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A Formalism for Optimal Search with Dynamic Heuristics (Extended Version)

Remo Christen, Florian Pommerening, Clemens Büchner, Malte Helmert

TL;DR

The paper addresses the limitations of static heuristics in A*-like search by formalizing dynamic heuristics that evolve with search information. It introduces a general information-source framework and a dynamic heuristic search algorithm (DYN-A*) that preserves optimality under dynamic admissibility, monotonicity, and related properties, with completeness guaranteed under converge conditions. The authors connect the framework to classical planning approaches (e.g., online abstraction refinement, landmark progression, LTL constraints, and lazy evaluation) by showing how they fit as instantiations and satisfy the necessary dynamic properties for optimality and non-reopening. The work provides a rigorous basis for claims of optimality in dynamic-heuristic search and clarifies when such claims hold, enabling direct application to existing planning techniques.

Abstract

While most heuristics studied in heuristic search depend only on the state, some accumulate information during search and thus also depend on the search history. Various existing approaches use such dynamic heuristics in $\mathrm{A}^*$-like algorithms and appeal to classic results for $\mathrm{A}^*$ to show optimality. However, doing so ignores the complexities of searching with a mutable heuristic. In this paper we formalize the idea of dynamic heuristics and use them in a generic algorithm framework. We study a particular instantiation that models $\mathrm{A}^*$ with dynamic heuristics and show general optimality results. Finally we show how existing approaches from classical planning can be viewed as special cases of this instantiation, making it possible to directly apply our optimality results.

A Formalism for Optimal Search with Dynamic Heuristics (Extended Version)

TL;DR

The paper addresses the limitations of static heuristics in A*-like search by formalizing dynamic heuristics that evolve with search information. It introduces a general information-source framework and a dynamic heuristic search algorithm (DYN-A*) that preserves optimality under dynamic admissibility, monotonicity, and related properties, with completeness guaranteed under converge conditions. The authors connect the framework to classical planning approaches (e.g., online abstraction refinement, landmark progression, LTL constraints, and lazy evaluation) by showing how they fit as instantiations and satisfy the necessary dynamic properties for optimality and non-reopening. The work provides a rigorous basis for claims of optimality in dynamic-heuristic search and clarifies when such claims hold, enabling direct application to existing planning techniques.

Abstract

While most heuristics studied in heuristic search depend only on the state, some accumulate information during search and thus also depend on the search history. Various existing approaches use such dynamic heuristics in -like algorithms and appeal to classic results for to show optimality. However, doing so ignores the complexities of searching with a mutable heuristic. In this paper we formalize the idea of dynamic heuristics and use them in a generic algorithm framework. We study a particular instantiation that models with dynamic heuristics and show general optimality results. Finally we show how existing approaches from classical planning can be viewed as special cases of this instantiation, making it possible to directly apply our optimality results.
Paper Structure (19 sections, 15 theorems, 8 equations, 2 figures, 2 algorithms)

This paper contains 19 sections, 15 theorems, 8 equations, 2 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Transition Systems
  3. Running Example
  4. Dynamic Heuristics
  5. Progression-Based Heuristics
  6. Dynamic Heuristic Search Framework
  7. Dynamic $\mathrm{A}^{\mkern-3mu\raisebox{.1ex}{$*$}}$
  8. Optimality of Solutions
  9. Reopening
  10. Applications
  11. Interleaved Search
  12. Online Cartesian Abstraction Refinement
  13. Landmark Progression
  14. LTL-$\mathrm{A}^{\mkern-3mu\raisebox{.1ex}{$*$}}$
  15. $\mathrm{A}^{\mkern-3mu\raisebox{.1ex}{$*$}}$ with Lazy Heuristic Evaluation
  16. ...and 4 more sections

Key Result

Lemma 1

Every state in $S_{\text{known}}$ in Algorithm alg:main is reachable.

Figures (2)

  • Figure 1: Transition system of our running example.
  • Figure 2: Example state space, showing that reopening can occur with $\mathrm{DYN}$-consistent heuristics. Edge costs are written on the edges, heuristic values are written inside the states. The heuristic is dynamic and the heuristic value of a state with label $x \to y$ changes from $x$ to $y$ once it is reached along the bold incoming edge.

Theorems & Definitions (37)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Definition 6
  • Definition 7
  • Definition 8
  • Definition 9
  • Lemma 1
  • ...and 27 more