GapP

GapP is a counting complexity class, consisting of all of the functions f such that there exists a polynomial-time non-deterministic Turing machine M where, for any input x, f(x) is equal to the number of accepting paths of M minus the number of rejecting paths of M. GapP is exactly the closure of #P under subtraction. It also has all the other closure properties of #P, such as addition, multiplication, and binomial coefficients.

The counting class AWPP is defined in terms of GapP functions.

References

S. Fenner, L. Fortnow, and S. Kurtz. Gap-definable counting classes, Journal of Computer and System Sciences 48(1):116-148, 1994.
Complexity Zoo: GapP

P ≟ NP

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v t e Complexity classes
Considered feasible	DLOGTIME AC⁰ ACC⁰ TC⁰ L SL RL FL NL NL-complete NC SC CC P P-complete ZPP RP BPP BQP APX FP
Suspected infeasible	UP NP NP-complete NP-hard co-NP co-NP-complete TFNP FNP AM QMA PH ⊕P PP #P #P-complete IP PSPACE PSPACE-complete
Considered infeasible	EXPTIME NEXPTIME EXPSPACE 2-EXPTIME ELEMENTARY PR R RE ALL
Class hierarchies	Polynomial hierarchy Exponential hierarchy Grzegorczyk hierarchy Arithmetical hierarchy Boolean hierarchy
Families of classes	DTIME NTIME DSPACE NSPACE Probabilistically checkable proof Interactive proof system
List of complexity classes