This paper presents new results on axiomatizations for fragments of Bounded Arithmetic which improve upon the author's dissertation. It is shown that (# i+1 )-PIND and strong # i -replacement are consequences of S 2 . Also # i+1 -IND is a consequence of T 2 . The latter result is proved by showing that S i+1 -conservative over 2 . Furthermore, S i+1 -replacement with respect to Boolean combinations of # i+1 -formulas. 1

This paper presents a new proof of the characterization of the provably recursive functions of the fragments I# n of Peano arithmetic. The proof method also characterizes the # k -definable functions of I# n and of theories axiomatized by transfinite induction on ordinals. The proofs are completely proof-theoretic and use the method of witness functions and witness oracles.

We present two proofs, one proof-theoretic and one model-theoretic, showing that

We intro du# a class of models of the bou arithmetic theory PV n . These models, which are generated by their tally part, have acu featu : they have end-extensions or satisfy B# b n only in case they are closeduose exponentiation. As an application, we show that if I#0 + exp # B#1 then the polynomial hierarchy does not collapse. This paper isc#-;A#Ijwith bounded theories ofarithmetic# following Buss (1986). Nonetheless, as opposed to Buss'c#ss'-;jA setting -- where the system of natural numbers forms the standard model -- we work with theories that aim todesc#jW e the language {0, 1} # .Henc## we shall use the notation of Ferreira (1990a & 1990b). To help the reader unfamiliar with the notation we briefly desc#y- e the (first-order) stringlanguage that we use. This stringlanguagec#tringl of threec#ree-# t symbols #, 0 and 1, two binary func#ry- symbols # (for concatenation, usually omitted) and , and a binary relation symbol # (for initialsitia dnes ). The interpretation of these symbols in the standard model 0 1} is c#s-I; exc#;# for thefunc##j# symbol : x y is the string xc#Ij#;-G;#;I with itself length of y times. Given an element e # 0 1} , we denote by e thec#e-## term of the language obtained by # This work wa spa69g supported by project 6E92 of CMAF (Portuga2 1 c#c#W-W-W-W-c (via thefunc#KjI symbol #) thec#e-qjA ts 0 or 1 ac#AjIA-G to the order of the bits in e (for determinateness, we always asso c#so- # to the left). We use the following abbreviations: x # # y (s dnes of x with respec# to y) abbreviates #z # y(z#x# y); x # y (the length of x is less than or equal to the length of y) abbreviates 1 x # 1 y; and x # y (x and y have the same length) abbreviates x # y # y # x. The theories studied in this paper are bu...