# On the Size of Logical Automata

## Abstract

The state complexity of simulating 1NFA by 2DFA is a long-standing open question, which is of particular interest also due to its connection to the DLOG vs. NLOG problem for Turing machines.

What makes proving lower bounds on the size of deterministic two-way automata particularly hard is the fact that one has to consider *any* automaton, and unlike the designer, one does not have any meaning of the states at hand. This motivates the notion of *logical automata* whose states are annotated by formulas representing the meaning of a state.

In the paper at hand, we first introduce the notion of logical automata and present a *general approach* to proving lower bounds on the number of states of logical automata. We then apply this approach to derive an *exponential* lower bound on the size of logical automata over formulas with a restricted set of atomic predicates. Finally, we complement the lower bound with an (also exponential) upper bound.

## Notes

### Acknowledgements

The author would like to thank Hans-Joachim Böckenhauer, Juraj Hromkovič, and the referees for their valuable comments and suggestions.

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