Allender's 10 #Complexity #OpenQuestions

(Each w/ $1000 reward)

Eric Allender has a new guest column in SIGACT News https://doi.org/10.1145/3586165.3586175, since he is officially retiring this year. It's a fun read, and in addition to the 10 Qs, highlights many others he thinks are approachable.

1. Exponential lower bounds on ACC⁰ circuits for PP
2. Is PRIMES TC⁰-hard under *uniform* AC⁰ Turing reductions?
3. Do all problems solvable by constant-width circuits of polylog genus lie in ACC⁰?
4. Is UL=coUL?
5. Improve the space upper bound on StUL. (Maybe StUL=L)
6. Is DET=NC¹(#\L)=AC⁰(#\L)?
7. Improve the NC-depth upper bound on #\NC¹. (Maybe #\NC¹=NC¹)
8. Is AC¹ in SAC¹[⊕,∧]=VP(F₂)?
9. Is the binary expansion of every irrational algebraic number equal to the characteristic function of a language in AC⁰? Does there exist such a number in PH?
10. Is Halt in \(P^{R_{C_U}}\) for some universal machine U? Is there universal U such that Halt is *not* in \(P^{R_{C_U}}\)?

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