VTC/sup O/: a second-order theory for TC/sup 0/

Abstract

We introduce a finitely axiomatizable second-order theory ÎÌ ¼ , which is associated with the class Ç-uniform Ì ¼ . It consists of the base theory Î ¼ for ¼ reasoning together with the axiom AEÍÅÇAE Ë, which states the existence of a "counting array" for any string : the th row of contains only the number of 1 bits up to (excluding) bit of . We introduce the notion of "strong ¡ ½ -definability" for relations in a theory, and use a recursive characterization of the Ì ¼ relations (rather than functions) to show that the Ì ¼ relations are strongly ¡ ½ -definable. It follows that the Ì ¼ functions are ¦ ½ -definable in ÎÌ ¼ .We prove a general witnessing theorem for second-order theories and conclude that the ¦ ½ theorems of ÎÌ ¼ are witnessed by Ì ¼ functions. We prove that ÎÌ ¼ is RSUV isomorphic to the first order theory ¡ ½ -Ê of Johannsen and Pollett (the "minimal theory for Ì ¼ "). ¡ ½ -Ê includes the ¡ ½ 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 ÎÌ ¼ , and therefore ¡ ½ -Ê, are finitely axiomatizable. Finally, we show that ¦ ¼ theorems of ÎÌ ¼ translate to families of tautologies which have polynomial-size constant-depth Ì ¼ -Frege proofs. We also show that ÈÀÈ is a ¦ ¼ theorem of ÎÌ ¼ . These together imply that the family of propositional tautologies associated with ÈÀÈ has polynomialsize constant-depth Ì ¼ -Frege proofs.

The counting hierarchy in binary notation

“…It was pointed out to us that ''substitution'' is not the standard designation for this scheme. This is the reason why we are using the more traditional epithet ''replacement'' ( [1], [10] and [15]).…”

“…Apropos bounded arithmetic theories related with counting, we should also mention the systems C 0 3 , C 0 2 and D b 1 -CR ( [10], [11]) by Johannsen and Pollett and the system VTC 0 ( [14], [15], [16]) by Phuong Nguyen and Stephen Cook. All these systems were developed in order to capture the computational power of particular classes of thresholds circuits.…”

“…D b 1 -CR is a ''minimal'' first-order theory to TC 0 . Another way to characterize TC 0 via formal systems is considering the finitely axiomatizable second-order theory VTC 0 (see [14], [15], [16]). It is known that VTC 0 is RSUV-isomorphic to D b 1 -CR and the former system appears to be weaker than D 0 1 ([16]).…”