TALLY: Tally Languages
Contained in SPARSE.
TC0: Constant-Depth Threshold Circuits
The class of decision problems solvable by polynomial-size, constant-depth circuits with unbounded fanin, which can use AND, OR, and NOT gates (as in AC0) as well as threshold gates. A threshold gate returns 1 if at least half of its inputs are 1, and 0 otherwise.
A uniformity requirement is sometimes also placed.
TC0 circuits of depth 3 are strictly more powerful than TC0 circuits of depth 2 [HMP+93].
[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].
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.
Θ2P: Alternate name for PNP[log]
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. Is contained in contained in Δ2P (in the BSS model over F; when F is a finite field this is the usual class Δ2P). As with Graph Isomorphism, the Tensor Isomorphism problem itself (say, over finite fields) is contained in NP as well as coAM (and indeed SZK). 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].
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.
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.