Exploration Space Theory: Formal Foundations for Prerequisite-Aware Location-Based Recommendation

TL;DR

This paper introduces Exploration Space Theory (EST), a formal lattice-theoretic framework for modeling prerequisite dependencies in location-based recommender systems, enabling structurally sound and explainable recommendations.

Contribution

It formalizes prerequisite-aware location recommendation using lattice theory, connecting it to Formal Concept Analysis, and develops the Exploration Space Recommender System with structural guarantees.

Findings

Valid user exploration states form a finite distributive lattice.

Recommendations are structurally sound with a validity certificate.

The system guarantees sub-path optimality and provides structural explanations.

Abstract

Location-based recommender systems have achieved considerable sophistication, yet none provides a formal, lattice-theoretic representation of prerequisite dependencies among points of interest -- the semantic reality that meaningfully experiencing certain locations presupposes contextual knowledge gained from others -- nor the structural guarantees that such a representation entails. We introduce Exploration Space Theory (EST), a formal framework that transposes Knowledge Space Theory into location-based recommendation. We prove that the valid user exploration states -- the order ideals of a surmise partial order on points of interest -- form a finite distributive lattice and a well-graded learning space; Birkhoff's representation theorem, combined with the structural isomorphism between lattices of order ideals and concept lattices, connects the exploration space canonically to Formal Concept Analysis. These structural results yield four direct consequences: linear-time fringe computation, a validity certificate guaranteeing that every fringe-guided recommendation is a structurally sound next step, sub-path optimality for dynamic-programming path generation, and provably existing structural explanations for every recommendation. Building on these foundations, we specify the Exploration Space Recommender System (ESRS) -- a memoized dynamic program over the exploration lattice, a Bayesian state estimator with beam approximation and EM parameter learning, an online feedback loop enforcing the downward-closure invariant, an incremental surmise-relation inference pipeline, and three cold-start strategies, the structural one being the only approach in the literature to provide a formal validity guarantee conditional on the correctness of the inferred surmise relation. All results are established through proof and illustrated on a fully traced five-POI numerical example.