Lofgren, MusbachCSci 5403— Homework 4Problem 1SupposeL∈PPSPACE. Then there exists a polynomialp:N→Nand ap(n)-space PTMMthat runs forexactly 2p(n)steps on every input (it counts its steps to ensure this), has a unique accepting configurationa(it clears its tape before accepting), and is such thatx∈L⇐⇒Pr[M(x)]≥1/2.We will constructan algorithm that decidesLin polynomial space, proving thatL∈PSPACE.Byδ(x, b), we mean theconfiguration that comes after configurationxwith random bitb.By [δ(u, b) =v], we mean the Iversonbracket: 1 ifδ(u, b) =v, and 0 otherwise. LetCbe the set of configurations ofM, so|C|=θ(2p(n)). Wedefine a recursive functionCount-Paths(u, v, i) which counts the number of computation paths fromutovof length 2i:Count-Paths(u, v, i) =([δ(u,0) =v] + [δ(u,
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