Improved Randomized Approximation of Hard Universality and Emptiness Problems

TL;DR

This work extends polynomial randomized approximation (PRAX) algorithms from NFA universality and (in)equivalence to a broad class of emptiness and universality problems across domains with tractable or samplable distributions. It proves a linear-in-$1/\delta$ sample size bound, replacing previous quadratic bounds and enabling efficient approximate decisions via Dirichlet-based sampling and truncation. The framework, including notions of tractable and locally tractable distributions, is instantiated for concrete problems such as tautology testing, 2D automata, and certain Diophantine equations, with tailored time complexities. The results offer a practical experimental mathematics approach to hard decision problems, delivering provable probabilistic guarantees while highlighting ongoing questions about constant factors and scope.

Abstract

We build on recent research on polynomial randomized approximation (PRAX) algorithms for the hard problems of NFA universality and NFA equivalence. Loosely speaking, PRAX algorithms use sampling of infinite domains within any desired accuracy $δ$. In the spirit of experimental mathematics, we extend the concept of PRAX algorithms to be applicable to the emptiness and universality problems in any domain whose instances admit a tractable distribution as defined in this paper. A technical result here is that a linear (w.r.t. $1/δ$) number of samples is sufficient, as opposed to the quadratic number of samples in previous papers. We show how the improved and generalized PRAX algorithms apply to universality and emptiness problems in various domains: ordinary automata, tautology testing of propositions, 2D automata, and to solution sets of certain Diophantine equations.

Figures (3)

• Figure 1: This random process returns an estimate of $p$ = the probability that an element $x$ selected from the distribution $D$ satisfies condition $C$. The parameter $n$ is the number of samples to select from the distribution $D$. Example 1: The condition $C$ is whether a truth assignment $x$ satisfies a certain CNF proposition $\alpha$. Hence, if $\alpha$ involves some $k$ variables then $D$ = the uniform distribution on $\{\mathsf{T}\xspace,\mathsf{F}\xspace\}^k$. This process can be used to determine whether the proposition $\alpha$ is close to being a tautology. Example 2: The condition $C$ refers to some NFA $\alpha$ over some alphabet $\Sigma$. The distribution is $D = \langle\mathsf{D}_{t,d}\xspace^F\rangle\xspace$ = the word distribution based on a truncated Dirichlet distribution. The condition is whether "$x=\bot\xspace$ or $x\in{\mathcal{L}}(\alpha\xspace)$". This condition is used in KoMaMoRo:2023 with regards to how close $\alpha$ is universal relative to $\langle\mathsf{D}_{t,d}\xspace\rangle\xspace$.
• Figure 2: PRAX algorithms for the emptiness problem $E_{T\xspace}$ (on the left) and for the universality problem $U_{T\xspace}$ (on the right), where $T$ is a family of locally tractable distributions. The time complexity is $O$-bounded by $\>M\cdot\mathrm{Cost}(\mathsf{prob}_{T\xspace}(M))+ (1/\varepsilon\xspace)\cdot(M+\mathrm{Cost}(\mathsf{sizeSelect}_{T\xspace}(M))+\mathrm{Cost}(x\in{\mathcal{L}}(\alpha\xspace)))$.
• Figure 3: PRAX algorithms for the emptiness problem $E_{T\xspace}$ (on the left) and for the universality problem $U_{T\xspace}$ (on the right), in which the instances are descriptions of finite subsets and $T$ is a polynomially samplable distribution family.

Theorems & Definitions (35)

• Example 2.1
• Definition 2.2
• Example 2.3
• Example 2.4
• Example 2.5
• Lemma 2.6
• Definition 3.1
• Definition 3.2
• Definition 3.3
• Example 3.4