Certified Hardness vs. Randomness for Log-Space

Let L be a language that can be decided in linear space and let ϵ > 0 be any constant. Let A be the exponential hardness assumption that for every n, membership in L for inputs of length n cannot be decided by circuits of size smaller than 2ϵ n. We prove that for every function f:{0,1}*→{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 A is false', followed by a (provenly correct) circuit of size smaller than 2ϵ n′ for membership in L for inputs of length n′, for some n′=Θ(log n); that is, a circuit that refutes 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)¹: 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 · log n (where C_R is a constant depending on R). Moreover, for every constant c ≥ 1, if BPL ⊆ SPACE[(log (n))c] then the space used by U is at most C_R ·(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.¹Our result is stated and proved for promise-BPL, but we ignore this difference in the abstract.