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# RE - Harvard CS 121 and CSCI E-207 Lecture 6 Optimality of...

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Harvard CS 121 and CSCI E-207 Lecture 6: Optimality of the Subset Construction Regular Expressions Harry Lewis September 22, 2009 Reading: Sipser, § 1.3.

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Harvard CS 121 & CSCI E-207 September 22, 2009 1976 – Michael O. Rabin See the ACM Author Profile in the Digital Library Citation For their joint paper "Finite Automata and Their Decision Problem," which introduced the idea of nondeterministic machines, which has proved to be an enormously valuable concept. Their (Scott & Rabin) classic paper has been a continuous source of inspiration for subsequent work in this field. Biographical Information Michael O. Rabin (born 1931 in Breslau, Germany) is a noted computer scientist and a recipient of the Turing Award, the most prestigious award in the field. 1
Harvard CS 121 & CSCI E-207 September 22, 2009 2

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Harvard CS 121 & CSCI E-207 September 22, 2009 The Subset Construction Causes Exponential Blowup The subset construction shows that any n -state NFA can be implemented as a 2 n -state DFA. NFA States DFA States 4 16 10 1024 100 2 100 1000 2 1000 the number of particles in the universe How to implement this construction on ordinary digital computer? NFA states DFA state bit vector 1 , . . . , n 1 0 1 0 . . . 1 1 2 n 3
Harvard CS 121 & CSCI E-207 September 22, 2009 Is this construction the best we can do? Could there be a construction that always produces an n 2 state DFA for example? Theorem: For every n 1 , there is a language L n such that 1. There is an ( n + 1) -state NFA recognizing L n . 2. There is no DFA recognizing L n with fewer than 2 n states. Conclusion: For finite automata, nondeterminism provides an exponential savings over determinism (in the worst case). 4

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Harvard CS 121 & CSCI E-207 September 22, 2009 Proving that exponential blowup is sometimes unavoidable (Could there be a construction that always produces an n 2 state DFA for example?) Consider (for some fixed n =17, say) L n = { w ∈ { a, b } * : the n th symbol from the right end of w is an a } There is an ( n + 1) -state NFA that accepts L n .
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