A note on the complexity of comparing succinctly represented integers, with an application to maximum probability parsing

Kousha Etessami, Alistair Stewart, Mihalis Yannakakis

Research output: Contribution to journalArticlepeer-review

Abstract

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, ...., a_n ; b_1,...., b_n ; c_1,....,c_m ; d_1, ...., 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} .... a_n^{b_n} = c_1^{d_1} c_2^{d_2} .... c_m^{d_m} .
Problem 2 (inequality testing): Decide whether a_1^{b_1} a_2^{b_2} ... a_n^{b_n} >= c_1^{d_1} c_2^{d_2} .... 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).
Original languageEnglish
Article number69
Number of pages21
JournalElectronic Colloquium on Computational Complexity (ECCC)
Volume20
Publication statusPublished - 2013

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