# Complexity Zoo:X

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Nonuniform

XOR-MIP*[2,1] -
XL -
XNL -
XP -
XP_{uniform}

##### XOR-MIP*[2,1]: MIP*[2,1] With 1-Bit Proofs

Same as MIP*[2,1], but with the further restriction that both provers send only a single bit to the verifier, and the verifier's output is a function of the exclusive-OR of those bits. There should exist 0<*a*<*b*<1 such that if the answer is "yes", then for some responses of the provers the verifier accepts with probability at least *b*, while if the answer is "no", then for all responses of the provers the verifier accepts with probability at most *a*.

Defined by [CHT+04], whose motivation was a connection to the Bell and CHSH inequalities of quantum physics.

XOR-MIP*[2,1] is contained in NEXP [CHT+04].

XOR-MIP*[2,1] is contained in QIP[2] [Weh06]

##### XL: Fixed-Parameter Logspace for Each Parameter

The class of decision problems of the form (x,k) (k a parameter) that are solvable in space O(^{f(k)}+log(n)) for some function f. The algorithm used may depend on k.

Analogous to XP, as L is to P.

A natural XL complete problem is p-MDFA-ACCEPTANCE: Given a finite two-way deterministic automation A with k heads, and given a string x, does A accept x?

Reference: [1]

##### XNL: Fixed-Parameter Nondeterministic Logspace for Each Parameter

The class of decision problems of the form (x,k) (k a parameter) that are solvable in space O(^{f(k)}+log(n)), by a nondeterministic Turing machine, for some function f. The algorithm used may depend on k.

Analogous to XP, as NL is to P. Nondeterministic version of XL.

A natural XNL complete problem is p-MNFA-ACCEPTANCE: Given a finite two-way nondeterministic automation A with k heads, and given a string x, does A accept x?

##### XP: Fixed-Parameter Tractable for Each Parameter

The class of decision problems of the form (x,k) (k a parameter) that are solvable in time O(|x|^{f(k)}) for some function f. The algorithm used may depend on k.

Defined in [DF99].

Contains XP_{uniform}.

Strictly contains FPT (by diagonalization).

##### XP_{uniform}: Uniform XP

Same as XP except that the algorithm used must be the same for each k (though it can take k as input).

Defined in [DF99].