On the Asymptotic Nullstellensatz and Polynomial Calculus Proof Complexity

Søren Riis
2008 Logic in Computer Science  
We show that the asymptotic complexity of uniformly generated (expressible in First-Order (FO) logic) propositional tautologies for the Nullstellensatz proof system (NS) as well as for Polynomial Calculus, (PC) has four distinct types of asymptotic behavior over fields of finite characteristic. More precisely, based on some highly non-trivial work by Krajicek, we show that for each prime p there exists a function l(n) ∈ Ω(log(n)) for NS and l(n) ∈ Ω(log(log(n)) for PC, such that the
more » ... l translation of any FO formula (that fails in all finite models), has degree proof complexity over fields of characteristic p, that behave in 4 distinct ways: (i) The degree complexity is bound by a constant. (ii) The degree complexity is at least l(n) for all values of n. (iii) The degree complexity is bound by a constant on an infinite set S, and is at least l(n) on the complement N \ S. Furthermore, membership n ∈ S is for some k ∈ N determined uniquely by the value of n modulo q k . (iv) The degree complexity fluctuates between constant and l(n) (and possibly beyond) in a very particular way. We leave it as an open question whether the classification remains valid for l(n) ∈ n Ω(1) or even for l(n) ∈ Ω(n). Finally, we show that for any non-empty proper subset A ⊆ {(i), (ii), (iii), (iv)} the decision problem of whether a given input FO formula ψ has type belonging to A -is undecidable.
doi:10.1109/lics.2008.30 dblp:conf/lics/Riis08 fatcat:hij6dbovtvgirjnlph24ztbifu