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Unifying lower bounds for algebraic machines, semantically

Thomas Seiller, Luc Pellissier, Ulysse Léchine

TL;DR

A new abstract method is presented that is used to prove that maxflow, a Ptime complete problem, is not computable in polylogarithmic time on parallel random access machines ( pram s) working with real numbers, and improves on a result of Mulmuley.

Abstract

This paper presents a new abstract method for proving lower bounds in computational complexity. Based on the notion of topological and measurable entropy for dynamical systems, it is shown to generalise three previous lower bounds results from the literature in algebraic complexity. We use it to prove that maxflow, a Ptime complete problem, is not computable in polylogarithmic time on parallel random access machines (prams) working with real numbers. This improves, albeit slightly, on a result of Mulmuley since the class of machines considered extends the class "prams without bit operations", making more precise the relationship between Mulmuley's result and similar lower bounds on real prams. More importantly, we show our method captures previous lower bounds results from the literature, thus providing a unifying framework for "topological" proofs of lower bounds: Steele and Yao's lower bounds for algebraic decision trees, Ben-Or's lower bounds for algebraic computation trees, Cucker's proof that NC is not equal to Ptime in the real case, and Mulmuley's lower bounds for "prams without bit operations".

Unifying lower bounds for algebraic machines, semantically

TL;DR

A new abstract method is presented that is used to prove that maxflow, a Ptime complete problem, is not computable in polylogarithmic time on parallel random access machines ( pram s) working with real numbers, and improves on a result of Mulmuley.

Abstract

This paper presents a new abstract method for proving lower bounds in computational complexity. Based on the notion of topological and measurable entropy for dynamical systems, it is shown to generalise three previous lower bounds results from the literature in algebraic complexity. We use it to prove that maxflow, a Ptime complete problem, is not computable in polylogarithmic time on parallel random access machines (prams) working with real numbers. This improves, albeit slightly, on a result of Mulmuley since the class of machines considered extends the class "prams without bit operations", making more precise the relationship between Mulmuley's result and similar lower bounds on real prams. More importantly, we show our method captures previous lower bounds results from the literature, thus providing a unifying framework for "topological" proofs of lower bounds: Steele and Yao's lower bounds for algebraic decision trees, Ben-Or's lower bounds for algebraic computation trees, Cucker's proof that NC is not equal to Ptime in the real case, and Mulmuley's lower bounds for "prams without bit operations".

Paper Structure

This paper contains 58 sections, 56 theorems, 41 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Computational Complexity
  3. Separation, Lower bounds and Barriers
  4. Algebraic models and Geometric Complexity Theory
  5. Lower bounds for pram s without bit operations
  6. Contributions.
  7. A more detailed view of the proof method
  8. Computational trees.
  9. Cucker's result.
  10. Mulmuley's result.
  11. Euclidean division.
  12. Contents of the paper
  13. Computation models as graphings.
  14. The general algebraic model
  15. Entropy
  16. ...and 43 more sections

Key Result

Theorem 1

Let $N$ be a natural number and $M$ be a real-valued pram with at most $2^{O((\log N)^c)}$ processors, where $c$ is any positive integer. Then $M$ does not compute euclidean division by $2$ on inputs of length $N$ in $O((\log N)^c)$ steps.

Figures (1)

  • Figure 1: Two curves that define the same partition of $\mathbf{Z}^2$

Theorems & Definitions (134)

  • Remark
  • Theorem 1
  • Proposition 2
  • Corollary 3: Steele and Yao SteeleYao82
  • Corollary 4: Mulmuley Mulmuley99
  • Lemma 5
  • Corollary 6: Ben-Or83
  • Corollary 7
  • Theorem 8
  • Theorem 8
  • ...and 124 more