This article appears in the following Journal of Logic and Computation issue: Special Issue: Logic and Computation in the Real World: CiE 2007 [View the issue table of contents]
Original Articles |
Existentially Closed Models and Conservation Results in Bounded Arithmetic
Facultad de Matemáticas. Universidad de Sevilla, C/ Tarfia s/n, 41012 Sevilla (Spain).
E-mail: acordon{at}us.es,afmargarit{at}us.es,fflara{at}us.es
Received 28 September 2007.
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Abstract |
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We develop model-theoretic techniques to obtain conservation results for first order Bounded Arithmetic theories, based on a hierarchical version of the well-known notion of an existentially closed model. We focus on the classical Buss' theories S
and T and prove that they are 
conservative over their inference rule counterparts, and 
conservative over their parameter-free versions. A similar analysis of the -replacement scheme is also developed. The proof method is essentially the same for all the schemes we deal with and shows that these conservation results between schemes and inference rules do not depend on the specific combinatorial or arithmetical content of those schemes. We show that similar conservation results can be derived, in a very general setting, for every scheme enjoying some syntactical (or logical) properties common to both the induction and replacement schemes. Hence, previous conservation results for induction and replacement can be also obtained as corollaries of these more general results.
Keywords: Bounded arithmetic; existentially closed models; conservation results; parameter-free schemes