Nepomnjascij's Theorem and Independence Proofs in Bounded Arithmetic

The use of Nepomnjaščǐi’s Theorem in the proofs of independence results for bounded arithmetic theories is investigated. Using this result and similar ideas, the following statements are proven: (1) At least one of S1 or TLS does not prove the Matiyasevich-Davis-RobinsonPutnam Theorem and (2) TLS does not prove Σ̂1,1 = Π̂ b 1,1. Here S1 is a conservative extension of the well-studied theory I∆0 and TLS is a theory whose ∆̂1,2-predicates are precisely LOGSPACE. The relation of TLS from this paper to previously studied theories is also developed and generalizations of the previous two results to quasi-linear settings are discussed as well. Mathematics Subject Classification: 03F30, 68Q15