Complexity classes as mathematical axioms

Abstract

Treating a conjecture, P #P 6 NP, on the separation of complex- ity classes as an an implication is found in three manifold topology with little obvious connection to complexity theory. This is reminiscent of Harvey Friedman's work on interpretations of large cardinal axioms. This short paper introduces a new subject with a simple example. The theory of computation defines a plethora of complexity classes. While the techniques of diagonalization and oracle relativization have produced important separation results, for nearly forty years the most interesting (absolute) separation conjectures, such as P 6 NP remain unproven, and with the invention of ever more complexity classes, analogous separation conjectures have multiplied in number. With no prospect in sight for proving these conjectures (within ZFC), we pro- pose considering them instead as potential axioms and looking for what implications they might 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 postu- late a weak separation axiom, say P 6 PSPACE, and prove the Riemann hypothesis, i.e. an important mathematical result apparently far removed from complexity theory. Of course, we should be more modest. We will assume a more technical but well accepted separation axiom P #P 6 NP, which we call Axiom A, and prove a theorem, Theorem A, in knot theory. The theorem is easily and briefly expressed in terms of classical notions such as girth and Dehn surgery and appears to be as close to current research topics in knot theory as the known implications of the large cardinal axioms are to research in Ramsey theory, to continue that analogy. Theorem A is extremely believable but seems to exist in a technique vacuum. What makes the theorem interesting is that it sounds both very plausible and impossible to prove.