Regularity Pinning: symmetric f:[gla→cT∈gk Pin(f)=g where g:[g]d-kC Vo E [q]d-k,g(o)=f(o1,...;od-k;TI,...,Tk) when q-2 write f=[fo,f1,...,fa] where f;=f(a)thatol1 =i a family F of symmetric functions is regular if ヨa finite C s.t.Vf∈F,fisC-regular fo.a-1 fal examples:bounded-arity d-k+1 equality [1,0,...01] counterexample:0.1.0..... at-most-one [1,1,0,...,0] cyclic [a,b.c,a,b.c,.]￾ ￾￾ ￾ d￾k+1 Regularity 8QV￾ (f) = g g : [q] d￾k ￾ C f : [q] symmetric d ￾ C where ￾￾ ￾ [q] g(￾) = f(￾1,..., ￾d￾k, ￾1,..., ￾k) d￾k , ￾ ￾ [q] k Pinning: f : [q] is C-regular if symmetric d ￾ C ￾ ￾ ￾ 8QV￾ (f) | ￾￾ ￾ [q] k￾￾ ￾0 ￾ k ￾ d, ￾ ￾ C a family of symmetric functions is F regular if ￾ a finite C s.t. ￾f ￾ F, f is C-regular counterexample: [0,..., 0 ￾ ￾￾ ￾ d 2 , 1, 0,..., 0 ￾ ￾￾ ￾ d 2 ] [f0, f1, f2,...,fi,...,fd￾1, fd] examples: equality [1,0,...,0,1] at-most-one [1,1,0,...,0] cyclic [a,b,c,a,b,c,...] bounded-arity when q=2 write f = [f0, f1,...,fd] where fi = f(￾) that ￾￾￾1 = i