[Paper Review] Lattice Gas Prediction is P-complete

Cristopher Moore, Mats G. Nordahl|ArXiv.org|Apr 17, 1997

Parallel Computing and Optimization Techniques7 references3 citations

TL;DR

This paper proves that predicting the HPP or FHP III lattice gas automaton for a finite time is P-complete, meaning it is as hard as solving any problem in P using a serial computer. The authors establish that no efficient parallel algorithm can significantly outperform explicit simulation unless P = NC or P = SP, implying inherent sequential complexity in lattice gas dynamics.

ABSTRACT

We show that predicting the HPP or FHP III lattice gas for finite time is equivalent to calculating the output of an arbitrary Boolean circuit, and is therefore P-complete: that is, it is just as hard as any other problem solvable by a serial computer in polynomial time. It is widely believed in computer science that there are inherently sequential problems, for which parallel processing gives no significant speedup. Unless this is false, it is impossible even with highly parallel processing to predict lattice gases much faster than by explicit simulation. More precisely, we cannot predict t time-steps of a lattice gas in parallel computation time O(log^k t) for any k, or O(t^α) for α< 1/2, unless the class P is equal to the class NC or SP respectively.

Motivation & Objective

• To determine the computational complexity of predicting lattice gas automata over finite time steps.
• To assess whether parallel computing can significantly accelerate lattice gas prediction beyond explicit simulation.
• To establish whether lattice gas prediction is inherently sequential, with no substantial parallel speedup possible.
• To relate lattice gas prediction to known complexity classes in theoretical computer science, particularly P, NC, and SP.
• To demonstrate that predicting lattice gas evolution is as hard as evaluating an arbitrary Boolean circuit.

Proposed method

• Reduction of an arbitrary Boolean circuit to a lattice gas configuration and time evolution.
• Construction of a lattice gas automaton (HPP or FHP III) that simulates the computation of a Boolean circuit over time.
• Mapping of circuit inputs and logic gates to particle configurations and collision rules in the lattice gas.
• Formal proof that predicting the lattice gas state after t time steps is equivalent to computing the output of the Boolean circuit.
• Use of standard complexity-theoretic reductions to show P-completeness under NC-reductions.
• Leveraging known results in computational complexity to infer that efficient parallel algorithms are unlikely unless P = NC or P = SP.

Experimental results

Research questions

• RQ1 Is predicting the state of a lattice gas automaton over finite time computationally as hard as solving any problem in P?
• RQ2 Can lattice gas prediction be efficiently parallelized, or is it inherently sequential?
• RQ3 Does the lattice gas model admit a polynomial-time parallel algorithm that outperforms explicit simulation?
• RQ4 What is the relationship between lattice gas prediction and the complexity class P, and how does it relate to NC and SP?
• RQ5 Can the prediction of lattice gas dynamics be reduced to evaluating an arbitrary Boolean circuit?

Key findings

• Lattice gas prediction for finite time is P-complete, meaning it is among the hardest problems in P.
• Predicting t time-steps of a lattice gas cannot be done in parallel computation time O(log^k t) for any k unless P = NC.
• No algorithm can predict lattice gas evolution in O(t^α) time for α < 1/2 unless P = SP.
• The result implies that no significant parallel speedup is possible for lattice gas prediction beyond explicit simulation.
• The proof relies on a reduction from an arbitrary Boolean circuit to a lattice gas configuration and dynamics.
• The HPP and FHP III lattice gas models are both shown to be P-complete under standard complexity-theoretic reductions.