Mittagsseminar (in cooperation with J. Lengler, A. Steger, and D. Steurer)

Date and Time: Tuesday, April 02, 2013, 12:15 pm

Duration: 30 minutes

Location: OAT S15/S16/S17

Speaker: Nemanja Skoric

On Stack Implementations on Turing Machines

Hennie and Stearns gave a construction of universal Turing machine (abbr. UTM) that can simulate any Turing machine with only logarithmic overhead. This result has not been improved for over 45 years, while, on the other hand, no superlinear lower bound is known. Since a TM tape can be simulated by two stacks, any stack implementation in TM model that allows executing n commands in t(n) time for any constant number of stacks implies UTM working in time O(t(n)). The main contribution of H&S can be viewed as such smart stack implementation in time O(n log n).

In this thesis we explore the time complexity of stack implementations on Turing machines. Two kinds of restrictions on implementations were investigated, both of which are satisfied by the implementation in UTM of H & S. We prove a lower bound of Omega(n log n) for both, implying that stack implementation of H & S is optimal under these restrictions.

Furthermore, we consider a natural probability distribution of command sequences and describe a stack implementation that can execute almost all of its sequences in time O(n log log n). This implies that showing a lower bound of Omega(n log n) (if possible at all) would require a more involved strategy than choosing a random sequence from our distribution.

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