[Paper Review] A universality theorem for allowable sequences with applications

Udo Hoffmann, Keno Merckx|arXiv (Cornell University)|Jan 18, 2018

Advanced Algebra and Logic15 references3 citations

TL;DR

This paper establishes a universality theorem for allowable sequences—refined combinatorial abstractions of point sets—proving their realization spaces are as complex as any semi-algebraic set, making realizability decision problems complete in the existential theory of the reals (∃ℝ). The result holds even when the induced order type is realizable, and it enables new ∃ℝ- hardness proofs for convex geometry realizability and visibility graph recognition of polygons with holes.

ABSTRACT

Order types are a well known abstraction of combinatorial properties of a point set. By Mnëv's universality theorem for each semi-algebraic set $V$ there is an order type with a realization space that is \emph{stably equivalent} to $V$. We consider realization spaces of \emph{allowable sequences}, a refinement of order types. We show that the realization spaces of allowable sequences are \emph{universal} and consequently deciding the realizability is complete in the \emph{existential theory of the reals} (\ER). This result holds even if the realization space of the order type induced by the allowable sequence is non-empty. Furthermore, we argue that our result is a useful tool for further geometric reductions. We support this by giving \ER-hardness proofs for the realizability of abstract convex geometries and for the recognition problem of visibility graphs of polygons with holes using the hardness result for allowable sequences. This solves two longstanding open problems.

Motivation & Objective

• To establish a universality theorem for realization spaces of allowable sequences, extending Mnëv’s theorem from order types to this refined combinatorial model.
• To show that realizability of allowable sequences is complete in the existential theory of the reals (∃ℝ), even when the induced order type is realizable.
• To demonstrate that the ∃ℝ-hardness of allowable sequence realizability serves as a powerful tool for geometric reductions.
• To resolve two longstanding open problems: ∃ℝ-completeness for convex geometry realizability and visibility graph recognition of polygons with holes.

Proposed method

• Construct a reduction from arbitrary semi-algebraic sets to realization spaces of allowable sequences via a combinatorial encoding of geometric constraints.
• Use the fact that allowable sequences refine order types, preserving orientation data while encoding dynamic point set configurations under line rotation.
• Prove that realization spaces of allowable sequences are stably equivalent to any given semi-algebraic set, generalizing Mnëv’s universality to this setting.
• Introduce a construction of simple allowable sequences (adjacent swaps only) to model point sets in general position with no parallel line pairs.
• Apply the universality result to reduce known ∃ℝ-complete problems to visibility graph and convex geometry realizability problems.
• Use geometric duality and sightline blocking arguments to model visibility constraints in polygons with holes, proving that ∃ℝ-hardness transfers via allowable sequence reductions.

Experimental results

Research questions

• RQ1 Can the universality principle of Mnëv’s theorem be extended from order types to allowable sequences?
• RQ2 Is the realizability problem for allowable sequences ∃ℝ-complete, even when the induced order type is realizable?
• RQ3 Can the ∃ℝ-hardness of allowable sequence realizability be leveraged to prove hardness for other geometric realizability problems?
• RQ4 Is the realizability problem for abstract convex geometries ∃ℝ-complete?
• RQ5 Is the recognition problem for visibility graphs of polygons with holes ∃ℝ-complete?

Key findings

• The realization space of any allowable sequence is stably equivalent to any given semi-algebraic set, proving a universality theorem for this model.
• The realizability problem for allowable sequences is complete in the existential theory of the reals (∃ℝ), even when the induced order type is realizable.
• The result holds for simple allowable sequences—those with only adjacent transpositions—corresponding to point sets in general position with no parallel line pairs.
• The ∃ℝ-hardness of allowable sequence realizability provides a new reduction tool for geometric problems.
• The paper proves ∃ℝ-completeness for the realizability of abstract convex geometries, resolving a longstanding open problem.
• The paper establishes ∃ℝ-completeness for the recognition problem of visibility graphs of polygons with holes, resolving another longstanding open problem.