Random Lattices and a Conjectured 0 - 1 Law about Their Polynomial Time Computable Properties

Authors:
Miklós Ajtai
Affiliations:
-
Venue:
FOCS '02 Proceedings of the 43rd Symposium on Foundations of Computer Science
Year:
2002

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Abstract

We formulate a conjecture about random n-dimensional lattices with a suitable distribution. The conjecture says that every polynomial time computable property of a random lattice holds with a probabiltiy either close to 0 or close to 1. Accepting the conjecture we get a large classof hard lattice problems. We describe an analogy between our conjecture and a set theoretical axiom, which cannot be proved in ZFC. This axiom says that there exists a nontrivial \sigma -additive 0 - 1 measure defined on the set of all subsets of some set S.