Theories for TC0 and Other Small Complexity Classes
<|reference_start|>Theories for TC0 and Other Small Complexity Classes: We present a general method for introducing finitely axiomatizable "minimal" two-sorted theories for various subclasses of P (problems solvable in polynomial time). The two sorts are natural numbers and finite sets of natural numbers. The latter are essentially the finite binary strings, which provide a natural domain for defining the functions and sets in small complexity classes. We concentrate on the complexity class TC^0, whose problems are defined by uniform polynomial-size families of bounded-depth Boolean circuits with majority gates. We present an elegant theory VTC^0 in which the provably-total functions are those associated with TC^0, and then prove that VTC^0 is "isomorphic" to a different-looking single-sorted theory introduced by Johannsen and Pollet. The most technical part of the isomorphism proof is defining binary number multiplication in terms a bit-counting function, and showing how to formalize the proofs of its algebraic properties.<|reference_end|>
@article{nguyen2005theories,
    title={Theories for TC0 and Other Small Complexity Classes},
    author={Phuong Nguyen and Stephen Cook},
    journal={Logical Methods in Computer Science, Volume 2, Issue 1 (March 7,
  2006) lmcs:2257},
    year={2005},
    doi={10.2168/LMCS-2(1:3)2006},
    archivePrefix={arXiv},
    eprint={cs/0505013},
    primaryClass={cs.LO cs.CC}
}