LH (complexity)

In computational complexity, the logarithmic time hierarchy (LH) is the complexity class of all computational problems solvable in a logarithmic amount of computation time on an alternating Turing machine with a bounded number of alternations. It is a particular case of a bounded alternating Turing machine hierarchy. It is equal to FO and to FO-uniform AC⁰.^[1]

The $i$ th level of the logarithmic time hierarchy is the set of languages recognised by alternating Turing machines in logarithmic time with random access and $i-1$ alternations, beginning with an existential state. LH is the union of all levels.

References

↑ Neil Immerman (1999). Descriptive Complexity. Springer. p. 85.

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[1] Neil Immerman (1999). Descriptive Complexity. Springer. p. 85.

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Important complexity classes
Considered feasible	DLOGTIME AC⁰ ACC⁰ TC⁰ L SL RL NL NL-complete NC SC CC P P-complete ZPP RP BPP BQP APX FP
Suspected infeasible	UP NP NP-complete NP-hard co-NP co-NP-complete AM QMA PH ⊕P PP #P #P-complete IP PSPACE PSPACE-complete
Considered infeasible	EXPTIME NEXPTIME EXPSPACE 2-EXPTIME ELEMENTARY PR R RE ALL
Class hierarchies	Polynomial hierarchy Exponential hierarchy Grzegorczyk hierarchy Arithmetical hierarchy Boolean hierarchy
Families of classes	DTIME NTIME DSPACE NSPACE Probabilistically checkable proof Interactive proof system
List of complexity classes