Qualitative Model for Object Rotations (QOR)

• The paper introduces a formal qualitative framework that maps real-valued 3D vectors to discrete sign-domains, enabling transparent reasoning of object rotations.
• It outlines methodologies using sign-tables, conceptual neighborhood graphs, and composition tables to infer both translational and rotational effects.
• Applied to tests like the Cube Comparison Test, QOR demonstrates efficient, explainable inference for physical reasoning and cognitive assessments.

A qualitative model for reasoning about object rotations (QOR) provides a formal, non-numeric framework for capturing, inferring, and explaining how rigid objects move and rotate in three-dimensional space. By reducing quantitative variables—such as velocities, angular velocities, forces, and orientations—to discrete qualitative sign-domains and symbolic transitions, QOR models enable transparent, explainable inference mechanisms. These models are particularly applicable to domains requiring ambiguity tolerance and transparency in reasoning, such as physical reasoning, forensics, and cognitive testing, including the Cube Comparison Test (CCT) (Ge et al., 2018, Falomir, 13 Jan 2026).

1. Formal Qualitative Representation of Rotational State

QOR models operate in a fixed world frame $(x, y, z)$ by mapping real-valued vector components to signs, using $S = \{+, 0, -\}$. Any 3D vector, such as velocity or force, becomes a sign-vector $q = (q_x, q_y, q_z) \in S^3$. Vector arithmetic—addition, subtraction, dot-product, cross-product—is discretized via sign tables, allowing fully qualitative composition of rotational and translational effects.

The qualitative state of a rigid object $O$ at time $t$ is defined as: $O_t = \langle qv_t, q\omega_t \rangle$ where $qv_t$ is the sign-vector for linear velocity and $q\omega_t$ for angular velocity. Thus, there are $27 \times 27 = 729$ distinct qualitative states. State changes are likewise defined using these qualitative representations: $$\Delta(O_{t_1}, O_{t_2}) = \langle Q(v_{t_2} - v_{t_1}), Q(\omega_{t_2} - \omega_{t_1}) \rangle$$ with $Q$ mapping real vectors to their sign-vectors.

2. Qualitative Laws and Inference Rules

Object motion in QOR is governed by qualitative analogs of physical laws. Key rules include:

• Vanishing Point: If two bodies separate at a contact point, there is no qualitative contact force at that location. For sign-vector contact normal $\hat{n}$, point velocity signs $x_i, x_j$, and their difference, the rule states: $Q(\hat{n}) \cdot (x_i \ominus x_j) \in \{- , 0\}$ implies zero contact force at point $p$.
• No Attraction: Qualitative contact forces must only push; they never pull.
• Newton's Third Law: Opposing forces at a contact are sign-inverses: $qd_i^{-1} = qd_j$.

Lemma 1 ensures that if observed state change $\Delta O$ can be generated by a subset $D$ of forces, then qualitative force compositions over all possible subsets of $D$ are sufficient: $$\Delta(O_{t_1}, O_{t_2}) \in \Delta_D \qquad \Delta_D = \bigcup_{D' \subseteq D} \langle \bigoplus_{(qd, qr) \in D'} qd, \bigoplus_{(qd, qr) \in D'} (qr \times qd) \rangle$$ This formalism is both sound and complete for sign-based physical inference (Ge et al., 2018).

3. Composition of Translational and Rotational Effects

In QOR, each qualitative force $\langle qd, qr, O \rangle$ induces effects:

• Translational: by vector sum of direction signs $qd$
• Rotational: via cross-product $qr \times qd$

Pure rotation is distinguished when all translation effects sum to zero but rotation effects do not: $$\bigoplus qd = (0,0,0), \quad \bigoplus(qr \times qd) \neq (0,0,0)$$ Enumeration of subsets and sign-sums yields all possible explanations consistent with observed qualitative changes.

4. Location and Orientation Reasoning: Faces, Rotations, and Features

In symbolic QOR extensions for tasks like the Cube Comparison Test (CCT), faces ($FACES = \{f, b, r, l, u, d\}$), allowed rotations ($ROT = \{r_x^+, r_x^-, r_y^+, r_y^-, r_z^+, r_z^-\}$), and feature orientations ($ORI = \{0q, 1q, 2q, 3q\}$) are formally defined (Falomir, 13 Jan 2026). The location-change function $L: FACES \times ROT \rightarrow FACES$ and orientation-change function $O: ORI \times (FACES \times ROT) \rightarrow ORI$ constitute the core mechanism for updating identities and attributes of visible features under allowed rotations.

A composite relation $Q$ is constructed: $$Q((i,o), r, (j,o')) \Longleftrightarrow L(i,r) = j \wedge O(o, i, r) = o'$$ encapsulating the qualitative effect of rotations on both position and orientation of features.

5. Conceptual Neighborhood Graph and Composition Tables

The conceptual neighborhood graph $CNG_{RLO}$ organizes all possible feature states $(i, o)$ as graph nodes, with directed arcs labeled by allowed rotations capturing reachable states. This graph is explicit and finite (maximum 24 nodes for a cube), supporting exhaustive enumeration of adjacency and path relations.

Composition tables for location ($L \circ L$), orientation ($O \circ O$), and combined rotation–location–orientation ($Q \circ Q$) enable efficient lookup and chaining of multi-step transformations. For example, a two-step table lists all double-rotation sequences that carry one face to another, while orientation increments obey the cyclic group structure of $\mathbb{Z}_4$.

6. Inference Algorithms and CCT Application

The central inference algorithm solves the transformation problem: given initial and goal feature states, find a minimal sequence of rotations connecting them. Because the $CNG_{RLO}$ graph is small, this is efficiently performed using forward or backward breadth-first search (BFS). The path labels directly provide the required sequence of allowable quarter-turns.

For the Cube Comparison Test, the QOR instantiation proceeds by:

• Extracting triples $(Feature, Location, Orientation)$ for all visible cube faces.
• Identifying repeated features between two cubes.
• Intersecting sets of minimal rotation sequences that achieve the observed correspondences.
• Applying candidate sequences to remaining features to test for occlusion or exposure.
• Declaring cubes "SAME" or "DIFFERENT" according to qualitative face and orientation consistency.

A worked example confirms that if the left cube's features can be rotated via a unique minimal sequence to match the right cube's observed state, and all unmatched features become hidden in the goal view, then the cubes are equivalent under QOR.

7. Discussion: Advantages, Limitations, and Research Directions

The QOR model offers several advantages:

• Entire inference operates on symbolic graphs and tables, ensuring explainability.
• Qualitative composition reduces computation to finite lookup and traversal.
• Direct mapping to physical reasoning and cognitive tests.

Limitations include:

• Restriction to 90° face-axis rotations (no diagonals or continuous angle rotation).
• Applicability only to rigid cubes with six faces and noiseless feature matching.
• Absence of quantitative metric constraints.

Possible extensions and open research problems:

• Inclusion of half-turns, diagonal or vertex-axis rotations.
• Fuzzy qualitative relations for perceptual robustness.
• Generalization to other polyhedra via neighborhood graphs.
• Learning QOR relations from 3D data rather than hand-crafting.
• Investigation of minimal qualitative sets for human mental rotation tasks.
• Empirical study of cognitive effort in relation to path-length in $CNG_{RLO}$ on CCT performance.

QOR thus constitutes a unified, transparent framework for capturing the symbolic dynamics of object rotations, supporting both computational and cognitive modeling domains (Ge et al., 2018, Falomir, 13 Jan 2026).