The consistency of "P = NP" and related problems with fragments of number theory

The main results of this paper demonstrate the consistency of “P &equil; NP” and a variant of “NP @@@@ coNP” with certain natural fragments of number theory to be defined precisely in the sequel.@@@@ Consistency results represent an approach to the lower bound problems of complexity theory which points to a number of interesting lines of inquiry. Our ultimate goal is to make precise the difficulty of proving certain nontrivial lower bounds. Among the possibilities which follow from this approach are: (1) that logical techniques may help us resolve the P &equil; NP question, (2) that showing why certain arguments must fail may lead to mathematical tools capable of resolving the problems, and (3) that the special character of model theoretic methods in complexity theory may lead to new results which are of purely logical interest. We will address these possibilities below.

[1]  G. Hardy.  Properties of Logarithmico‐Exponential Functions , 2022 .

[2]  S. C. Kleene,et al.  Introduction to Metamathematics , 1952 .

[3]  J. Paris.  A Mathematical Incompleteness in Peano Arithmetic , 1977 .

[4]  Jeff B. Paris,et al.  Some independence results for Peano arithmetic , 1978, Journal of Symbolic Logic.

[5]  John L. Bell,et al.  A course in mathematical logic , 1977 .

[6]  P. J. Cohen.  Set Theory and the Continuum Hypothesis , 1966 .

[7]  Abraham Robinson,et al.  Forcing in Model Theory , 1975 .

[8]  R. Statman.  Lower bounds on Herbrand’s theorem , 1979 .

[9]  Paul Young,et al.  Independence Results in Computer Science? , 1981, J. Comput. Syst. Sci..

[10]  T. Skolem.  Über die Nicht-charakterisierbarkeit der Zahlenreihe mittels endlich oder abzählbar unendlich vieler Aussagen mit ausschliesslich Zahlenvariablen , 1934 .

[11]  M. E. Szabo,et al.  The collected papers of Gerhard Gentzen , 1969 .

[12]  Richard J. Lipton,et al.  Model theoretic aspects of computational complexity , 1978, 19th Annual Symposium on Foundations of Computer Science (sfcs 1978).

[13]  Stephen A. Cook,et al.  Feasibly constructive proofs and the propositional calculus (Preliminary Version) , 1975, STOC.

[14]  John Gill,et al.  Relativizations of the P =? NP Question , 1975, SIAM J. Comput..

[15]  Henryk Wozniakowski,et al.  Complexity of linear programming , 1982, Oper. Res. Lett..

[16]  Abraham Robinson.  Nonstandard Arithmetic and Generic Arithmetic1 , 1973 .

[17]  Vaughan R. Pratt,et al.  Every Prime has a Succinct Certificate , 1975, SIAM J. Comput..

[18]  Richard J. Lipton,et al.  Some connections between mathematical logic and complexity theory , 1979, STOC.

[19]  Th. Skolem,et al.  Peano's Axioms and Models of Arithmetic , 1955 .

[20]  R. Statman.  Herbrand's Theorem and Gentzen's Notion of a Direct Proof , 1977 .

[21]  R. G. Cooke,et al.  Abstract Set Theory , 1954, Nature.

[22]  Michael O'Donnell,et al.  A programming language theorem which is independent of Peano Arithmetic , 1979, STOC.