Unconditional $V_1^0$-independence of a certified hitting-set principle

Abstract

We show that a certified formalization of the hitting-set-existence axiom of Atserias and Tzameret, instantiated on the parity-based Nisan-Wigderson compression class of Khaniki, is independent of the two-sorted theory $V_1^0$ of ${\mathrm{AC}}^0$-reasoning, unconditionally: $V_1^0$ proves neither it nor its negation. The same holds for the corresponding certified dual weak pigeonhole principle, whose refutation is witnessed by a single seed that certified-computes every string of the model simultaneously. The mechanism is a bounded-arithmetic transfer of Atserias-Tzameret's reduction from hitting sets to the dual weak pigeonhole principle: the amplification half of that reduction, the sole source of its NP-oracle, is unnecessary at the native stretch of the Nisan-Wigderson map, and the compression half becomes a $V_1^0$-provable implication once circuit evaluation is replaced by its certified ${\Sigma }_0^B$ unfolding. This is, to our knowledge, the first independence result for a derandomization-flavoured existence principle at the ${\mathrm{AC}}^0$-reasoning level, and it makes explicit the bridge between the Khaniki Nisan-Wigderson line and the Atserias-Tzameret reverse mathematics of hitting sets.