Theories for TC0 and Other Small Complexity Classes
Phuong Nguyen2006, Logical Methods in Computer Science
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Abstract
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 differentlooking 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.
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