Hartmanis and Stearns' original hierarchy results show that below logarithmic space, the gap conditions for proper separation become insufficient to absorb simulation overhead, meaning the limit condition on s1/s2 alone does not guarantee separation in sub-log regimes.

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Key Terms

Gap conditions(as computational complexity theory): Mathematical requirements that create a meaningful difference or separation between two things, like a cushion that keeps them distinct.
Hartmanis and Stearns(as founders of computational complexity theory): Two computer scientists who developed an important framework for understanding how much computational effort (time or memory) different types of problems require to solve.
Hierarchy results(as computational complexity theory): Mathematical proofs showing that problems can be organized into different levels or tiers based on how hard they are to solve.
Limit condition(as used in mathematical logic and computability theory): A mathematical boundary or endpoint that a formula or process approaches but may never quite reach—like how 1/n gets closer and closer to 0 the bigger n gets.

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Logarithmic space(as a measure of computational resources)

A very small amount of computer memory (measured in a specific mathematical way) needed to solve a problem—much smaller than the problem's input size.

Proper separation(as computational complexity theory)

A clear, definite distinction between two different classes or categories of problems, showing they are genuinely different from each other.

Simulation overhead(as computational complexity theory)

The extra computing power or memory needed when one type of computer simulation tries to mimic another type, like the cost of translation.

Sub-log regimes(as computational complexity theory)

Problem scenarios that require even less memory than logarithmic space—extremely tight memory constraints.

s1/s2(as mathematical notation)

A mathematical ratio comparing two space measurements (s1 divided by s2), often used to compare memory requirements of different computational approaches.

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Hartmanis and Stearns' original hierarchy results show that below logarithmic space, the gap conditions for proper separation become insufficient to absorb simulation overhead, meaning the limit condition on s1/s2 alone does not guarantee separation in sub-log regimes.

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