# Combinatorics of nondeterministic walks of the Dyck and Motzkin type

## TL;DR

This paper introduces nondeterministic walks, a new model where multiple walk paths are explored in parallel, and analyzes their combinatorial properties, including generating functions and asymptotic probabilities for Dyck and Motzkin step types.

## Contribution

It presents the first formalization and analysis of nondeterministic walks, extending classical walk models with parallel exploration and deriving their generating functions and asymptotic behaviors.

## Key findings

- Derived generating functions for nondeterministic walks with specific step sets.
- Computed asymptotic probabilities for bridges, meanders, and excursions in Dyck and Motzkin cases.
- Established connections to network protocols involving encapsulation and decapsulation.

## Abstract

This paper introduces nondeterministic walks, a new variant of one-dimensional discrete walks. At each step, a nondeterministic walk draws a random set of steps from a predefined set of sets and explores all possible extensions in parallel. We introduce our new model on Dyck steps with the nondeterministic step set {{--1}, {1}, {--1, 1}} and Motzkin steps with the nondeterministic step set {{--1}, {0}, {1}, {--1, 0}, {--1, 1}, {0, 1}, {--1, 0, 1}}. For general lists of step sets and a given length, we express the generating function of nondeterministic walks where at least one of the walks explored in parallel is a bridge (ends at the origin). In the particular cases of Dyck and Motzkin steps, we also compute the asymptotic probability that at least one of those parallel walks is a meander (stays nonnegative) or an excursion (stays nonnegative and ends at the origin). This research is motivated by the study of networks involving encapsulations and decapsulations of protocols. Our results are obtained using generating functions and analytic combinatorics.