Exploring Convex Terrains: Deterministic Automata with Pebbles Achieve Full Coverage

In the realm of theoretical computer science, a fascinating advancement has emerged in the study of autonomous agent navigation. Researchers Mohamed Anouar Baaziz and Andrzej Pelc have introduced groundbreaking algorithms enabling a deterministic finite automaton—essentially a memoryless computational model—to systematically explore convex terrains using a minimal number of pebbles. This work, detailed in their recent publication, addresses a core challenge in robotics and distributed computing: how simple agents can map unknown continuous environments without infinite memory.

Convex terrains represent idealized yet practically relevant models of navigable spaces. Defined as open convex subsets of the Euclidean plane, these are regions where any line segment connecting two points within the terrain lies entirely inside it. Unlike complex polygonal environments with obstacles, convex terrains simplify boundary interactions but introduce unique exploration hurdles due to their potentially infinite extent and lack of distinctive landmarks.

🔍 The Challenge of Terrain Exploration

Exploration requires that every point in the terrain appears in at least one snapshot taken by the agent. The agent, modeled as a Deterministic Finite Automaton (DFA), operates with finite internal states. Positioned at an arbitrary unknown starting point, it captures a 'snapshot'—the intersection of the terrain with a unit disk centered at its current location—before deciding on a move: a direction and distance less than one unit. This local visibility model mimics real-world sensors like LiDAR or cameras with fixed range.

Without markers, many convex terrains defy exploration by a single DFA. For instance, unbounded terrains resembling half-planes lack reference points, causing the agent to loop indefinitely without covering new ground. Pebbles serve as movable markers: the agent can drop them at its position, detect their presence in future snapshots within the unit disk, and retrieve them. This finite marking capability emulates dropping beacons or tokens in robotic swarms.

Classifying Convex Terrains

Baaziz and Pelc classify all convex terrains into five distinct types based on two properties: boundedness (whether the terrain is finite in extent) and the presence of an infinite straight line entirely contained within it. This classification is pivotal, as the agent knows the terrain type beforehand but not its precise shape or starting position.

• Type 1: Bounded terrains without infinite lines—compact regions like disks or polygons.
• Type 2: Bounded with infinite lines—rare, but theoretically possible in limiting cases.
• Type 3: Unbounded without infinite lines—expansive regions like quadrants.
• Type 4: Unbounded with one infinite line.
• Type 5: Unbounded with multiple parallel or diverging infinite lines.

This typology allows tailored algorithms, optimizing pebble usage per category. Detailed proofs accompany each classification, ensuring universality across all terrains of a given type.

Minimal Pebble Requirements

The core contribution determines the exact minimal number of pebbles sufficient—and necessary—for exploration in each type. Algorithms are constructed using finite states that systematically spiral outward, backtrack to pebbles, and adjust paths based on boundary curvatures detected in snapshots.

Note: Exact counts derived from matching upper and lower bounds; positive algorithms halt after finite steps, negative proofs via adversarial terrains showing failure with fewer pebbles. For full details, access the paper on Springer.

Photo by Callum Blacoe on Unsplash

Algorithmic Innovations

The exploration strategies leverage geometric properties of convexity. In bounded terrains, the agent circumnavigates the boundary counterclockwise, using pebbles to resolve ambiguities at 'flat' sections mimicking lines. For unbounded cases, it constructs expanding spirals, placing pebbles at key reversal points to measure curvature and detect infinite extents.

Step-by-step process for a one-pebble unbounded terrain without lines: 1) Initial boundary detection via snapshot; 2) Follow boundary while monitoring radius; 3) Drop pebble at maximal curvature point; 4) Retrace to pebble, veer inward; 5) Repeat with increasing amplitude until infinity simulated by no new boundaries. This ensures coverage without redundancy.

Authors and Institutional Context

Mohamed Anouar Baaziz, a PhD candidate at the Université du Québec en Outaouais (UQO), collaborates with Professor Andrzej Pelc, a renowned expert in distributed algorithms and graph exploration. Pelc's extensive portfolio, including over 200 publications and Google Scholar citations exceeding 11,000, underscores UQO's strength in theoretical computing. Baaziz's work builds on prior studies in grid exploration with pebbles, bridging discrete and continuous models.

Their paper was accepted at SIROCCO 2025, a premier conference on structural information and communication complexity held in Delphi, Greece. Proceedings appear in Springer's Lecture Notes in Computer Science, with journal version in Theoretical Computer Science (2026). Read more at the conference site.

Connections to Broader Research

This work extends pebble automata from graphs to continua, echoing discrete results where finite automata explore trees or grids with markers. Related efforts include wedge exploration on oriented grids and faulty graph traversal. In robotics, it informs minimalist agents for planetary rovers or drones in convex obstacle-free zones, minimizing hardware via software markers.

Implications for Robotics and AI

Beyond theory, pebble models simulate bio-inspired navigation—ants using pheromones or bacteria with chemical trails. In practice, pebbles equate to RFID tags or virtual markers in multi-robot systems. For higher education, this highlights algorithmic efficiency's role in scalable autonomy, relevant to CS curricula in algorithms and AI.

Challenges remain: extending to non-convex terrains or dynamic obstacles. Yet, tight bounds provide benchmarks for approximation algorithms.

Photo by Bryan Brittos on Unsplash

Future Directions and Open Problems

Future research may minimize states alongside pebbles or incorporate probabilistic automata for robustness. Integrating with machine learning for snapshot interpretation could bridge theory and practice. UQO's ongoing projects in exploration promise further innovations.

For the full paper, visit ScienceDirect or Andrzej Pelc's Google Scholar profile.

This research exemplifies how pure theory drives real-world progress, empowering resource-constrained agents to conquer infinite spaces. Aspiring researchers can draw inspiration for theses in automata and geometry.