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TR23-040 | 28th March 2023 21:51

#### Certified Hardness vs. Randomness for Log-Space

TR23-040
Authors: Edward Pyne, Ran Raz, Wei Zhan
Publication: 28th March 2023 22:09
Keywords:

Abstract:

Let $\mathcal{L}$ be a language that can be decided in linear space and let $\epsilon >0$ be any constant. Let $\mathcal{A}$ be the exponential hardness assumption that for every $n$, membership in $\mathcal{L}$ for inputs of length~$n$ cannot be decided by circuits of size smaller than $2^{\epsilon n}$.
We prove that for every function $f :\{0,1\}^* \rightarrow \{0,1\}$, computable by a randomized logspace algorithm $R$, there exists a deterministic logspace algorithm $D$ (attempting to compute $f$), such that on every input $x$ of length $n$, the algorithm $D$ outputs one of the following:

1: The correct value $f(x)$.

2: The string: I am unable to compute $f(x)$ because the hardness assumption $\mathcal{A}$ is false'', followed by a (provenly correct) circuit of size smaller than $2^{\epsilon n'}$ for membership in $\mathcal{L}$ for inputs of length~$n'$, for some $n' = \Theta (\log n)$; that is, a circuit that refutes $\mathcal{A}$.

Moreover, $D$ is explicitly constructed, given $R$.

We note that previous works on the hardness-versus-randomness paradigm give derandomized algorithms that rely blindly on the hardness assumption. If the hardness assumption is false, the algorithms may output incorrect values, and thus a user cannot trust that an output given by the algorithm is correct. Instead, our algorithm $D$ verifies the computation so that it never outputs an incorrect value. Thus, if $D$ outputs a value for $f(x)$, that value is certified to be correct. Moreover, if $D$ does not output a value for $f(x)$, it alerts that the hardness assumption was found to be false, and refutes the assumption.

Our next result is a universal derandomizer for $BPL$ (the class of problems solvable by bounded-error randomized logspace algorithms): We give a deterministic algorithm $U$ that takes as an input a randomized logspace algorithm $R$ and an input $x$ and simulates the computation of $R$ on $x$, deteriministically. Under the widely believed assumption $BPL=L$, the space used by $U$ is at most $C_R \cdot \log n$ (where $C_R$ is a constant depending on~$R$). Moreover, for every constant $c \geq 1$, if $BPL\subseteq SPACE[(\log(n))^{c}]$ then the space used by $U$ is at most $C_R \cdot (\log(n))^{c}$.

Finally, we prove that if optimal hitting sets for ordered branching programs exist then there is a deterministic logspace algorithm that, given a black-box access to an ordered branching program $B$ of size $n$, estimates the probability that $B$ accepts on a uniformly random input. This extends the result of (Cheng and Hoza CCC 2020), who proved that an optimal hitting set implies a white-box two-sided derandomization.

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