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Welcome to Honors Intro to CS Theory

Welcome to Honors Intro to CS Theory. Introduction to CS Theory (Honors & Traditional): formalization of computation various models of computation (increasing difficulty/power) what can / cannot be done ?. Why a theory course ?

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Welcome to Honors Intro to CS Theory

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  1. Welcome to Honors Intro to CS Theory • Introduction to CS Theory (Honors & Traditional): • formalization of computation • various models of computation (increasing difficulty/power) • what can / cannot be done ? • Why a theory course ? • relevant to practice (grammars for programming languages, finite automata & regular expressions for pattern matching of strings, NP-completeness to determine required time complexity – e.g. for cryptography) • problem solving skills independent of current technology (specific programming languages, etc.), ability to express ideas clearly, succinctly, and correctly

  2. Welcome to Honors Intro to CS Theory • Honors vs. Traditional Intro to CS Theory • Book: M. Sipser, Introduction to the Theory of Computation • faster speed through introductory topics and simpler models, more time for advanced topics (decidability, complexity) • more challenging and non-traditional homeworks • Honors vs. Traditional Intro to CS Theory • Book: M. Sipser, Introduction to the Theory of Computation • faster speed through introductory topics and simpler models, more time for advanced topics (decidability, complexity) • more challenging and non-traditional homeworks • You should: • be very comfortable with discrete math • have fun (a course full of puzzles! )

  3. [Chapter 0] Introduction Automata Theory – mathematical models of computation Computability Theory – what can be computed ? Complexity Theory – which problems are computationally hard / easy ? • Need math background • review Chapter 0 • discrete math quiz next class

  4. [Chapter 0] Strings and Languages Alphabet – non-empty finite set of symbols, typically denoted by § or ¡, e.g. §1 = { 0,1 }, §2 = { a,b,c,d }, ¡ = { #,$,0,1,2 } String over an alphabet – a finite sequence of symbols from the alphabet, e.g. w1 = 00101 over §1, w2 = badcab over §2 The length of a string w over § (the number of symbols in w) is denoted |w|. The string with no symbols is called the empty string and denoted ε.

  5. [Chapter 0] Strings and Languages • Operations on strings (let w = w1w2…wn): • reverse: wR = wnwn-1…w1 • substring: wiwi+1…wj • concatenation of w with a string z = z1z2…zm: • wz = w1w2…wnz1z2…zm • wk means concatenation of k copies of w • lexicographic ordering of strings: first by length, then “alphabetically,” e.g for § = {0,1}: • ε,0,1,00,01,10,11,000,…

  6. [Chapter 0] Strings and Languages Language: a set of strings over an alphabet §, e.g. { a, ab, bab }, ;, { ε }, { w over {0,1} | w contains more 1’s than 0’s } [throughout the book, e.g. page 44]

  7. [Chapter 0] Strings and Languages • Language: a set of strings over an alphabet §, e.g. • { a, ab, bab }, ;, { ε }, • { w over {0,1} | w contains more 1’s than 0’s } • Operations on languages: • typical set operations: [, Å, etc. • concatenation: L1.L2 = { w1w2 | w12 L1, w22 L2 } [throughout the book, e.g. page 44]

  8. [Chapter 0] Strings and Languages • Language: a set of strings over an alphabet §, e.g. • { a, ab, bab }, ;, { ε }, • { w over {0,1} | w contains more 1’s than 0’s } • Operations on languages: • typical set operations: [, Å, etc. • concatenation: L1.L2 = { w1w2 | w12 L1, w22 L2 } • Kleene’s star: L* = [k=0…1 Lk • reverse: LR = { wR | w 2 L } • Note: a language: L µ§* [throughout the book, e.g. page 44]

  9. [Chapter 0] Strings and Languages • Language: a set of strings over an alphabet §, e.g. • { a, ab, bab }, ;, { ε }, • { w over {0,1} | w contains more 1’s than 0’s } • Operations on languages: • - Kleene’s star: L* = [k=0…1 Lk • Note: a language: L µ§* [throughout the book, e.g. page 44]

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