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l1-handout

# l1-handout - Outline What is this course about What is it...

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Lecture 1: Introduction Prof. David L. Dill Department of Computer Science 1 Outline What is this course about? What is it good for? Course administration Basic concepts: Strings, languages, and problems. Proof expectations. Reading: Chapter 1 of the textbook 2 Representing Sets (discussion) Supppose you, as a programmer, need to represent a small, finite, set. What does “represent” mean? (what questions/operations?) Answer: You can answer questions about it. Simple common question: Is x S ? Other questions: Is S = ? Is S T = ? Etc. What representations would be appropriate? Ok, suppose you want to represent infinite sets. How do you do it? What does “represent” mean? (what questions/operations?) Answer: You can answer questions about it. Simple common question: Is x S ? What representations would be appropriate? That’s what the course is about. 3 One view of formal language theory Automata and complexity theory is concerned with properties of formal languages . 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.) Basically, any object or value that is of interest to computer science can be represented as a string. So a set of anything can be considered a language. 4

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What is a representation? Suppose you have representation that can be stored in a computer. Can all sets be represented? [Poll: how many students know about countably infinite vs. uncountably infinite?] No: Compare the number of possible strings (which is countable) with the number of sets of string (uncountable). A particular set that cannot be represented is the set of all irrational numbers – there are “too many” irrational numbers.
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