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Understanding Context-Free Grammars and Algorithmic Complexity for Mid-Term Exam

This study guide covers essential concepts for the upcoming mid-term exam, focusing on Context-Free Grammars (CFGs), the Chomsky Hierarchy, and algorithmic complexity. Topics include types of grammars (Type 0 to Type 3), examples of regular and context-free languages, and practical applications of algorithms. Additionally, it explores concepts such as recursion, dynamic programming, and symbolic processing using various programming tools. Review the provided links and examples to reinforce understanding of these key computational theories and algorithms.

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Understanding Context-Free Grammars and Algorithmic Complexity for Mid-Term Exam

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Presentation Transcript

  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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