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Abstract:
We give a direct proof by generic reduction that a decidable
rudimentary theory of finite typed sets [Henkin 63, Meyer 74,
Statman 79, Mairson 92] requires space exceeding infinitely
often an exponentially growing stack of twos. This gives
the highest currently known lower bound for a decidable
logical theory and affirmatively answers to Problem 10.13
of [Compton & Henson 90]:
Is there a `natural' decidable theory with a lower bound of the
form $\exp_\infty(f(n))$, where $f$ is not linearly bounded?
The highest previously known lower and upper bounds for `natural'
decidable theories, like WS1S, S2S, are `just' linearly growing
stacks of twos.
