l1 - Lecture 1: Introduction Prof. David L. Dill Department...

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Lecture 1: Introduction Prof. David L. Dill Department of Computer Science 1
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Outline What is this course about? What is it good for? Course administration. Basic concepts: Strings and languages. Guidelines for proofs. Reading: Chapter 1 of the textbook 2
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Representing Sets Supppose you, as a programmer, need to represent a small, finite, set. What does “represent” mean? Answer: You can answer questions about it, perform computations on it. Simple common question (membership): Is x S ? Other questions: Is S = ? Is S T = ? Etc. Ok, suppose you want to a finite representation of infinite sets. How do you do it? 3
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One view of formal language theory Automata and complexity theory is concerned with properties of formal languages . (A “formal language” is just an impressive name for a set of strings.) In formal language, automata, and complexity theory, a language is just a set of strings. Like many mathematical definitions, this leaves behind most of what we think of as “languages,” but can be made precise. And it leads to very profound results. Formal languages are only interesting when they are infinite. I claim: Something can be represented in a computer iff it can be written as a string (e.g., integers, floating point numbers, graph structures, etc.) 4
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Self-reference Formal language theory becomes almost surrealistic because of “self reference.” Finite representation = a string, so we can talk about languages of other (finitely representable) languages . E.g., we can talk about the language of all Turing machines that halt, and then wonder about the properties of that language. This leads to special proof techniques, and can tie your neurons into knots. 5
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What is a representation? Suppose you have representation that can be stored in a computer. Can all sets be represented? No: Compare the number of possible strings (which is countable) with the number of sets of strings (uncountable). A particular set that cannot be represented is the set of all irrational numbers – there are “too many” irrational numbers. This raises profound questions: Which sets can be represented on a computer and which can’t? 6
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Questions from formal language theory What (infinite) sets are representable? What can a computer do with the representations, in theory?
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l1 - Lecture 1: Introduction Prof. David L. Dill Department...

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