VTC/sup O/: a second-order theory for TC/sup 0/
Phuong M. Nguyen, Stephen A Cook · 2004
We introduce a finitely axiomatizable second-order theory, which is VTC/sup 0/ associated with the class FO-uniform TC/sup 0/. It consists of the base theory V/sup 0/ for AC/sup 0/ reasoning together with the axiom NUMONES, which states the existence of a "counting array" Y for any string X: the ith row of Y contains only the number of 1 bits up to (excluding) bit i of X. We introduce the notion of "strong /spl Delta//sub 1//sup B/-definability" for relations in a theory, and use a recursive characterization of the TC/sup 0/ relations (rather than functions) to show that the TC/sup 0/ relations are strongly /spl Delta//sub 1//sup B/-definable. It follows that the TC/sup 0/ functions are /spl Sigma//sub 1//sup B/-definable in VTC/sup 0/. We prove a general witnessing theorem for second-order theories and conclude that the/spl Sigma//sub 1//sup B/ theorems of VTC/sup 0/ are witnessed by TC/sup 0/ functions. We prove that VTC/sup 0/ is RSUV isomorphic to the first order theory /spl Delta//sub 1//sup b/-CR of Johannsen and Pollett (the "minimal theory for TC/sup 0/"), /spl Delta//sub 1//sup b/-CR includes the /spl Delta//sub 1//sup b/ comprehension rule, and J and P ask whether there is an upper bound to the nesting depth required for this rule. We answer "yes", because VTC/sup 0/ , and therefore /spl Delta//sub 1//sup b/-CR, are finitely axiomatizable. Finally, we show that /spl Sigma//sub 1//sup B/ theorems of VTC/sup 0/ translate to families of tautologies which have polynomial-size constant-depth TC/sup 0/-Frege proofs. We also show that PHP is a /spl Sigma//sub 0//sup B/ theorem of VTC/sup 0/. These together imply that the family of propositional tautologies associated with PHP has polynomial-size constant-depth TC/sup 0/-Frege proofs.