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Algorithms

Algorithms. Homework. None Next Week: Mid-Term Open book / Open laptop. Old Business. CFGs: Context Free Grammars (Ch8). Ambiguity. The Chomsky Hierarchy Type 0 > Type 1 > Type 2 > Type 3 Recursively Enumerable > CS > CF > Regular Examples Type 3: Regular (Finite State):

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Algorithms

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  1. Algorithms

  2. Homework • None • Next Week: Mid-Term • Open book / Open laptop

  3. Old Business

  4. CFGs: Context Free Grammars (Ch8)

  5. Ambiguity

  6. The Chomsky Hierarchy • Type 0 > Type 1 > Type 2 > Type 3 • Recursively Enumerable > CS > CF > Regular • Examples • Type 3: Regular (Finite State): • Grep & Regular Expressions • Right-Branching: A  a A • Left-Branching: B  B b • Type 2: Context-Free (CF): • Center-Embedding: C  …  x C y • Parenthesis Grammars: <expr>  ( <expr> ) • w wR • Type 1: Context-Sensitive (CS): w w • Type 0: Recursively Enumerable • Beyond Type 0: Halting Problem

  7. New Business

  8. http://ocw.mit.edu/OcwWeb/Electrical-Engineering-and-Computer-Science/6-006Spring-2008/LectureNotes/index.htmhttp://ocw.mit.edu/OcwWeb/Electrical-Engineering-and-Computer-Science/6-006Spring-2008/LectureNotes/index.htm

  9. Hash Tables n steps n2 time Linear Time n steps n time Constant Time

  10. Profiling

  11. O(n) << O(n2) Time n = |words|

  12. Excel

  13. Group Problems by Time Bounds • log(n) • Binary Search, Fibonacci • n • tr, cat, cut, uniq, egrep • n log n • Sort • n2 • Edit Distance (http://www.csse.monash.edu.au/~lloyd/tildeAlgDS/Dynamic/Edit/) • n3 • Matrix Multiplication n2.376 • Context-Free Parsing • 2n • Factoring • Satisfiability

  14. http://ocw.mit.edu/NR/rdonlyres/Electrical-Engineering-and-Computer-Science/6-046JFall-2005/LectureNotes/lec3.pdfhttp://ocw.mit.edu/NR/rdonlyres/Electrical-Engineering-and-Computer-Science/6-046JFall-2005/LectureNotes/lec3.pdf

  15. Summary (for Mid-Term) • Symbolic Processing v. Stat Packages • Practical • Symbolic: LISP, Wolfram Alpha • Stats: Newton’s Method, R • Fundamental • Stuff you can’t do with stat packages • LISP: Recursion, Eval, Symbolic Differentiation • Lambda Calculus • Small is Beautiful (beyond reason) • Church’s Thesis & Computability Theory

  16. Summary (continued) • Unix for Poets • Pipes: tr | sort | uniq –c • Python & NLTK • Unix for Poets (without Unix) • Objects • Polymorphism • Equivalence Classes • Partial Orders • Algorithms • Group problems by time & space bounds

  17. Introduction to Programming Traditional (Start with Definitions) Non-Traditional (Start with Examples) Recursion def fact(x): if(x <= 1): return 1 else: return x * fact(x-1) Streams: Unix Pipes Briefly mentioned Everything else • Constants: 1 • Variables:x • Objects: • lists, strings, arrays, matrices • Expressions: x == 1 • Statements: Side Effects • x = 1 (assignment) • Conditionals: • If (x<=1) return 1; • Iteration: for loops • Functions • Recursion • Streams

  18. Examples • Factorial • Fibonacci • Counting Words in a Text

  19. Python def fact(x): if(x <= 1): return 1 else: return x * fact(x-1) def fact2(x): result=1 for i in range(x): result *=(i+1); return result Recursion Iteration

  20. Out-takes

  21. Cosine Distance

  22. Inner Products

  23. Inner Product in Python def bag_of_words(words): D = {} for w in words: if(D.has_key(w)): D[w] = D[w] + 1; else: D[w]=1; return D def inner_product(D1,D2): sum = 0.0 for key in D1: if key in D2: sum += D1[key] * D2[key] return sum

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