Abstract The use of Nepomnjaščiǐ's Theorem in the proofs of independence results for bounded arithmetic theories is investigated. Using this result and similar ideas, it is shown that at least one ofS1orTLSdoes not prove the Matiyasevich-Robinson-Davis-Putnam Theorem. It is also established thatTLSdoes not prove a statement that roughly means nondeterministic linear time is equal to co-nondeterministic linear time. HereS1is a conservative extension of the well-studied theoryIΔ0andTLSis a theory for LOGSPACE reasoning.

For any q>1, let MOD/sub q/ be a quantum gate that determines if the number of 1's in the input is divisible by q. We show that for any q, t>1, MODP is equivalent to MOD/sub t/ (up to constant depth). Based on the case q=2, C. Moore (1999) has shown that quantum analogs of AC/sup (0)/, ACC[q], and ACC, denoted QAC/sub wf//sup (0)/ QACC[2], QACC respectively, define the same class of operators, leaving q>2 as an open question. Our result resolves this question, proving that QAC/sub wf//sup (0)/=QACC[q] QACC for all q. We also develop techniques for proving upper bounds for QACC in terms of related language classes. We define classes of languages EQACC, NQACC and BQACC/sub Q/. We define a notion of log-planar QACC operators and show the appropriately restricted versions of QACC and BQACC are contained in P/poly. We also define a notion of log-gate restricted QACC operators and show the appropriately restricted versions of QACC and NQACC are contained in TC/sup (0)/. To do this last proof; we show that TC/sup (0)/ can perform iterated addition and multiplication in certain field extensions. We also introduce the notion of a polynomial-size tensor graph and we show that families of such graphs can encode the amplitudes resulting from applying an arbitrary QACC operator to an initial state.

We define theories of Bounded Arithmetic characterizing classes of functions computable by constant-depth threshold circuits of polynomial and quasipolynomial size. Then we define certain second-order theories and show that they characterize the functions in the Counting Hierarchy. Finally we show that the former theories are isomorphic to the latter via the so-called RSUV-isomorphism.