CSci 5403, Spring 2010 Hannah Jaber Problem 4.2 Proposition. BPL ⊆ P where BPL is deﬁned as follows: the language A is in BPL iﬀ there exists a logspace PTM M such that Pr[ M ( x ; r ) = ( x ∈ A )] ≥ 2 / 3 . Proof. With only logarithmic space, M can have at most n c states, where n is the size of the input. More states would require the machine would loop states and not halt. We can therefore build an adjacency matrix K to descibe the possible conﬁgurations of M on input A . Iterate through all possible states s . If s j is adjacent to s i , the i,j entry in the matrix is 1, otherwise it is 0. s j is considered to be adjacent to state s i iﬀ M can reach s j from s i in a single step. If s i requests a random bit, there will be two adjacent states for the two “output” edges. Random bit 1 will lead to one state, random bit 0 will lead to a diﬀerent state. Consider the matrix K x . Entry i,j will list the number of paths from
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