# Complexity Zoo:H

Complexity classes by letter: Symbols - A - B - C - D - E - F - G - H - I - J - K - L - M - N - O - P - Q - R - S - T - U - V - W - X - Y - Z

Lists of related classes: Communication Complexity - Hierarchies - Nonuniform

##### HalfP: RP With Exactly Half Acceptance

The class of decision problems solvable by an NP machine such that

1. If the answer is 'yes,' exactly 1/2 of computation paths accept.
2. If the answer is 'no,' all computation paths reject.

Significantly, the number of candidate witnesses is restricted to be a power of 2. (This is implicit if they are binary strings.)

Contained in RP, EP, and ModkP for every odd k. Contained in EQP by the Deutsch-Jozsa algorithm.

Defined in [BB92], where it was called C==P[half]. The name used here is from [BS00]. There it was shown that HalfP is contained in every similar class in which 1/2 is replaced by some other dyadic fraction.

##### HeurBPP: Heuristic BPP

The class of problems for which a 1-1/poly(n) fraction of instances are solvable by a BPP machine.

[FS04] showed a strict hierarchy theorem for HeurBPP; thus, HeurBPP does not equal HeurBPTIME(nc) for any fixed c.

##### HeurBPTIME(f(n)): Heuristic BPTIME(f(n))

The class of problems for which a 1-1/poly(n) fraction of instances are solvable by a BPTIME(f(n)) machine. Thus, HeurBPTIME(f(n)) has the same relationship with BPTIME as HeurDTIME.

Thus HeurBPP is the union of HeurBPTIME(nc) over all c.

##### HeurDTIME${\displaystyle \delta }$(f(n)): Heuristic DTIME

For functions ${\displaystyle f(n)}$ and ${\displaystyle \delta (n)}$, we say that tuple ${\displaystyle (L,D)\in {\mathsf {HeurDTIME}}_{\delta }(f(n))}$, where ${\displaystyle L}$ is a language and ${\displaystyle D}$ is a distribution of problem instances, if there exists a heuristic deterministic algorithm ${\displaystyle A}$ such that for all ${\displaystyle x}$ in the support of ${\displaystyle D}$, ${\displaystyle A}$ runs in time bounded by ${\displaystyle f(n)}$ and with failure probability bounded by ${\displaystyle \delta (n)}$ [BT06].

##### HeurP: Heuristic P

The class of distributional problems solvable by a P machine. Defined in [Imp95], though he calls the class HP.

Alternately, [BT06] define HeurP as being the set of tuples ${\displaystyle (L,D)}$, where ${\displaystyle L}$ is a language and ${\displaystyle D}$ is a distribution of problem instances, such that there exists an algorithm ${\displaystyle A}$ satisfying two properties:

• For every ${\displaystyle n\in \mathbb {N} }$, for every ${\displaystyle x}$ in the support of ${\displaystyle D}$, and for every ${\displaystyle \delta >0}$, ${\displaystyle A(x;n,\delta )}$ runs in time bounded by ${\displaystyle \mathrm {poly} (n/\delta )}$.
• For every ${\displaystyle \delta >0}$, ${\displaystyle A(\cdot ;\cdot ,\delta )}$ is a heuristic algorithm for ${\displaystyle (L,D)}$ whose error probability is bounded by ${\displaystyle \delta }$.

##### HeurPP: Heuristic PP

The class of distributional problems solvable by a PP machine. Defined in [Ill95], though he calls the class HPP.

##### HeurNTIME${\displaystyle \delta }$(f(n)): Heuristic NTIME

Defined as Heur${\displaystyle \delta }$DTIME, but for non-deterministic heuristic algorithms.

NP is not contained in HeurNTIME${\displaystyle 1/2+1/{n^{a}}}$(${\displaystyle n^{c}}$) for any constants ${\displaystyle a,c}$ [Per07].

##### HkP: High Hierarchy In NP

The class of problems A in NP such that ΣkPA = Σk+1P; that is, adding A as an oracle increases the power of the kth level of the polynomial hierarchy by a maximum amount.

For all k, Hk is contained in Hk+1.

H0 consists exactly of the problems complete for NP under Cook reductions.

H1 consists exactly of the problems complete for NP under strong non-deterministic Turing reductions [Sch83].

Defined in [Sch83].

##### HO: High-Order logic

High order logic is an extension of Second order, First order where we add quantification over higher order variables.

We define a relation of order ${\displaystyle o}$ and arity ${\displaystyle k}$ to be a subset of ${\displaystyle k}$-tuple of relation of order ${\displaystyle o-1}$ and arity ${\displaystyle k}$. When ${\displaystyle o=1}$ it is by extension a first order variable. The quantification of formula in HO is over a given order (which is a straightforward extension of SO where we add quantification over constant (first-order variable) and relation (second-order variables). The atomic predicates now can be general application of relation of order ${\displaystyle o}$ and arity ${\displaystyle k}$ to ${\displaystyle k}$ relations of order ${\displaystyle o-1}$ and arity ${\displaystyle a}$ and test of equality between two relations of the same order and arity.

${\displaystyle HO^{o}}$ is the set of formulae with quantification up to order O. ${\displaystyle \Sigma _{j}^{i}}$(resp. ${\displaystyle \Pi _{j}^{i}}$) is defined as the set of formula in ${\displaystyle HO^{i+1}}$ beginning by an existantial (resp universal) quantifier followed by at most ${\displaystyle j-1}$ alternation of quantifiers.

This class was define in [HT06], and it was proved that ${\displaystyle \Sigma _{j}^{i}=\exp _{2}^{i-1}(n^{O}(1))^{\Sigma _{j-1}^{P}}}$ where ${\displaystyle \Sigma _{j-1}^{P}}$ is the ${\displaystyle j}$th level of the polynomial hierarchy.

##### HVPZK: Honest-Verifier PZK

The class of decision problems that have PZK protocols assuming an honest verifier (i.e. one who doesn't try to learn more about the problem by deviating from the protocol).

Contained in PP [BHCTV17]. There is an oracle where it is not closed under complement [BHCTV17].

##### HVSZK: Honest-Verifier SZK

The class of decision problems that have SZK protocols assuming an honest verifier (i.e. one who doesn't try to learn more about the problem by deviating from the protocol).

Equals SZK [Oka96].