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Axiomatizations and Conservation Results for Fragments of Bounded Arithmetic
, 1990
"... 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 ..."
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Cited by 27 (3 self)
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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
The Witness Function Method and Provably Recursive Functions of Peano
 Logic, Methodology and Philosophy of Science IX
, 1994
"... 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 complete ..."
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Cited by 5 (0 self)
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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 prooftheoretic and use the method of witness functions and witness oracles.
Abstract. A CONSERVATION RESULT CONCERNING BOUNDED THEOFUES AND THE COLLECTION AXIOM
, 1985
"... We present two proofs, one prooftheoretic and one modeltheoretic, showing that ..."
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We present two proofs, one prooftheoretic and one modeltheoretic, showing that
Binary Mod Generated By Their Tally Part
 Archive for Math. Logic
"... 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 endextensions 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 polyno ..."
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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 endextensions 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 (firstorder) 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#sI; 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#WWWWc (via thefunc#KjI symbol #) thec#eqjA ts 0 or 1 ac#AjIAG 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...