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The 2-Dimensional Constraint Loop Problem is Decidable

Quentin Guilmant, Engel Lefaucheux, Joël Ouaknine, James Worrell

TL;DR

This work proves that the termination problem for linear constraint loops over the real numbers is decidable in dimension two. By introducing a witness framework $\Wcal(K)=(M,C,v,w)$, the authors reduce termination to the (decidable) theory of real closed fields with a bounded quantifier, leveraging MW-convex geometry to ensure projections preserve the recession cone. They develop accumulation-expansion techniques to analyze unbounded executions and show tight equivalences: a non-terminating run exists iff a suitable witness exists, with constructive sufficiency and dimension-specific necessity results. The approach unifies bounded and unbounded behaviors via recession cones, convexity theory, and recurrence relations, yielding a polynomial-space decision procedure for the real-parameter setting in dimension $2$ and advancing the understanding of termination beyond linear ranking functions. Overall, the results give a concrete, geometry-driven certificate for non-termination and a path to decidability for a class of non-deterministic, linear-constraint loops in low dimensions.

Abstract

A linear constraint loop is specified by a system of linear inequalities that define the relation between the values of the program variables before and after a single execution of the loop body. In this paper we consider the problem of determining whether such a loop terminates, i.e., whether all maximal executions are finite, regardless of how the loop is initialised and how the non-determinism in the loop body is resolved. We focus on the variant of the termination problem in which the loop variables range over $\mathbb{R}$. Our main result is that the termination problem is decidable over the reals in dimension~2. A more abstract formulation of our main result is that it is decidable whether a binary relation on $\mathbb{R}^2$ that is given as a conjunction of linear constraints is well-founded.

The 2-Dimensional Constraint Loop Problem is Decidable

TL;DR

This work proves that the termination problem for linear constraint loops over the real numbers is decidable in dimension two. By introducing a witness framework , the authors reduce termination to the (decidable) theory of real closed fields with a bounded quantifier, leveraging MW-convex geometry to ensure projections preserve the recession cone. They develop accumulation-expansion techniques to analyze unbounded executions and show tight equivalences: a non-terminating run exists iff a suitable witness exists, with constructive sufficiency and dimension-specific necessity results. The approach unifies bounded and unbounded behaviors via recession cones, convexity theory, and recurrence relations, yielding a polynomial-space decision procedure for the real-parameter setting in dimension and advancing the understanding of termination beyond linear ranking functions. Overall, the results give a concrete, geometry-driven certificate for non-termination and a path to decidability for a class of non-deterministic, linear-constraint loops in low dimensions.

Abstract

A linear constraint loop is specified by a system of linear inequalities that define the relation between the values of the program variables before and after a single execution of the loop body. In this paper we consider the problem of determining whether such a loop terminates, i.e., whether all maximal executions are finite, regardless of how the loop is initialised and how the non-determinism in the loop body is resolved. We focus on the variant of the termination problem in which the loop variables range over . Our main result is that the termination problem is decidable over the reals in dimension~2. A more abstract formulation of our main result is that it is decidable whether a binary relation on that is given as a conjunction of linear constraints is well-founded.
Paper Structure (14 sections, 24 theorems, 10 equations, 1 figure)

This paper contains 14 sections, 24 theorems, 10 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Key notations
  4. Some sets
  5. Landau Notations
  6. Convex Sets
  7. Minkowski-Weyl Convex Sets
  8. Accumulation Expansions
  9. Deciding the Constraint Loop Problem
  10. Deciding the Existence of a Bounded Sequence
  11. A Sufficient Condition for the Existence of a Sequence
  12. Necessary Condition for the Existence of a 1-Dimensional Sequence
  13. Necessary Condition for the Existence of a 2-Dimensional Sequence
  14. Detailed proof for the Proposition \ref{['prop:dim2nec']}

Key Result

theorem 1

Let $E$ be a Euclidean space of dimension at most $2$. Let $K\subseteq E^2$ be MW-convex. There is a sequence $\suiten\in E^\Nbb$ such that $(u_n,u_{n+1})\in K$ for all $n\in\Nbb$ if and only if there exists a witness $\Wcal(K)$.

Figures (1)

  • Figure 1: The case disjunction structure

Theorems & Definitions (51)

  • definition 1
  • theorem 1
  • definition 2
  • definition 3
  • proposition 1
  • proposition 2
  • definition 4
  • lemma 1
  • proof
  • proposition 3
  • ...and 41 more