PolyL
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In computational complexity theory, polyL is the complexity class of decision problems that can be solved on a deterministic Turing machine by an algorithm whose space complexity is bounded by a polylogarithmic function in the size of the input. In other words, polyL = DSPACE((log n)O(1)), where n denotes the input size, and O(1) denotes a constant.cite-ref-1[1]
One proof that polyL ≠ P is that P has a complete problem under logarithmic space many-one reductions but polyL does not due to the space hierarchy theorem. The space hierarchy theorem guarantees that DSPACE(logd n) ⊊ DSPACE(logd + 1 n) for all integers d > 0. If polyL had a complete problem, call it A, it would be an element of DSPACE(logk n) for some integer k > 0. Suppose problem B is an element of DSPACE(logk + 1 n) but not of DSPACE(logk n). The assumption that A is complete implies the following O(logk n) space algorithm for B: reduce B to A in logarithmic space, then decide A in O(logk n) space. This implies that B is an element of DSPACE(logk n) and hence violates the space hierarchy theorem.cite-ref-frontiers-4-0[4]
The lack of complete problems for polyL under logarithmic space many-one reductions has led Ferrarotti et al. to define a different notion of completeness for this class, involving transformations from parameterized problems to polylog-space machines that solve the problems for specific parameter values.cite-ref-frontiers-4-1[4]
An interesting subclass is SC (Steve's Class, named in honor of Stephen Cook and in analogy with Nick's Class.): The class of decision problems solvable by a Turing machine that simultaneously uses polynomial time and polylogarithmic space. It is obviously a subset of P ∩ polyL, and might even be strictly smaller than it, since for the latter, it suffices to have two separate algorithms: one polynomial-time and the other polylogarithmic-space, whereas for SC, there must be a single algorithm that satisfies both constraints.
Deterministic context-free languages can be recognized in SC.cite-ref-5[5] SC contains Randomized L and Bounded-Error Probabilistic L.cite-ref-6[6]
References
cite-note-11. ↑ citerefpapadimitriou1994Papadimitriou, Christos H. (1994), Computational Complexity, Addison-Wesley, p. 405, ISBN 9780201530827
cite-note-33. ↑ Complexity Zoo: polyL
cite-note-55. ↑ citerefcook1979Cook, Stephen A. (1979-04-30). "Deterministic CFL's are accepted simultaneously in polynomial time and log squared space". Proceedings of the eleventh annual ACM symposium on Theory of computing - STOC '79. New York, NY, USA: Association for Computing Machinery. pp. 338–345. doi:10.1145/800135.804426. ISBN 978-1-4503-7438-5.