INTRODUCTION TO COMPUTER THEORY PDF
and the theory of computation / John C. Martin.—4th ed. p. cm. Introduction to Languages and the Browse's Introduction to the Symptoms & Signs of Surgical . Introduction to Computer Theory by Daniel I. a. Cohen - Second Edition - Ebook download as PDF File .pdf), Text File .txt) or read book online. Introduction to. Library of Congress Cataloging-in-Publication Data: Cohen, Daniel 1. A., Introduction to computer theory. Includes Index. 1. Electronic digital computers.
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How do I view solution manuals on my smartphone? Given any algebraic problem having a specified number of linear equations, in a specified set of unknowns, with specified coefficients, a system had been developed called linear algebra that would guarantee one could decide weather the equations had any simultaneous solution at all, and find the solutions if they did exist.
Thi s would have been an even more satisfactory situation than existed in Euclidean geometry at the time.
If we are presented with a correct Euclidean proposition relating line segments and angles in a certain diagram, we have no guidance as to how to proceed to pro duce a mathematically rigorous proof of its truth. We have to be creative - we may make false starts, we may get completely lost, frustrated, or angry. We may never fi n d the proof. Linear algebra guarantees that none of this will ever happen with equations.
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As long as we are tireless and precise in following the rules, we must prevail, no matter how little imagination we ourselves possess.
Notice how well this de scribes the nature of a computer. Today, we might rephrase Hilbert's request as a demand for a set of computer programs to solve mathematical problems. When we input the problem, the machine generates the proof.
It was not easy for mathematicians to figure out how to follow Hilbert 's plan. Math ematicians are usually in the business of creating the proofs themselves, not the proof-gener ating techniques. What had to be invented was a whole field of mathematics that dealt with algorithms or procedures or programs we use these words interchangeably. From this we see that even before the first computer was ever built, some people were asking the question of what programs can be written.
It was necessary to codify the universal language in which algorithms could be stated. Addition and circumscribing circles were certainly allowable steps in an algorithm, but such activities as guessing and trying infinitely many possibilities at once were definitely prohibited. The language of algorithms that Hilbert required evolved in a natural way into the language of computer programs. The road to studying algorithms was not a smooth one.
The first bump occurred in when Kurt Godel proved that there was no algorithm to provide proofs for all the true state ments in mathematics. In fact, what he proved was even worse.
He showed that either there were some true statements in mathematics that had no proofs, in which case there were cer tainly no algorithms that could provide these proofs, or else there were some false state ments that did have proofs of their correctness, in which case the algorithm would be disas trous. Mathematicians then had to retreat to the question of what statements do have proofs and how can we generate these proofs?
The people who worked on this problem. They each fashioned various but simi lar versions of a univer sal model for all algorithms- what, from our perspective, we wou ld call a un iversal al gorithm machine.
Turing then went one step farther. He proved that there were mathematically definable fundamental questions about the machine itself that the ma chine could not answer. On the one hand, this theorem completely destroyed all hope of ever achieving any part of Hilbert's program of mechanizing mathematics, or even of deciding which classes of problems had mechanical answers. On the other hand, Turing 's theoretical model for an al gorithm machine employing a very simple set of mathematical structures held out the possi bility that a physical model of Turing's idea could actually be constructed.
If some human could figure out an algorithm to solve a particular class of mathematical problem, then the machine could be told to follow the steps in the program and execute this exact sequence of instructions on any inserted set of data tirelessly and with complete precision.
The electronic discoveries that were needed for the implementation of such a dev ice in cluded vacuum tubes, which just coincidentally had been developed recently for engineering purposes completely unrelated to the possibility of building a calculating machine.
This was another fortuitous phenomenon of this period of history.
Introduction To Computer Theory By Daniel I. A Cohen 2nd Edition
All that was required was the impe tus for someone with a vast source of money to be motivated to invest in this highly specula tive project. It is practically sacrilegious to maintain that World War II had a serendipitous impact on civil ization no matter how unintentional, yet it was exactly in this way that the first computer was born - sponsored by the Allied military to break the German secret code, with Turing himself taking part in the construction of the machine.
What started out as a mathematical theorem about mathematical theorems-an abstrac tion about an abstraction - became the single most practically applied invention since the wheel and axle. Not only was this an ironic twist of fate, but it all happened within the re markable span of IO years. It was as incredible as if a mathematical proof of the existence of intelligent creatures in outer space were to provoke them to land immediately on Earth.
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Independently of all the work being done in mathematical logic, other fields of science and social science were beginning to develop mathematical models to describe and analyze difficult problems of their own. As we have noted before, there is a natural correspondence between the study of models of computation and the study of linguistics in an abstract and mathematical sense.
It is also natural to assume that the study of thinking and learning branches of psychology and neurology-play an important part in understanding and facili tating computer theory. What is again of singular novelty is the historical fact that, rather than turning their attention to mathematical models to computerize their own applications, their initial development of mathematical models for aspects of their own science directly aided the evolution of the computer itself.
It seems that half the intel lectual forces in the world were leading to the invention of the computer, while the other half were producing ap plications that were desperate for its arrival. Two neurophysiologists, Warren McCulloch and Walter Pitts, constructed a mathemati cal model for the way in which sensory receptor organs in animals behave. The model they constructed for a "neural net" was a theoretical machine of the same nature as the one Turing invented, but with certain limitations.
Modern linguists, some influenced by the prevalent trends in mathematical logic and some by the emerging theories of developmental psychology, had been investigating a very similar subject: What is language in general? How could primitive humans have developed language? How do people understand it? How do they learn it as children?
How do people construct sentences from the ideas in their minds? Noam Chomsky created the subject of mathematical models for the description of lan guages to answer these questions.
His theory grew to the point where it began to shed light on the study of computer languages. The languages humans invented to communicate with one another and the languages necessary for humans to communicate with machines shared many basic properties.
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Although we do not know exactly how humans understand language, we do know how machines digest what they are told. Thus, the formulations of mathematical logic became usefu l to linguistics, a previously nonmathematical subject.
Metaphorical ly, we could say that the computer then took on linguistic abilities. It became a word processor, a translator, and an interpreter of simple grammar, as well as a compiler of computer lan guages. The software invented to interpret programming languages was applied to human languages as well. One point that will be made clear in our studies is why computer lan guages are easy for a computer to understand, whereas human languages are very difficult.
Because of the many influences on its development, the subject of this book goes by various names. It includes three major fundamental areas: the theory of automata, the the ory of formal languages, and the theory of Turing machines. This book is divided into three parts corresponding to these topics. Our subject is sometimes called computation theory rather than computer theory, be cause the items that are central to it are the types of tasks algorithms or programs that can be performed, not the mechanical nature of the physical computer itself.
However, the name "computation" is misleading, since it popularly connotes arithmetical operations which com prise only a fraction of what computers can do. The term computation is inaccurate when de scribing word processing, sorting, and searching and awkward in discussions of program verification.
Just as the term "number theory" is not limited to a description of calligraphic displays of number systems but focuses on the question of which equations can be solved in integers, and the term "graph theory" does not include bar graphs, pie charts, and his tograms, so too "computer theory" need not be limited to a description of physical machines but can focus on the question of which tasks are possible for which machines.
We shall study different types of theoretical machines that are mathematical models for actual physical processes. By considering the possible inputs on which these machines can work, we can analyze their various strengths and weaknesses.
We then arrive at what we may believe to be the most powerful machine possible.
When we do, we shall be surprised to find tasks that even it cannot perform. This will be our ultimate result, that no matter what machine we build, there will always be questions that are simple to state that it cannot an swer. Along the way, we shall begin to understand the concept of computability, which is the foundation of further research in this field. This is our goal. Computer theory extends further to such topics as complexity and verification, but these are beyond our intended scope.
Even for the topics we do cover- automata, languages, Turing machines - much more is known than we present here. As intriguing and engaging as the field has proven so far, with any luck the most fascinating theorems are yet to be discovered.
There is a certain parallelism between the fact that groups of letters make up words and the fact that groups of words make up sentences. Not all collections of letters form a val id word, and not all collections of words form a valid sentence. The analogy can be continued.
Certain groups of sentences make up coherent paragraphs, certain groups of paragraphs make up coherent stories, and so on.
What is more important to note is that, to a large degree, humans agree on which sequences are valid and which are not. How do they do that? This situation also exists with computer languages. Certain strings of words are recognizable commands. Certain sets of commands become a program with or without data that can be compiled, which means translated into machine commands.We begin.
These definitions have the same self-referential sense.
If the very next letter is another a. Just as the term "number theory" is not limited to a description of calligraphic displays of number systems but focuses on the question of which equations can be solved in integers. Those strings that are permissible in the language we call words.
Transition graphs were invented by John Myhill in 1 for reasons revealed in the next chapter. This means that any picture that represents an FA can be interpreted as a picture of a TG.
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