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2018, Complexity Theory is Inconsistent

Abstract

The paper presents a counter-example to the celebrated Cook-Levin theorem which is stated as (copy-pasted from Cook’s 1971 paper): “Given an input w for M we will construct a proposition formula A(w) in conjunctive normal form such that A(w) is satisfiable if and only if M accepts w.”

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References (17)

  1. Theorem 9.1: SAT is (NOT) NP-complete. Proof: 1. M accepts w ⇐⇒ ["True" ⇐⇒ "False"].
  2. The celebrated Cook-Levin theorem stated (copy-pasted from Cook's 1971 paper): "Given an input w for M , we will construct a proposition formula A(w) in conjunctive normal form such that A(w) is satisfiable if and only if M accepts w." It can be written as: 3. ∀w ∈ L ∈ NP, ∃A(w) : A(w) is satisfiable ⇐⇒ M accepts w.
  3. Item [1] and [3] =⇒ ∀w ∈ L ∈ NP, ∃A(w): A(w) is satisfiable ⇐⇒ T r(r) = ["True" ⇐⇒ "False"].
  4. =⇒ [∃A(w) : A(w) is satisfiable] ⇐⇒ A(w) is unsatisfiable].
  5. =⇒ [ ∃A(w) : A(w) is satisfiable]. Thus:
  6. A(w) is satisfiable CAN NEVER be constructed in the case of a paradoxical input for a Paradox Recognizer Turing machine.
  7. =⇒ SAT is (NOT) NP-complete Theorem 9.2: SAT is NP-complete ⇐⇒ SAT is (NOT) NP-complete Proof:
  8. Cook's 1971 is still correct, so one has: SAT is NP-complete.
  9. From the above theorem: SAT is (NOT) NP-complete.
  10. =⇒ SAT is NP-complete ⇐⇒ SAT is (NOT) NP-complete.
  11. Some NP-complete Problems in Logic Other than SAT 1. Modal Logic S5-Satisfiability.
  12. Michael Garey, David Johnson:"Computers and Intractability: A Guide to the Theory of NP-completeness".
  13. Richard Karp: "Reducibility Among Combinatorial Problems", In Com- plexity of Computer Computations, edited by: R. E. Miller, J. W. Thatcher, pp. 85-103, (1972).
  14. Stephen Cole Kleene, Barkley Rosser:"The inconsistency of certain for- mal logics". Annals of Mathematics 36 (3): 630636.
  15. Stephen Cook: "The Complexity of Theorem-Proving Procedures", STOC 1971, downloaded from: http://rjlipton.wordpress.com.
  16. Stephen Cook: "P versus NP, Official Problem Description", http://www.claymath.org, 2004.
  17. Sanjeev Arora, Boaz Baraak: "Computational Complexity: A Modern Approach", Cambridge University Press, http://www.cs.princeton.edu.