# 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

##### 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.

##### TC: Threshold Circuits

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

##### 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 UD-uniform TC0. Indeed division is complete for this class under AC0 reductions.

##### 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].

##### TC1: Log-depth Threshold Circuits

See TC1 for definition.

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

##### 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.

##### 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].

##### Tower: Tetrational Time

Defined in [Sch16] as the union of DTIME(F3(p(n))) over all elementary functions p(n), where F3 is a single application of the third level of the extended Grzegorczyk hierarchy. 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.

##### 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.

##### 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.