Sampling Markov Models under Constraints: Complexity Results for Binary Equalities and Grammar Membership

TL;DR

This paper investigates the computational complexity of sampling sequences from Markov models under constraints like Binary Equalities and Grammar Membership, identifying both intractable cases and new tractable classes.

Contribution

It proves #P-completeness for key constrained sampling problems and introduces a broader class of grammars where sampling remains computationally feasible.

Findings

#P-completeness for binary equalities and grammar membership constraints

Identification of polynomial cases for unambiguous grammars

Introduction of a new class of grammars with tractable sampling

Abstract

We aim at enforcing hard constraints to impose a global structure on sequences generated from Markov models. In this report, we study the complexity of sampling Markov sequences under two classes of constraints: Binary Equalities and Grammar Membership Constraints. First, we give a sketch of proof of #P-completeness for binary equalities and identify three sub-cases where sampling is polynomial. We then give a proof of #P-completeness for grammar membership, and identify two cases where sampling is tractable. The first polynomial sub-case where sampling is tractable is when the grammar is proven to be unambiguous. Our main contribution is to identify a new, broader class of grammars for which sampling is tractable. We provide algorithm along with time and space complexity for all the polynomial cases we have identified.