Fragments of approximate counting
Seminar Room 1, Newton Institute
We pose the question: are there any simple sentences, provable using bounded induction, that cannot already be proved using approximate counting of simple sets? More precisely, we study the long-standing open problem of giving $\forall \Sigma^b_1$ separations for fragments of bounded arithmetic in the relativized setting, but rather than considering the usual fragments defined by the amount of induction they allow, we study Jerabek's theories for approximate counting and their subtheories. We prove separations for some of the subtheories, and also give new propositional translations, in terms of random resolution refutations, for the consequences of $T^1_2$ augmented with a certain weak pigeonhole principle. Joint work with Sam Buss and Leszek Kolodziejczyk.