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# [email protected] - Removing Nondeterminism from Two-Way Automata...

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Removing Nondeterminism from Two-Way Automata Giovanni Pighizzini Dipartimento di Informatica e Comunicazione Università degli Studi di Milano Porto – June 22, 2010

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Outline The Question of Sakoda and Sipser Quasi Sweeping Automata and Quasi Sweeping Simulation Sadoka&Sipser Question vs L ? = NL Making Unary 2NFAs Unambiguous Conclusion
Finite State Automata i n p u t . . . 6 - Base version: one-way deterministic finite automata (1DFA) I one-way input tape I deterministic transitions Possibile variants allowing: I nondeterministic transitions one-way nondeterministic finite automata (1NFA) I input head moving forth and back two-way deterministic finite automata (2DFA) two-way nondeterministic finite automata (2NFA) I alternation I ...

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Two-Way Automata: Technical Details i n p u t . . . a 6 - I Input surrounded by the endmarkers and a I Transition function δ : Q × ∪ {‘ , a} ) 2 Q ×{- 1 , 0 , + 1 } where - 1 , 0 , + 1 are the possible movements of the input head I w Σ * accepted iff there is a computation with input tape w a from the initial state q 0 , scanning the left endmarker reaching a final state
1DFA, 1NFA, 2DFA, 2NFA What about the power of these models? They share the same computational power, namely they characterize the class of regular languages , however... ...some of them are more succinct

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Example: L = ( a + b ) * a ( a + b ) n - 1 I L is accepted by a 1NFA with n + 1 states q 0 q 1 q 2 q 3 q n @ @ R - a - a , b - a , b - a , b - a , b I The minimum 1DFA accepting L requires 2 n states I We can get a deterministic automaton for L with n + 2 states, which reverses the input head direction just one time I Hence L is accepted by a 1NFA and a 2DFA with approx. the same number of states a minimum 1DFA exponentially larger
Example: L = ( a + b ) * a ( a + b ) n - 1 a ( a + b ) * I L is accepted by a 1NFA with n + 2 states q 0 q 1 q 2 q 3 q n q f @ R - a - a , b - a , b - a , b - a - a , b a , b I The minimum 1DFA accepting L uses 3 · 2 n - 1 + 1 states I Using head reversals the number of states becomes linear I Even in this case L is accepted by a 1NFA and a 2DFA with linearly related numbers of states a minimum 1DFA exponentially larger

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Example: L = ( a + b ) * a ( a + b ) n - 1 a ( a + b ) * b b a b a a b a a a a n = 4 while input symbol 6 = a do move to the right move n squares to the right if input symbol = a then accept else move n - 1 cells to the left repeat from the first step Exception: if input symbol = a then reject I This can be implemented by a 2DFA with O ( n ) states I By a different algorithm, L can be also accepted by a 2DFA with O ( n ) states which changes the direction of its input head only at the endmarkers
Costs of the Optimal Simulations Between Automata 1DFA 1NFA 2DFA 2NFA @ @ @ @ @ @ @ R ? 2 n O ( 2 n log n ) O ( 2 n 2 ) - ? ? [Rabin&Scott ’59, Shepardson ’59, Meyer&Fischer ’71, . . . ] Question How much the possibility of moving the input head forth and back is useful to eliminate the nondeterminism?

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Costs of the Optimal Simulations Between Automata 1DFA 1NFA 2DFA 2NFA @ @ @ @ @ @ @ R ?
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[email protected] - Removing Nondeterminism from Two-Way Automata...

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