input, patterns that describe changes to themselves and other patterns, and that causes the described changes to occur (Hayes 1997)

3 Computation 3 Central concept for computer science and for philosophical reflections about computer science Cluster conceptthat can be characterized in different ways (Scheutz 2002)s Intuitive perspective Historical perspective Philosophical perspective

4 Origins 4 Computation derives from the Latin com-putare which means to determine or to calculate especially by mathematical means Within computer science, computation means the execution of algorithms What is an algorithm? What does it mean to execute an algorithm?

5 Algorithm (intuitive definition) 5 Finite set of instructions, which operate on certain entities (symbols, representations of numbers, etc.) and can be implemented in some mechanism To execute an algorithm means to have the mechanismto carry out the instructions for any given input in a deterministic, discrete, stepwise fashion The mechanism goes through a sequence of atomic stepsin such a way that (one or more of) these steps correspond to some instruction, for all the instructions specified by the algorithm

6 Algorithms and mechanisms 6 Algorithmic specification will take different forms (depending on the kind of mechanism) In the case of computers it is expressed as a programming language In the case of humans instructions may be given in ordinary language(individual steps must be clearly distinguishable and described with sufficient precision) The concept of algorithm is tightly connected to that of mechanically realizable processes What is a mechanism Precise specification of the notion of algorithm

7 Historical perspective 7 Philosophers have been always analyzing the possibility to realize mechanical systemsto help humans making calculations (Davies 2012) First mechanical calculators in XVII century (Pascal) Leibniz and his calculus of thought (calculus ratiocinator) The idea of reasoningas a mechanical processis at the roots of the notion of computation From the beginning computation is connected to the idea of mechanically manipulated representations

8 The turning point 8 These ideas are formally developed during XX century and are concretely applied in the construction of the first electronic calculators developed starting from 1940s Major progresses in the construction of computersand the conception of computing are due to two independent developments The logical analysisof the notions formal systemand formal proof(leading to further studies of notions as effectively computable function and algorithm) The rapid progression in the engineering of electronic components

9 A logico-philosophical perspective 9 In the 1930s logicians start working for a well-defined formal notion of computationin order to make the intuitive notion of computation formally precise They focus on the class of functionsthat can be effectively calculated in principle(being logicians and digital computers not existing yet) Mathematician Alonzo Church (1936) is the first to give this class of effective calculable functions a formal characterization through a definition postulate In the same year Alan Turingintroduces his machine model to define A notion of effective procedure or algorithm A notion of function computed by an algorithm

10 On Computable Numbers(Turing 1936) 10 How to demonstrate that a procedure able to perform the requested task in an automatic way has not been invented yet and will not be invented in the future? The concept of mechanical process needs to be conceived in a precise and rigorous way The Turing Machine

11 Computability and algorithms 11 A function fis computablemeans by definition that there exists an algorithm that computes f An algorithm for a given problem Pcan be characterized as a finite procedure(a finite set of instructions) to solve P such that The algorithm is not ambiguousfor both the computer and the human being that will perform it The algorithm is effective, namely it must terminate and give back a correct solution to P

12 Turing machine 12 Unbounded tapedivided into squares, each of which can hold exactly one symbol, a tape headfor reading and writing symbols from a given alphabet on the square, and a controller, which is exactly one of finitely many states at any given time Each computational step involves, first, reading the symbol under the tape headand then, depending on the current state of the controller, writing a new symbol on the square, possibly switching to another state and possibly moving the tape head one square to the left or to the right

13 Action and configuration 13

14 Turing s approach 14 The notion of algorithm receives a satisfactory account only after Turing (1936) has introduced his machine model of a (human) computer This model results from Turing s analysis of the possible processes a human( the computer ) can go through while performing a calculation using paper and pencil applying rules from a given finite set The human computer follows the rules blindly, without using any insight or ingenuity

15 Turing: at the basis of computation 15 Five major constraints for doing automatic computations 1. Only a finite numberof symbolscan be written down and used in any computation 2. There is a fixed boundon the amount of scratch paper(and the symbols on it) that a human can take in at a time in order to decide what to do next 3. At any time a symbolcan be written downor erased(in a certain area on the scratch paper called cell ) 4. There is a upper limitto the distance between cells that can be considered in two consecutives computational steps 5. There is an upper boundto the number of states of mind a human can be in, and the current state of mind together with the last symbol written or erased determine what to do next

16 The type of model 16 Mathematicalmodel of an imagined mechanical device satisfying conditions 1-5 Idealizedmodel of human computing since it could process and store arbitrarily long, finite sequences of symbols Abstractmodel since it only captures high-level processes that take place in humans when they compute (as opposed to low-level neuronal processes, for example)

17 Turing computability 17 Any function computable by a humanbeing following fixed rules can be computed by a Turing machine Every function computed by a Turing machinecould (in principle) be computed by a human computer This equivalence does not preclude humans from being able to find answers to problems which no Turing machine can compute (for example using intuition)

18 Computational model 18 Three similar, but different expressions To be computable To be computational To be a computer

19 To be computable 19 X is computable Xis a functionfrom a numerable domain to a numerable codomain For each argument n of X,the corresponding value X(n) can be obtained by adopting a mechanical calculus(algorithm)

20 A task to be computable 20 A task is computableif it is possible to specify a sequence of instructionswhich will result in the completion of the taskwhen they are carried out by some machine Set of instructions called effective procedure or algorithm What counts as an effective procedure may dependon the capabilitiesof the machineused to carry out the instructions

21 Again on Turing computability 21 Turing s proposal of a class of devices known as Turing machines and lead to the formal notion of computation (Turing s computability) A task is Turing computable if it can be carried out by some Turing machine

22 Church-Turing thesis 22 The proposition that Turing s notion captures exactly the intuitive idea of effective procedure is called the Church- Turing thesis This proposition is not provable, since it is a claim about the relationship between a formal concept and intuition The thesis would be refuted by an intuitively accepted algorithm for a task that is not Turing-computable No such counterexample has been found

23 On the naturalness of computability 23 Other independently defined notions of computability Lambda calculus, Recursive functions, Post machines They have been shown to be equivalent to Turingcomputability This indicates that there is at least something natural about this notion of computability

24 To be computational 24 X is computational X is a model(representing something) Model X is formulated in terms of computable functions By extension Computational is a disciplinedeveloping computational models

25 To be a computer 25 X is a computer (Copeland 1996) (Hayes 1997) X is a physical system executing programs In the structure of the systems two distinct but connected components are identifiable Representation of a program Executor of a program

26 References 26 Church, A. (1936) An unsolvable problem of elementary number theory, American Journal of Mathematics 58: Hayes, P. (1997) WhatIsa Computer? An Electronic Discussion, The Monist 80:3 Davies, M. (2012) The Universal Computer: The Road from Leibniz to Turing, Taylor & Francis Group Scheutz, M. (2002) Philosophical Issues about Computation, Encyclopedia of Cognitive Science, Macmillan, London Turing, A. (1936) On computablenumbers, with an applicationto the Entscheidungsproblem, Proceedings of the London Mathematical Society, series 2, 42: