Imai Laboratory Introduction Studies on Computational Complexity

1. Imai Laboratory Introduction Studies on Computational Complexity Time versus Space in the Computation Universe • Complexity Classes • L = the set of languages computable in logarithm space. • P = the set of languages computable in polynomial time. • PSPACE = the set of language computable in polynomial space. Function Complexity Classes Reversal versus Access • FP = the set of functions computable • in polynomial time. • FPSPACE = the set of functions computable • in polynomial space. • P=PSPACE ⇔FP=FPSPACE 1 13 2 3 1 4 5 2 3 6 4 5 6 7 3 8 9 10 11 12 Recursion Theoretic Operators • Reversal complexity is the total number of tape head reversals. • Access complexity is the maximum number of accesses • among all tape cells. Comp*(C) = the smallest class containing C and closed under Comp Space Bounded Reversal and Access Complexity Classes Reversal Access PSPACE PSPACE exp exp L P L P poly poly P versus PSPACE log log L • Theorem • FPSPACE=Comp*(BRec(FP)) • Corollary • FP is closed under BRec • ⇔P=PSPACE 1 1 1 log poly Space 1 log poly Space [2] [1] PSPACE Random Combinatorial Structures FPSPACE-completeness Probabilistic analyses of reversal and access complexity using the Balls-into-Bins model. P We introduce a notion of FPSPACE-completeness and show some function is in FP iff P=PSPACE. L • Balls ⇒Time Complexity • Bins ⇒Space Complexity To clarify the notion of efficient algorithms and the limit of computation, we give structural analyses on the fundamental models of computation. References [1] Kenya Ueno: "Recursion Theoretic Operators for Function Complexity Classes," The 16th Annual International Symposium on Algorithms and Computation (ISAAC 2005), Sanya, Hainan, China, December, 2005. (LNCS 3827, pp.748-756) [2] Kenya Ueno: "Reversal versus Access: Complexity Classes and Random Combinatorial Structures," The First AAAC Annual Meeting (AAAC 2008), Hong Kong, China, April, 2008.