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

Research in the last decade has shown that to prove lower bounds or to derandomize polynomial identity testing (PIT) it suffices to solve these questions for restricted circuits. In this work, we study the possibly most restricted class of circuits, within depth-4, which would yield such results for general circuits (that is, the complexity class VP). We show that if, for some μ, there is a poly(s, 2, μ(a))-time blackbox PIT for size-s Σ ∧ ΣΠ, where the number of variables, or arity, is n, then blackbox PIT for VP is in QuasiP. Further, in a strengthening of (Kabanets & Impagliazzo, STOC’03), the former algorithm implies that either E6⊆#P/poly or that VNP requires size-2 VP circuits. Parameters a, b & n, for which the time-complexity above is poly(s), define the model of tiny diagonal depth-4. Note that these are merely polynomials with arity O(log s) whose degree ab is arbitrarily close to log s. In fact, we show that one only needs, for some μ′, a poly(s, 2, μ′(a′))-time blackbox PIT for individual-degree-a′ arity-n homogeneous polynomials computed by a size-s depth-3 circuit. Alternatively, we claim that, to understand VP we barely need a fixed-parameter tractable (FPT) blackbox PIT for depth-3 (parameters being arity & individual-degree). Almost any FPT PIT is welcome: We show that if, for some μ, there is a poly(s, μ(n))time blackbox PIT for size-s arity-n ΣΠΣ∧ circuits (resp. Σ ∧ ΣΠ), then blackbox PIT for VP is in QuasiP, and one proves the claimed lower bound. (Fig.1 lists more results.) Finally, our methods prove a stunning arity reduction for PIT– to solve the general problem in poly(sd)-time it suffices to find a blackbox PIT with poly(sd, exp◦c(n))-time, where exp◦c refers to a c = O(1) fold composition of exp (eg. exp◦2(n) := 2 n ). This suggests that, in algebraic-geometry terms, PIT is an ‘extremely low’ dimensional problem. One expects that with this severe restriction (or tinyness) on n, a, b and the semantic individual-degree, it should be “super-exponentially” easier to design hitting-sets. Indeed, we give several examples of (log s)-variate circuits where a new measure (called cone-size) helps in devising poly(s)-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. The latter models are very well studied, following (Nisan & Wigderson, FOCS’95), but no sd2-time PIT algorithm was known before us. We introduce a novel concept, called cone-closed basis isolation, and provide example models where it occurs, or can be achieved by a small shift. This refines the previously studied notions of low-support (resp. low-cone) rank concentration and least basis isolation in certain ABP models. Cone-closure holds special relevance in the low-arity regime. 2012 ACM CCS concept: Theory of computation– Algebraic complexity theory, Fixed parameter tractability, Pseudorandomness and derandomization; Computing methodologies– Algebraic algorithms; Mathematics of computing– Combinatoric problems.