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Why the classes P and NP are not well-defined finitarily

Bhupinder Singh Anand

Abstract

We distinguish finitarily between algorithmic verifiability, and algorithmic computability, to show that Goedel's 'formally' unprovable, but 'numeral-wise' provable, arithmetical proposition [(Ax)R(x)] can be finitarily evidenced as: algorithmically verifiable as 'always' true, but not algorithmically computable as 'always' true. Hence, though [R(x)] is algorithmically verifiable as a tautology, it is not algorithmically computable as a tautology by any Turing machine, whether deterministic or non-deterministic. By interpreting the PvNP problem arithmetically, rather than set-theoretically, we conclude that the clkasses P and NP are not well-defined finitarily since it immediately follows that SAT is neither in P nor in NP.

Why the classes P and NP are not well-defined finitarily

Abstract

We distinguish finitarily between algorithmic verifiability, and algorithmic computability, to show that Goedel's 'formally' unprovable, but 'numeral-wise' provable, arithmetical proposition [(Ax)R(x)] can be finitarily evidenced as: algorithmically verifiable as 'always' true, but not algorithmically computable as 'always' true. Hence, though [R(x)] is algorithmically verifiable as a tautology, it is not algorithmically computable as a tautology by any Turing machine, whether deterministic or non-deterministic. By interpreting the PvNP problem arithmetically, rather than set-theoretically, we conclude that the clkasses P and NP are not well-defined finitarily since it immediately follows that SAT is neither in P nor in NP.
Paper Structure (20 sections, 30 theorems)

This paper contains 20 sections, 30 theorems.

Table of Contents

  1. Introduction
  2. Revisiting an evidence-based paradigm
  3. Reviewing Tarski's inductive assignment of truth-values under an interpretation
  4. Tarski's inductive definitions
  5. The ambiguity in the classical standard interpretation of PA over $\mathbb{N}$
  6. The weak, algorithmically verifiable, standard interpretation $\mathcal{I}_{PA(\mathbb{N},\ SV)}$ of PA
  7. The PA axioms are algorithmically verifiable as true under $\mathcal{I}_{PA(\mathbb{N},\ SV)}$
  8. The strong, algorithmically computable, interpretation $\mathcal{I}_{PA(\mathbb{N},\ SC)}$ of PA
  9. The PA axioms are algorithmically computable as true under $\mathcal{I}_{PA(\mathbb{N},\ SC)}$
  10. Bridging PA Provability and Turing-computability
  11. Gödel's 'undecidable' formula $[\neg(\forall x)R(x)]$ is provable in PA
  12. The significance of the Provability Theorem for PA for the P$v$NP problem
  13. Interpreting the standard definition of the P$v$NP problem finitarily
  14. SAT $\notin$ P and SAT $\notin$ NP
  15. Appendix: An implicit ambiguity in the 'official' definition of P
  16. ...and 5 more sections

Key Result

Lemma 2..1

Under the interpretation $\mathcal{I}_{PA(\mathbb{N},\ S)}$, an atomic formula $A^{*}(x_{1}, x_{2}, \ldots, x_{n})$ would be evidenced as both algorithmically verifiable and algorithmically computable in $\mathbb{N}$ by any witness $\mathcal{W}_{(\mathbb{N},\ S)}$.

Theorems & Definitions (40)

  • Definition 12
  • Definition 13
  • Definition 14
  • Definition 15
  • Definition 16
  • Definition 17
  • Lemma 2..1
  • Definition 18
  • Theorem 2..2
  • Lemma 2..3
  • ...and 30 more