TR13-069 Authors: Kousha Etessami, Alistair Stewart, Mihalis Yannakakis

Publication: 3rd May 2013 10:42

Downloads: 1882

Keywords:

The following two decision problems capture the complexity of

comparing integers or rationals that are succinctly represented in

product-of-exponentials notation, or equivalently, via arithmetic

circuits using only multiplication and division gates, and integer

inputs:

Input instance: four lists of positive integers:

$a_1, \ldots , a_n; \ b_1, \ldots ,b_n; \ c_1, \ldots ,c_m; \ d_1, \ldots ,d_m;$

where each of the integers is represented in binary.

Problem 1 (equality testing): Decide whether

$a_1^{b_1} a_2^{b_2} \ldots a_n^{b_n} = c_1^{d_1} c_2^{d_2} \ldots c_m^{d_m}$.

Problem 2 (inequality testing): Decide whether

$a_1^{b_1} a_2^{b_2} \ldots a_n^{b_n} \geq c_1^{d_1} c_2^{d_2} \ldots c_m^{d_m}$ .

Problem 1 is easily decidable in polynomial time using a simple

iterative algorithm. Problem 2 is much harder. We observe that the

complexity of Problem 2 is intimately connected to deep conjectures

and results in number theory. In particular, if a refined form of the

ABC conjecture formulated by Baker in 1998 holds, or if the older

Lang-Waldschmidt conjecture (formulated in 1978) on linear forms in

logarithms holds, then Problem 2 is decidable in P-time (in the

standard Turing model of computation). Moreover, it follows from the

best available quantitative bounds on linear forms in logarithms,

e.g., by Baker and W\"{u}stholz (1993) or Matveev (2000), that if m

and n are fixed universal constants then Problem 2 is decidable in

P-time (without relying on any conjectures).

We describe one application: P-time

maximum probability parsing for *arbitrary* stochastic

context-free grammars (where \epsilon-rules are allowed).