Small hitting-sets for tiny arithmetic circuits or: How to turn bad designs into good

We show that if we can design poly(s)-time hitting-sets for Σ∧^aΣΠ^O(log s) circuits of size s, where a=ω(1) is arbitrarily small and the number of variables, or arity n, is O(log s), then we can derandomize blackbox PIT for general circuits in quasipolynomial time. This also establishes that either E⊈#P/poly or that VPVNP. In fact, we show that one only needs a poly(s)-time hitting-set against individual-degree a'=ω(1) polynomials that are computable by a size-s arity-(log s) ΣΠΣ circuit (note: Π fanin may be s). Alternatively, we claim that, to understand VP one only needs to find hitting-sets, for depth-3, that have a small parameterized complexity. Another tiny family of interest is when we restrict the arity n=ω(1) to be arbitrarily small. We show that if we can design poly(s,μ(n))-time hitting-sets for size-s arity-n ΣΠΣ∧ circuits (resp. Σ∧^aΣΠ), where function μ is arbitrary, then we can solve PIT for VP in quasipoly-time, and prove the corresponding lower bounds. Our methods are strong enough to prove a surprising arity reduction for PIT– to solve the general problem completely it suffices to find a blackbox PIT with time-complexity sd2^O(n). We give several examples of (log s)-variate circuits where a new measure (called cone-size) helps in devising poly-time hitting-sets, but the same question for their s-variate versions is open till date: For eg., diagonal depth-3 circuits, and in general, models that have a small partial derivative space. We also introduce a new concept, called cone-closed basis isolation, and provide example models where it occurs, or can be achieved by a small shift.