Complexity Zoo:T

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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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TALLY - TC - TC⁰ - TC⁰(FOLL) - TC¹ - TFNP - Θ₂P - TI - TOWER - TreeBQP - TREE-REGULAR

TALLY: Tally Languages

The class of decision problems for which every 'yes' instance has the form 0ⁿ (i.e. inputs are encoded in unary). If TALLY intersects NPC then P = NP [Mah82].

Contained in SPARSE.

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TC: Threshold Circuits

TCⁱ is the class of decision problems solvable by polynomial-size, depth $O(\log ^{i}n)$ circuits with unbounded fanin AND, OR, and majority (MAJ) gates. A majority gate returns 1 if at least half of its inputs are 1, and 0 otherwise. Other gates that can be used in place of majority (up to polynomial size equivalence) are threshold gates (THR) and MOD_pn, where p_n is the nth prime.

A uniformity requirement is sometimes also placed.

Each TCⁱ contains ACⁱ (in fact ACCⁱ) and is contained in NCⁱ⁺¹. Thus NC = AC = TC.

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TC⁰: Constant-Depth Threshold Circuits

See TC for definition.

TC⁰ contains ACC⁰, and is contained in NC¹.

TC⁰ circuits of depth 3 are strictly more powerful than TC⁰ circuits of depth 2 [HMP+93].

TC⁰ circuits of depth 3 and quasipolynomial size can simulate all of ACC⁰ [Yao90].

There is a function in AC⁰ (explicitly given in [She08]), whose computation with TC⁰ circuits of depth 2 requires an exponential number of gates.

[NR97] give a candidate pseudorandom function family computable in TC⁰, that is secure assuming a subexponential lower bound on the hardness of factoring. (See also [NRR01] for an improvement of this construction, as well as [Kha93].)

One implication is that, assuming such a bound, there is no natural proof in the sense of [RR97] separating TC⁰ from P/poly. (It is important for this that a function family, and not just a candidate pseudorandom generator, is computable in TC⁰.) Another implication is that functions in TC⁰ are likely to be difficult to learn.

The permanent of a 0-1 matrix cannot be computed in uniform TC⁰ [All99].

In a breakthrough result [Hes01] (building on [BCH86] and [CDL01]), integer division was shown to be in U_D-uniform TC⁰. Indeed division is complete for this class under AC⁰ reductions.

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TC⁰(FOLL): Languages that are TC⁰ Turing reducible to FOLL

The class of decision problems that are reducible to FOLL languages under TC⁰-computable Turing reductions.

For an Abelian group G and an arbitrary group H given by their multiplication tables, deciding whether G and H are isomorphic belongs to TC⁰(FOLL). [CTW13].

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TC¹: Log-depth Threshold Circuits

See TC for definition.

In addition to the different gatesets allowed in place of AND, OR, THR, also equivalent to arithmetic/counting analogue of AC¹ modulo p_n (the nth prime) [RT92].

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TFNP: Total Function NP

The class of function problems of the following form:

Given an input x and a polynomial-time predicate F(x,y), output any y satisfying F(x,y). (Such a y is promised to exist.)

Can be considered as the functional analogue of NP ∩ coNP. Defined in [MP91].

Contained in FNP.

Subclasses include PPA, PPP, and PLS.

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TI: Tensor Isomorphism

The class of problems that are polynomial-time Turing reducible to Tensor Isomorphism. Defined in [GQ19]. Can depend on the field, and the relationship for TI over different fields is an open question, but many reductions hold for TI over any field (over finite fields or the rationals this can be done in the usual model of Turing machines; over arbitrary fields one can use the BSS model to formalize this).

Over any field F, contains GI. As with Graph Isomorphism, the Tensor Isomorphism problem itself (say, over finite fields) is contained in NP as well as coAM (and indeed SZK; the same results hold over arbitrary fields in the BSS model). So in particular, if Tensor Isomorphism is NP-complete, then PH collapses.

Many natural problems are TI-complete, such as isomorphism of d-tensors for any fixed d ≥ 3, isomorphism of algebras, conjugacy of spaces of matrices, (pseudo-)isometry of alternating matrix spaces, isomorphism of matrix p-groups of class 2 and exponent p, and equivalence of cubic forms [GQ19]. This was extended to include p-groups of class c<p and exponent p [GQ21]. Analogous classes were also defined under other group actions such as unitary, orthogonal, and symplectic groups [CGQ+24].

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TOWER: Iterated Exponential Time

Defined in [Sch16] as the union of DTIME(F₃(p(n))) over all elementary functions p(n), where F₃ is the third level of the fast growing hierarchy, i.e., the iterated exponential function. The author describes this as the class of problems that are not in ELEMENTARY but only barely so. Unlike ELEMENTARY and PR, which have no complete problems, Tower has several natural examples of complete problems (under elementary reductions). Specifically, the Star-Free Expression Equivalence (SFEq) problem is complete for this class, as is the satisfiability of the Weak Monadic Theory of One Successor (WS1S).

Strictly contains ELEMENTARY and is strictly contained in PR.

Note that the same class is obtained if DTIME is replaced by NTIME or DSPACE in the definition.

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TreeBQP: BQP Restricted To Tree States

The class of languages accepted by a BQP machine subject to the constraint that at every time step t, the machine's state is exponentially close to a tree state -- that is, a state expressible by a polynomial-size tree of additions and tensor products (together with complex constants and |0> and |1> leaf nodes).

More formally, a uniform classical polynomial-time algorithm generates a sequence of gates g⁽¹⁾, ... ,g^(p(n)). Each g^(t) can be either be selected from some finite universal basis of unitary gates (the choice turns out not to matter), or can be a 1-qubit measurement. When we perform a measurement, the state evolves to one of two possible pure states, with the usual probabilities, rather than to a mixed state. We require that the final gate g^(p(n)) is a measurement of the first qubit. If at least one intermediate state was more than distance 2^-Ω(n) away from the nearest state of tree size at most p(n), then the outcome of the final measurement is chosen adversarially; otherwise it is given by the usual Born probabilities. The measurement must return 1 with probability at least 2/3 if the input is in the language, and with probability at most 1/3 otherwise.

Contains BPP, and is contained in BQP.

Defined in [Aar03b], where it was also shown that TreeBQP is contained in the third level of PH, which might provide weak evidence that TreeBQP does not equal BQP.

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TREE-REGULAR: Regular Tree-Valued Languages

Same as REG, except that now the inputs are trees (say, binary trees) instead of strings. Each vertex is labeled with a symbol from a fixed alphabet. Evaluation begins at the leaves and proceeds to the root. The state of the finite automaton at each vertex v is a function of (1) the states at v's children (if any), and (2) the symbol at v. The tree is in the language if and only if the automaton is in an 'accept' state at the root.

See [Koz92] for example.