Fragments of Approximate Counting

We study the long-standing open problem of giving ∀Σ1 separations for fragments of bounded arithmetic in the relativized setting. Rather than considering the usual fragments defined by the amount of induction they allow, we study Jeřábek’s theories for approximate counting and their subtheories. We show that the ∀Σ1 Herbrandized ordering principle is unprovable in a fragment of bounded arithmetic that includes the injective weak pigeonhole principle for polynomial time functions, and also in a fragment that includes the surjective weak pigeonhole principle for FP functions. We further give new propositional translations, in terms of random resolution refutations, for the consequences of T 1 2 augmented with the surjective weak pigeonhole principle for polynomial time functions. ∗Supported in part by NSF grant DMS-1101228. †Partially supported during the early stages of this work by Polish Ministry of Science and Higher Education grant N201 382234. Part of the work was carried out while the author was visiting the University of California, San Diego, supported by Polish Ministry of Science and Higher Education programme “Mobilność Plus”. ‡Partially supported by institutional research plan AV0Z10190503 and grant IAA100190902 of GA AV ČR and by grant 1M0545 (ITI) of MŠMT. Part of this work was done on visits to the University of Warsaw and to the University of California, San