Analogical Reasoning¶
Overview¶
Analogical reasoning addresses new problems by transferring knowledge of relationships from known problems, potentially across domains. It generalizes the transfer mechanism introduced in explanation-based learning and extends case-based reasoning to cross-domain settings.
The process has five phases: retrieval, mapping, transfer, evaluation, and storage.
Similarity¶
Similarity between two situations can be measured along multiple dimensions:
Relationships between objects (e.g., “climbing up”)
Objects themselves (e.g., woman, ladder)
Features of objects (e.g., size, shape)
Values of features (e.g., tall, short)
Two situations may share objects but differ in relationships (a woman climbing a ladder vs. painting a ladder), or share relationships but differ in objects (a woman climbing a ladder vs. an ant climbing a wall). Relationship similarity is generally more important for analogy than object similarity.
Spectrum of Similarity¶
At one extreme, the target problem and source case are identical (all dimensions match). At the other, nothing is shared. Between these:
Recording cases: relationships, objects, features, and values all similar (same domain, same objects)
Case-based reasoning: relationships and objects similar, but features/values may differ (same domain)
Analogical reasoning: only relationships similar; objects, features, and values may differ entirely (cross-domain)
Cross-Domain Analogy¶
Cross-domain analogy occurs when the target problem and source case come from different domains, sharing deep relational structure but not surface features.
Duncker’s radiation problem: A physician must destroy a stomach tumor with a laser, but a full-intensity beam would kill healthy tissue. The analogous source: a rebel army must capture a fortified king, but marching a full army down any mined road would trigger explosions. Solution: decompose the resource (army/laser) into smaller units and converge on the goal from multiple directions simultaneously.
The objects differ entirely (physician/king, laser/army, tissue/mines), but the relational pattern — decompose resource, converge from multiple directions to overcome obstacle — transfers across domains.
Three Types of Similarity¶
Semantic similarity — conceptual overlap between objects (woman/woman, ladder/step-ladder)
Pragmatic similarity — shared external factors, especially goals (killing tumor ≈ capturing fort)
Structural similarity — isomorphism between representational graphs; the relational structure matches even when objects and goals differ (solar system ≈ atomic structure)
Different theories of analogy weight these differently.
Analogical Retrieval¶
Retrieval finds a source case relevant to the target problem. Key distinction:
Superficial similarity: shared features, object counts, or objects themselves
Deep similarity: shared relationships between objects, or relationships between relationships (higher-order relations)
Higher-order relationships indicate deeper similarity. For example, in Raven’s Progressive Matrices:
Unary: features of individual objects (size, fill)
Binary: relationships between two objects (X is outside Y)
Tertiary: relationships between relationships (the pattern of change from A→B matches C→D)
The mind judges two situations as more similar when similarity operates at the relational level rather than the object/feature level.
Analogical Mapping¶
Mapping solves the correspondence problem: what in the target corresponds to what in the source?
Without constraints, mapping is combinatorially explosive (m × n possibilities). The key heuristic: prioritize higher-order relationships. Instead of mapping king↔patient (superficial — both are people), we map king↔tumor (both are goals to be neutralized via a resource despite an obstacle). The deeper relational structure — goal/resource/obstacle — determines correct alignment.
Example — solar system ↔ atomic structure:
Sun ↔ nucleus (both are central, massive bodies)
Planet ↔ electron (both revolve around the central body)
Gravitational force ↔ electrostatic force (both cause attraction and revolution)
Correct mapping requires deep models of both systems.
Analogical Transfer¶
Once alignment is established, the agent:
Abstracts a relational pattern from the source (e.g., “decompose resource into smaller units, send from multiple directions”)
Instantiates that pattern in the target using the mapping (laser → smaller beams, converge on tumor)
Transfer depends on correct mapping, which depends on successful retrieval. Goals often drive the process (pragmatic similarity).
In the solar system → atom example, the transferred knowledge is: “the electron revolves around the nucleus” (inferred from “the planet revolves around the sun” via structural similarity, even without semantic or pragmatic similarity).
Evaluation and Storage¶
Analogical reasoning provides no correctness guarantees — proposed solutions must be evaluated (e.g., via simulation or prototyping). If evaluation succeeds, the target problem and solution are encapsulated as a new case for future reuse, enabling incremental learning.
If evaluation fails, the agent can revisit transfer (change what is transferred), mapping (realign), or retrieval (find a different source).
Design by Analogy¶
Biologically inspired design (biomimicry) applies analogical reasoning to engineering:
Shinkansen 500 bullet train: nose shape inspired by kingfisher beak — both transition between media (air/water, outside/inside tunnel) while minimizing shock waves
Basilisk lizard → water-walking robot: functional model (walk on water) provides pragmatic similarity; structural model of the lizard’s locomotion is mapped and transferred to robot design
Design by analogy uses structure-behavior-function (SBF) models. Mapping proceeds bottom-up: structural alignment enables behavioral transfer, which enables functional transfer. This is compositional analogy — analogy at multiple levels of abstraction.
Compound and Compositional Analogy¶
Compound analogy: multiple source cases contribute to a single design (e.g., copepod for slow underwater motion + squid jet propulsion for fast motion → stealth microbot)
Compositional analogy: mapping at one level (structure) supports transfer at the next level (behavior, then function)
The process is iterative — transfer may trigger new retrieval, mapping may trigger new transfer, evaluation may send the agent back to any earlier phase.
Advanced Issues¶
Common vocabulary: cross-domain transfer may require shared terms or an alignment mechanism for different vocabularies (“revolve” vs. “rotate”)
Problem abstraction: the agent may need to abstract/transform the problem itself to enable retrieval
Compound analogy: combining knowledge from multiple source cases
Visuospatial analogies: analogies where causality is implicit and transfer is primarily spatial/perceptual
Conceptual combination: creating new concepts by merging parts of existing ones via analogy
Cognitive Connection¶
Analogy is considered a core cognitive process. Metaphors are everyday analogies (“we had grown far apart” — spatial metaphor for emotional distance; “All the world’s a stage” — theater as metaphor for life). Raven’s Progressive Matrices, a widely used intelligence test, is fundamentally based on analogical reasoning.