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Automation of Triangle Ruler-and-Compass Constructions Using Constraint Solvers

Milan Banković

TL;DR

The paper addresses automating ruler-and-compass triangle constructions by recasting them as finite-domain constraint-satisfaction problems solved via MiniZinc, guided by automated planning to sequence construction steps. It demonstrates how geometric knowledge can be encoded as rich relational constraints and how states, plans, and transitions are modeled to produce valid constructions, with optimization available to minimize construction length. An empirical evaluation on 74 solvable Wernick problems shows the approach is competitive with the dedicated tool ArgoTriCS and often yields shorter constructions due to solver-based optimization, albeit with a few instances where the dedicated solver performs better. The work highlights the practicality and extensibility of a solver-based framework for geometric construction problems and suggests avenues for broader application to additional problem sets.

Abstract

In this paper, we present an approach to automated solving of triangle ruler-and-compass construction problems using finite-domain constraint solvers. The constraint model is described in the MiniZinc modeling language, and is based on the automated planning. The main benefit of using general constraint solvers for such purpose, instead of developing dedicated tools, is that we can rely on the efficient search that is already implemented within the solver, enabling us to focus on geometric aspects of the problem. We may also use the solver's built-in optimization capabilities to search for the shortest possible constructions. We evaluate our approach on 74 solvable problems from the Wernick's list, and compare it to the dedicated triangle construction solver ArgoTriCS. The results show that our approach is comparable to dedicated tools, while it requires much less effort to implement. Also, our model often finds shorter constructions, thanks to the optimization capabilities offered by the constraint solvers.

Automation of Triangle Ruler-and-Compass Constructions Using Constraint Solvers

TL;DR

The paper addresses automating ruler-and-compass triangle constructions by recasting them as finite-domain constraint-satisfaction problems solved via MiniZinc, guided by automated planning to sequence construction steps. It demonstrates how geometric knowledge can be encoded as rich relational constraints and how states, plans, and transitions are modeled to produce valid constructions, with optimization available to minimize construction length. An empirical evaluation on 74 solvable Wernick problems shows the approach is competitive with the dedicated tool ArgoTriCS and often yields shorter constructions due to solver-based optimization, albeit with a few instances where the dedicated solver performs better. The work highlights the practicality and extensibility of a solver-based framework for geometric construction problems and suggests avenues for broader application to additional problem sets.

Abstract

In this paper, we present an approach to automated solving of triangle ruler-and-compass construction problems using finite-domain constraint solvers. The constraint model is described in the MiniZinc modeling language, and is based on the automated planning. The main benefit of using general constraint solvers for such purpose, instead of developing dedicated tools, is that we can rely on the efficient search that is already implemented within the solver, enabling us to focus on geometric aspects of the problem. We may also use the solver's built-in optimization capabilities to search for the shortest possible constructions. We evaluate our approach on 74 solvable problems from the Wernick's list, and compare it to the dedicated triangle construction solver ArgoTriCS. The results show that our approach is comparable to dedicated tools, while it requires much less effort to implement. Also, our model often finds shorter constructions, thanks to the optimization capabilities offered by the constraint solvers.
Paper Structure (17 sections, 3 figures, 1 table)

This paper contains 17 sections, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Background
  3. Ruler-and-Compass Constructions
  4. Constraint Solving
  5. Automated Planning
  6. Model Description
  7. Encoding of Geometric Knowledge
  8. Encoding objects.
  9. Encoding relations.
  10. Encoding of the Planning Problem
  11. Encoding of states.
  12. Encoding the plan.
  13. Encoding the state transitions.
  14. Encoding the goal.
  15. Evaluation
  16. ...and 2 more sections

Figures (3)

  • Figure 1: Per-instance comparison of different setups. Times are given in seconds
  • Figure 2: Survival plot for all three setups, compared to ArgoTriCS. Times are given in seconds
  • Figure 3: A per-instance comparison of construction (plan) lengths between ArgoTriCS and our approach