On closure properties of GapP

We study the closure properties of the function classes GapP and GapP₊. We characterize the property of GapP₊ being closed under decrement and of GapP being closed under maximum, minimum, median, or division by seemingly implausible collapses among complexity classes, thereby giving evidence that these function classes don't have the stated closure properties.We show a similar result concerning operations we callbit cancellation andbit insertion: Given a functionf GapP and a polynomialtime computable function , we ask whether the functionsf^*(x) andf⁺(x) are in GapP or not, wheref^*(x) is obtained fromf(x) by cancelling the (x)-th bit in the binary representation off(x), andf⁺(x) is obtained fromf(x) by inserting a bit at position (x) in the binary representation off(x). We give necessary and sufficient conditions for GapP being closed under bit cancellation and bit insertion, respectively.