[Paper Review] The Last Paper on the Halpern-Shoham Interval Temporal Logic

Jerzy Marcinkowski, Jakub Michaliszyn|arXiv (Cornell University)|Oct 21, 2010

Logic, Reasoning, and Knowledge13 references4 citations

TL;DR

This paper proves that the Halpern-Shoham logic fragment restricted to the 'during' operator (D), known as the logic of subintervals, is undecidable over both finite and discrete linear orders. The authors establish this by reducing the satisfiability problem of a tiling system to the D-fragment, demonstrating that even this seemingly simple fragment inherits the undecidability of full HS logic, contrary to prior belief in its decidability over discrete structures.

ABSTRACT

The Halpern-Shoham logic is a modal logic of time intervals. Some effort has been put in last ten years to classify fragments of this beautiful logic with respect to decidability of its satisfiability problem. We contribute to this effort by showing - what we believe is quite an unexpected result - that the logic of subintervals, the fragment of the Halpern-Shoham where only the operator "during", or D, is allowed, is undecidable over discrete structures. This is surprising as this logic is decidable over dense orders and its reflexive variant is known to be decidable over discrete structures.

Motivation & Objective

• To resolve the open problem of whether the Halpern-Shoham logic fragment using only the 'during' operator (D) is decidable over discrete linear orders.
• To demonstrate that the logic of subintervals, despite its limited expressive power, is undecidable over finite and discrete models.
• To unify and subsume previous undecidability results for richer fragments (e.g., ABE, BE, BD, AĀD) by showing they are implied by the undecidability of the D-fragment.
• To provide a novel reduction from a tiling system to the D-fragment using a 'cloud' construction to simulate infinite configurations.

Proposed method

• Construct a formula Φ_cloud that defines a 'cloud'—a set of intervals labeled with p that are closed under shifting by one time unit, ensuring uniformity and infinite extent.
• Define Φ_orient and Φ_LA to encode a linear order with a designated starting point and a linear structure resembling a path, enabling simulation of a tiling system.
• Use the cloud construction to simulate the infinite tape of a tiling system, where intervals labeled with p represent positions in the tape.
• Introduce Φ_length to enforce length constraints: configurations and shades must be long enough to contain n leaves (when above the cloud) and short enough (when below) to avoid overcounting.
• Combine all formulas into Ψ = Φ_orient ∧ Φ_LA ∧ Φ_cloud ∧ Φ_length, showing that Ψ is satisfiable iff a valid tiling exists.
• Prove that the satisfiability of Ψ reduces to the satisfiability of a D-fragment formula, establishing undecidability via reduction from the undecidable tiling problem.

Experimental results

Research questions

• RQ1 Is the Halpern-Shoham logic fragment using only the 'during' operator (D) decidable over discrete linear orders?
• RQ2 Can the logic of subintervals, despite its apparent simplicity, express undecidable problems?
• RQ3 Does the undecidability of richer fragments (e.g., ABE, BE, BD) imply the undecidability of the minimal D-fragment?
• RQ4 Can a tiling system be encoded within the D-fragment using structural constraints like 'clouds' and length guards?

Key findings

• The satisfiability problem for the D-fragment of Halpern-Shoham logic is undecidable over finite models.
• The satisfiability problem for the D-fragment is also undecidable over all discrete linear orders.
• The logic of subintervals, previously believed to be decidable, is in fact undecidable due to its ability to simulate tiling systems via a 'cloud' construction.
• The result subsumes all prior undecidability results for richer fragments such as ABE, BE, BD, AĀD, and ADB.
• The construction uses a novel encoding of infinite configurations via a 'cloud' of intervals, ensuring that only intervals of specific length can be labeled with p.
• The length constraints Φ_length ensure that configurations and shades are correctly positioned relative to the cloud, enabling faithful simulation of tiling systems.