  We prove a complexity dichotomy theorem for Holant problems over an arbitrary
set of complex-valued symmetric constraint functions F on Boolean variables.
This extends and unifies all previous dichotomies for Holant problems on
symmetric constraint functions (taking values without a finite modulus). We
define and characterize all symmetric vanishing signatures. They turned out to
be essential to the complete classification of Holant problems. The dichotomy
theorem has an explicit tractability criterion expressible in terms of
holographic transformations. A Holant problem defined by a set of constraint
functions F is solvable in polynomial time if it satisfies this tractability
criterion, and is #P-hard otherwise. The tractability criterion can be
intuitively stated as follows: A set F is tractable if (1) every function in F
has arity at most two, or (2) F is transformable to an affine type, or (3) F is
transformable to a product type, or (4) F is vanishing, combined with the right
type of binary functions, or (5) F belongs to a special category of vanishing
type Fibonacci gates. The proof of this theorem utilizes many previous
dichotomy theorems on Holant problems and Boolean #CSP. Holographic
transformations play an indispensable role as both a proof technique and in the
statement of the tractability criterion.
