## COMPLEXITY CLASSES AS MATHEMATICAL AXIOMS (810)

### BibTeX

@MISC{Freedman810complexityclasses,

author = {Michael H. Freedman},

title = {COMPLEXITY CLASSES AS MATHEMATICAL AXIOMS},

year = {810}

}

### OpenURL

### Abstract

Abstract. Treating a conjecture, P #P ̸ = NP, on the separation of complexity classes as an axiom, an implication is found in three manifold topology with little obvious connection to complexity theory. This is reminiscent of Harvey Friedman’s work on finitistic interpretations of large cardinal axioms. 1.

### Citations

2348 | Computational Complexity
- Papadimitriou
- 1994
(Show Context)
Citation Context ...ing finite quantification. Toda proved that PH ⊆ P PP [11]. Finally, PSPACE is the class of decision problems solvable in an arbitrary amount of time, but using only a polynomial memory resource. See =-=[7]-=- for more background. We use Axiom A, P #P ̸= NP, to prove Theorem A. Failure of Axiom A would imply a large collapse of the polynomial hierarchy PH down to NP, so Axiom A must be considered extremely... |

301 |
The undecidability of the domino problem
- Berger
- 1966
(Show Context)
Citation Context ... than two crossings or two critical points can be added to a diagram per unit weight step. This completes the proof of Theorem A in ZF ∪ Axiom A. 5. Conclusion Mathematical structures such as tilings =-=[1]-=-, groups [8], and, in several contexts, links [3] are known to encode quite general computations. If transformations are6 MICHAEL H. FREEDMAN found which preserve the computational “content” of the s... |

273 |
Invariants of 3-manifolds via Link polynomials and quantum
- Reshetikhin, Turaev
- 1991
(Show Context)
Citation Context ...between links L and complexity is the writhe unnormalized Jones polynomial [12] (also known as the Kauffman bracket) which we write as JL(q). Evaluations of JL at roots of unity ω = e 2πi/r are known =-=[13]-=- to be computed as the partition function Z SU(2),k(S 3 , L) of the topological quantum field theory (TQFT) associated with the Lie group SU(2) at level k = r − 2. What will be of critical importance ... |

108 | VYALYI: Classical and Quantum Computation
- KITAEV, SHEN, et al.
(Show Context)
Citation Context ... added to a diagram per unit weight step. This completes the proof of Theorem A in ZF ∪ Axiom A. 5. Conclusion Mathematical structures such as tilings [1], groups [8], and, in several contexts, links =-=[3]-=- are known to encode quite general computations. If transformations are6 MICHAEL H. FREEDMAN found which preserve the computational “content” of the structure, then it may be expected that axioms sta... |

105 |
A.(4-OX) On the computational complexity of the Jones and Tutte polynomials
- Jaeger, Vertigan, et al.
- 1990
(Show Context)
Citation Context ...as a lower bound to the Euler characteristic (the exponent in VF) for any z = constant slice of the link complement in R3 . 1 Actually, applying Lagrangian interpolation, these functions are shown in =-=[5]-=- to be FP #P - complete6 MICHAEL H. FREEDMAN Thus, there is a prospect of replacing the oracle Jr entirely with a classical polynomial time computation in this small Hilbert space, by representing cr... |

60 |
On the computational power of PP and ⊕P , in
- Toda
- 1989
(Show Context)
Citation Context ...so weakening #P to a language does not affect its oracular power. PH denotes the polynomial time hierarchy, a game theoretic extension of NP allowing finite quantification. Toda proved that PH ⊆ P PP =-=[11]-=-. Finally, PSPACE is the class of decision problems solvable in an arbitrary amount of time, but using only a polynomial memory resource. See [7] for more background. We use Axiom A, P #P ̸= NP, to pr... |

36 |
The globalization of communication
- Thompson
- 2003
(Show Context)
Citation Context ...er 2 Reidemeister 3 Birth Death Level Crossing Figure 2. Before turning to the proof, let us see that it is the equivalence relation ∼r that creates the “technique vacuum.” In the 1990’s, A. Thompson =-=[10]-=- pointed out to me that girth, by itself, can sometimes be computed exactly. Claim. Let k be the (p, q)-torus knot. Then g(k) = 2 min(p, q). Proof. So, k ⊂ T ⊂ R 3 , where T is an unknotted torus whic... |

28 | Finite functions and the necessary use of large cardinals
- Friedman
- 1998
(Show Context)
Citation Context ...ight have in mathematics as a whole. This program is analogous to the search for interesting “finitistic” consequences of large cardinal axioms, an area explored by Harvey Friedman and collaborators (=-=[4]-=-). (Although, in the latter case, the large cardinal axioms are actually known to be independent of ZFC.) What would be the best possible theorem in this subject? It would be to postulate a very weak ... |

28 |
A spanning tree expansion of the Jones polynomial
- Thistlethwaite
(Show Context)
Citation Context ...inting out that Fox [2] considered a relation similar to ∼r in the 1950’s and that Lackenby’s theorem 2.1 [5] contains lemma 4.1. □ It is a theorem of Vertigan ([14] or [15] assisted by the result of =-=[9]-=-) that all nonzero algebraic evaluations of the Jones polynomial JL(q) are #P-complete functions of the input L with the exceptions of those q satisfying q 4 = 1 or q 6 = 1. Thus, in oracle notation, ... |

12 |
Quantum invariants of knots and 3-manifolds, volume 18 of de Gruyter
- Turaev
- 1994
(Show Context)
Citation Context ...olynomial hierarchy PH down to NP, so Axiom A must be considered extremely safe. 4. Axiom A Implies Theorem A The connection between links L and complexity is the writhe unnormalized Jones polynomial =-=[12]-=- (also known as the Kauffman bracket) which we write as JL(q). Evaluations of JL at roots of unity ω = e 2πi/r are known [13] to be computed as the partition function Z SU(2),k(S 3 , L) of the topolog... |

11 |
The computational complexity of Tutte invariants for planar graphs
- Vertigan
- 1992
(Show Context)
Citation Context ...2πi/r). I thank Ian Agol for pointing out that Fox [2] considered a relation similar to ∼r in the 1950’s and that Lackenby’s theorem 2.1 [5] contains lemma 4.1. □ It is a theorem of Vertigan ([14] or =-=[15]-=- assisted by the result of [9]) that all nonzero algebraic evaluations of the Jones polynomial JL(q) are #P-complete functions of the input L with the exceptions of those q satisfying q 4 = 1 or q 6 =... |

8 |
Non-recursive functions, knots “with thick ropes”, and self-clenching “thick” hyperspheres
- Nabutovsky
- 1995
(Show Context)
Citation Context ...inks L cannot be made, via ∼r, extremely thin except possibly by an extraordinarily elaborate sequence of moves. It would be a surprise if the second alternative actually occurred. In high dimensions =-=[6]-=-, unsolvability of the triviality problem for groups implies that geometric landscapes, for example that of the 5-sphere in S 6 , are extremely (non-recursively) rough. However, this phenomenon has no... |

8 |
The word problem and the isomorphism problem for groups
- Stillwell
- 1982
(Show Context)
Citation Context ...ossings or two critical points can be added to a diagram per unit weight step. This completes the proof of Theorem A in ZF ∪ Axiom A. 5. Conclusion Mathematical structures such as tilings [1], groups =-=[8]-=-, and, in several contexts, links [3] are known to encode quite general computations. If transformations are6 MICHAEL H. FREEDMAN found which preserve the computational “content” of the structure, th... |

7 |
Congruence classes of knots
- Fox
- 1958
(Show Context)
Citation Context ... figure 4 to the ai labeled particle line crossing ∆. Since θ(ai) 4r = 1, Z(Gi) does not change under a 8πr twist. Consequently, JL(e2πi/r ) = JL ′(e2πi/r). I thank Ian Agol for pointing out that Fox =-=[2]-=- considered a relation similar to ∼r in the 1950’s and that Lackenby’s theorem 2.1 [5] contains lemma 4.1. □ It is a theorem of Vertigan ([14] or [15] assisted by the result of [9]) that all nonzero a... |

7 |
Fox’s congruence classes and the quantum-SU(2) invariants of links in 3-manifolds
- Lackenby
- 1996
(Show Context)
Citation Context ...ot change under a 8πr twist. Consequently, JL(e2πi/r ) = JL ′(e2πi/r). I thank Ian Agol for pointing out that Fox [2] considered a relation similar to ∼r in the 1950’s and that Lackenby’s theorem 2.1 =-=[5]-=- contains lemma 4.1. □ It is a theorem of Vertigan ([14] or [15] assisted by the result of [9]) that all nonzero algebraic evaluations of the Jones polynomial JL(q) are #P-complete functions of the in... |

3 |
On the computational complexity of Tutte, Jones, Homfly and Kauffman invariants, D.Phil
- Vertigan
- 1991
(Show Context)
Citation Context ...= JL ′(e2πi/r). I thank Ian Agol for pointing out that Fox [2] considered a relation similar to ∼r in the 1950’s and that Lackenby’s theorem 2.1 [5] contains lemma 4.1. □ It is a theorem of Vertigan (=-=[14]-=- or [15] assisted by the result of [9]) that all nonzero algebraic evaluations of the Jones polynomial JL(q) are #P-complete functions of the input L with the exceptions of those q satisfying q 4 = 1 ... |