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- W77620528 abstract "Samuel Buss' theory S$sbsp{2}{1}$ is that bounded analog of the theory I$sumsbsp{1}{0}$ in which induction for NP-predicates plays the role of induction for recursively enumerable predicates. A new bounded theory, the bounded analog of primitive recursive arithmetic, is defined. In this theory, called PTCA, induction of arithmetic that is used is the language ${0,1}$*, as opposed to the natural numbers $omega$. The provably recursive functions of this theory are characterized and a certain class of thin models is defined, the elements of which do not have end extensions. In a small digression we consider the subtheory of PTCA with induction restricted to strictly rudimentary predicates: it is shown that the complement function is not provably total in this subtheory.By permitting NP-induction on notation we obtain a theory that is $forallsumsbsp{1}{rm b}$-conservative over PTCA. An intuitionistic version of this new system is studied and, via a realizability argument within a suitable infinitary theory, a question of Buss is partially settled.Finally, second-order theories that are $forallsumsbsp{1}{rm b}$-conservative over PTCA are introduced. These theories may contain (1) a particular case of the $Deltasbsp{1}{0}$-comprehension scheme that implies the existence of NP $cap$ co-NP sets, (2) weak Konig's lemma for trees in the Meyer-Stockmeyer hierarchy of predicates, and (3) the scheme of bounded collection. It is shown that the Heine-Borel theorem and the uniform continuity theory in the Cantor space 2$sp{omega}$ are provable in suitable $forallsumsbsp{1}{rm b}$-conservative extensions of PTCA. It is argued that the amount of induction necessary to prove the theorem on the existence of a maximum for total continuous functions (with domain and range 2$sp{omega}$) is slow induction for NP-predicates." @default.
- W77620528 created "2016-06-24" @default.
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- W77620528 date "1988-01-01" @default.
- W77620528 modified "2023-09-27" @default.
- W77620528 title "Polynomial time computable arithmetic and conservative extensions" @default.
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