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**Extra info for An introduction to differentiable manifolds and riemannian geometry**

**Example text**

PSPACE ⊂ ≠ DSPACE(f (n)) for all superpolynomial It follows that ???? ⊂≠ EXP and LOGSPACE ⊂≠ PSPACE ⊂≠ EXPSPACE. The above time and space hierarchy theorems can also be extended to nondeterministic time- and space-bounded complexity classes. The proofs, however, are more involved, because acceptance and rejection in NTMs are not symmetric. We only list the simpler results on nondeterministic space-bounded complexity classes, which can be proved by a straightforward diagonalization. 28. 26 (a) NLOGSPACE ⊂≠ NPSPACE.

24 P ⊂≠ DTIME(f (n)) for all superpolynomial functions f . Proof. If f (n) is superpolynomial, then so is g(n) = (f (n))1∕2 . By the time hierarchy theorem, P ⊆ DTIME(g(n)) ⊂≠ DTIME(f (n)). 25 functions f . PSPACE ⊂ ≠ DSPACE(f (n)) for all superpolynomial It follows that ???? ⊂≠ EXP and LOGSPACE ⊂≠ PSPACE ⊂≠ EXPSPACE. The above time and space hierarchy theorems can also be extended to nondeterministic time- and space-bounded complexity classes. The proofs, however, are more involved, because acceptance and rejection in NTMs are not symmetric.

17 There exists a TM Mu , which, on input ⟨x, y⟩, simulates the machine Mx on input y so that L(Mu ) = {⟨x, y⟩ ∶ y ∈ L(Mx )}. Furthermore, for each x, there is a constant c such that ????????????????Mu (⟨x, y⟩) ≤ c ⋅ (????????????????Mx (y))2 . Proof. We first construct a four-tape machine Mu . The machine Mu first decodes ⟨x, y⟩ and copies the string x to the second tape and copies string y to the third tape. 2) with 0 ≤ j ≤ 2, 0 ≤ ???? ≤ 2 and 0 ≤ h ≤ 1. Furthermore, it verifies that it encodes a deterministic TM by verifying that no two ????????????????p and ????????????????q begin with the same initial segment 0i+1 10j+1 1.