Arithmetizing Uniform NC

Allen, Bill

doi:10.1016/0168-0072(91)90057-s

Cited by 33 publications

(12 citation statements)

References 13 publications

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“…An alternative approach to characterizing N C 1 is given by Bloch [4] based on the methods of this paper and the work of Allen [1]. Recently, Leivant and Marion [19], [20] have used a method similar to ours and [18] to characterize the Kalmar elementary functions.…”

Section: Directions For Researchmentioning

confidence: 99%

A new recursion-theoretic characterization of the polytime functions

1992

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Section: Directions For Researchmentioning

confidence: 99%

A new recursion-theoretic characterization of the polytime functions

1992

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“…Case (1) f(tl,..., tn). We treat only the case that t is tl~#t~ 2 since other cases can be treated similarly.…”

Section: Fl __~ 4l 3y < 2tl+i(fl({x}(x < T L a Bit(x Y) = 1))) []mentioning

confidence: 98%

“…This isomorphism is called the RSUV isomorphism between two theories. In [10], it is proved among many other things that R~ is isomorphic to U~(BD), where U~I(BD) is obtained from U~(BD) by deleting # and [~ is a reformulation of D 1 in B. Allen [1] and also a system in Clote and Takeufi [3] which is a subsystem of S~ and corresponds to NC.…”

Section: Is In L Iff It Is Expressed By An Esb Formula In Tls a Funcmentioning

confidence: 99%

“…We investigate the second order bounded arithmetical systems which is isomorphic to TAC ~, TNC i or TLS.In [1], S. Buss has introduced the first order systems of Bounded Arithmetic S~ and T~, which corresponds to computational classes in polynomial hierachy A~' and AiP+I respectively. Especially S~ corresponds to the polynomial time computable class P. There he has also introduced a bounded second order system U~(BD) which corresponds to the computational class PSPACE.…”

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confidence: 99%

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RSUV isomorphisms for TAC i , TNC i and TLS

Takeuti

1995

Arch Math Logic

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Summary. We investigate the second order bounded arithmetical systems which is isomorphic to TAC ~, TNC i or TLS.In [1], S. Buss has introduced the first order systems of Bounded Arithmetic S~ and T~, which corresponds to computational classes in polynomial hierachy A~' and AiP+I respectively. Especially S~ corresponds to the polynomial time computable class P. There he has also introduced a bounded second order system U~(BD) which corresponds to the computational class PSPACE.There are many important computational classes below P e.g. AC i, NC i, and L. The classes NC i and AC' are those predicates or functions computable by a uniform family of polynomial size, O(log i n) depth circuits with constant and unbounded fanin respectively. The class L is those predicates or functions logspace computable by a deterministic Turing machine.In Clote and Takeuti MSP(a,i + l)= llMSP(a,i) j •In order to formulate these theories the notion "essentially sharply bounded" (abbreviated by esb) plays a central role. Let T be a theory in Bounded Arithmetic. A formula is said to be esb in T if it belongs to the smallest family ._~7" satisfying the following conditions.(1) Every atomic formula belongs to ~° (2) ~ is closed under Boolean connectives.(3) ~" is closed under sharply bounded quantifications.

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“…In Sect. 3 we illustrate this framework on Buss' witnessing theorem. Construction A (Theorem 1) developed there manages not to introduce new lengths but does not appear to be easily amenable to other situations.…”

Section: Introductionmentioning

confidence: 99%

Construction of models of bounded arithmetic by restricted reduced powers

Garlík

2016

Arch. Math. Logic

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We present two constructions of models of bounded arithmetic, both in the form of a generalization of the ultrapower construction, that yield nonelementary extensions but do not introduce new lengths. As an application we show, assuming the existence of a one-way permutation g hard against polynomial-size circuits, that strictR 1 2 (g) is weaker than R 1 2 (g). In particular, if such a permutation can be defined by a term in the language of R 1 2 , then strictR 1 2 is weaker than R 1 2 .

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Arithmetizing Uniform NC

Cited by 33 publications

References 13 publications

A new recursion-theoretic characterization of the polytime functions

A new recursion-theoretic characterization of the polytime functions

RSUV isomorphisms for TAC i , TNC i and TLS

Construction of models of bounded arithmetic by restricted reduced powers

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