Complexity Zoo:H

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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

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HalfP - HeurBPP - HeurBPTIME(f(n)) - HeurDTIME_$\delta$(f(n)) - HeurP - HeurPP - HeurNTIME_$\delta$(f(n)) - H_kP - HVSZK

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 Mod_kP 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.

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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(n^c) for any fixed c.

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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(n^c) over all c.

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HeurDTIME_$\delta$(f(n)): Heuristic DTIME

For functions $f(n)$ and $\delta (n)$, we say that tuple $(L,D)\in {\mathsf {HeurDTIME}}_{\delta }(f(n))$, where $L$ is a language and Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle D} is a distribution of problem instances, if there exists a heuristic deterministic algorithm $A$ such that for all $x$ in the support of Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle D} , $A$ runs in time bounded by $f(n)$ and with failure probability bounded by $\delta (n)$ [BT06].

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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 $(L,D)$, where $L$ is a language and Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle D} is a distribution of problem instances, such that there exists an algorithm $A$ satisfying two properties:

• For every Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle n\in \mathbb {N} } , for every $x$ in the support of Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle D} , and for every $\delta >0$, $A(x;n,\delta )$ runs in time bounded by $\mathrm {poly} (n/\delta )$.
• For every $\delta >0$, $A(\cdot ;\cdot ,\delta )$ is a heuristic algorithm for $(L,D)$ whose error probability is bounded by $\delta$.

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HeurPP: Heuristic PP

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

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HeurNTIME_$\delta$(f(n)): Heuristic NTIME

Defined as Heur_$\delta$DTIME, but for non-deterministic heuristic algorithms.

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

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H_kP: High Hierarchy In NP

The class of problems A in NP such that Σ_kP^A = Σ_k+1P; that is, adding A as an oracle increases the power of the k^th level of the polynomial hierarchy by a maximum amount.

For all k, H_k is contained in H_k+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].

See also L_kP.

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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 Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle o} and arity $k$ to be a subset of $k$-tuple of relation of order $o-1$ and arity $k$. When $o=1$ 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 Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle o} and arity $k$ to $k$ relations of order $o-1$ and arity Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a} and test of equality between two relations of the same order and arity.

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle HO^o} is the set of formulae with quantification up to order O. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Sigma^i_j} (resp. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Pi_j^i} ) is defined as the set of formula in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle HO^{i+1}} beginning by an existantial (resp universal) quantifier followed by at most Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j-1} alternation of quantifiers.

This class was define in [HT06], and it was proved that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Sigma^i_j=\exp_2^{i-1}(n^O(1))^{\Sigma^P_{j-1}}} where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Sigma^P_{j-1}} is the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j} th level of the polynomial hierarchy.

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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].

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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].