Binary models generated by their tally part

We introduce a class of models of the bounded arithmetic theory PVn. These models, which are generated by their tally part, have a curious feature : they have end-extensions or satisfy BΣn only in case they are closed under exponentiation. As an application, we show that if I∆0 + ¬exp BΣ1 then the polynomial hierarchy does not collapse. This paper is concerned with bounded theories of arithmetic, following Buss (1986). Nonetheless, as opposed to Buss’ classical setting – where the system of natural numbers forms the standard model – we work with theories that aim to describe the language {0, 1}∗. Hence, we shall use the notation of Ferreira (1990a & 1990b). To help the reader unfamiliar with the notation we briefly describe the (first-order) stringlanguage that we use. This stringlanguage consists of three constant symbols , 0 and 1, two binary function symbols (for concatenation, usually omitted) and ×, and a binary relation symbol ⊆ (for initial subwordness). The interpretation of these symbols in the standard model {0, 1}∗ is clear, except for the function symbol ×: x× y is the string x concatenated with itself length of y times. Given an element e ∈ {0, 1}∗, we denote by ē the closed term of the language obtained by ∗This work was partially supported by project 6E92 of CMAF (Portugal)