Root finding in TC^0 and open induction
Seminar Room 1, Newton Institute
It is known that elementary arithmetic operations are computable in uniform TC^0, and some (multiplication, division) are even complete for this complexity class. The corresponding theory of bounded arithmetic, VTC^0, duly defines addition, multiplication, and ordering, and proves that integers form a discretely ordered ring under these operations. It is a natural question what other first-order properties of elementary arithmetic operations are provable in VTC^0.
In particular, we are interested whether VTC^0 (or a theory of similar
strength) can prove open induction in the basic language of arithmetic (Shepherdson's theory IOpen). This turns out equivalent to the problem whether there are TC^0 root-finding algorithms for constant-degree polynomials whose soundness is provable in VTC^0. In this talk, we will establish that such root-finding algorithms exist in the real (or rather, complex) world, and we will discuss the prospects of their formalization in bounded arithmetic.