Bongard Problem

A concept-learning puzzle that presents positive and negative diagram sets and asks for a rule true of every positive case and no negative case.

Version

v1 · 2026-09-28 · History

Domain-specific #

8242

Domain group

Applied Sciences & Engineering

Origin domain

Computer Science & Software Engineering

Subdomains

Pattern Recognition, Concept Learning → Computer Science & Software Engineering

Core Idea

A Bongard problem presents positive and negative sets of simple diagrams and asks for a rule true of every positive and no negative. Solving requires inventing a visual vocabulary, using near misses to reject overbroad hypotheses, and articulating a general concept rather than memorizing panels. The challenge is underdetermination. The challenge is underdetermination.

How would you explain it like I'm…

Secret Rule Picture Puzzle

A Bongard problem is a picture puzzle with two sides. All the drawings on one side share a secret rule, and none of the drawings on the other side follow it. Your job is to figure out the secret rule, like "all these shapes are closed" or "all these have three corners."

Spot the Rule Between Two Sides

A Bongard problem is a puzzle made of two groups of simple drawings. Every drawing on the left has some shared feature, and every drawing on the right is missing it. You have to figure out the rule that separates the two sides and say it clearly. It's tricky because lots of small details might happen to differ between the sides, so a good puzzle includes drawings that are very different on the same side and drawings on the other side that are almost the same but just miss the rule. A real solution should work for new drawings too, not just the ones shown.

Contrastive Concept-Learning Puzzle

A Bongard problem is a small puzzle for learning a concept by contrast. It shows two panels of simple diagrams: every diagram on the positive side has some intended property, and every diagram on the negative side lacks it. The solver must state that property as a general rule. The difficulty is underdetermination: with only a few examples, many accidental features could separate the two sides, so well-designed problems vary the positive examples and include near misses on the negative side. Solving one means inventing a vocabulary for describing the pictures, testing guesses against both sides, and choosing the rule that best explains every example. It began in pattern-recognition research and is now used as a benchmark for few-shot concept learning and visual reasoning, where the goal is a rule that can also classify new diagrams, not just sort the given ones.

 

A Bongard problem is a contrastive concept-learning puzzle consisting of two panels of relatively simple diagrams: all diagrams on the positive side share an intended property, and all on the negative side lack it. The solver must formulate the distinguishing property as an explicit general rule. The core difficulty is underdetermination: with a finite sample, many accidental features could separate the sets, so good problems vary the positive examples and include near misses among the negatives to rule out spurious hypotheses. Solving requires inventing a suitable visual vocabulary, generating and testing hypotheses against both classes, and preferring a coherent rule that explains why each example was included. The format originated in pattern-recognition research and is now a benchmark for few-shot concept learning and visual reasoning. Success is not mere image classification: the rule should be expressible, compositional where needed, and able to classify new diagrams from the same concept family.

Scope of Application

The format applies to pattern-recognition research, cognitive studies, education, and benchmarks for explainable few-shot reasoning. Use the format for human or machine studies of sparse visual concept learning, contrastive reasoning, and explainable classification.

• Human concept learning. Studies hypothesis formation from sparse contrasts.
• Artificial intelligence. Benchmarks visual relational reasoning.
• Cognitive modeling. Tests representation and analogy mechanisms.
• Education. Practices classification and rule articulation.
• Puzzle design. Constructs informative positives and near-miss negatives.

Clarity

The two-panel structure separates resemblance from definition. A valid rule must classify every example, expose the relation doing the work, and survive negative near misses; merely naming a salient object or aesthetic impression is insufficient. The closest near miss sets the boundary: Raven's Progressive Matrices are the closest near miss: they infer a missing item from row and column transformations rather than separate a positive class from a negative class.

Manages Complexity

A few simple shapes can support a combinatorial number of object, count, topology, orientation, and relation hypotheses. Positive variation and negative contrast prune that space, while explicit rule formulation makes the remaining inductive leap inspectable. The central finite fit–intended concept tradeoff is this: Many rules separate a small sample, but only some capture the designer's generalization. A second perceptual salience–relational depth tension matters because Obvious shapes can distract from topology, nesting, or role relations. The human interpretability–benchmark difficulty tension adds that Harder tasks may demand richer composition while becoming ambiguous or culturally dependent.

Abstract Reasoning

Use three linked moves: inventory objects, attributes, counts, and relations without committing to the first salient feature; find invariants across all positive diagrams and violations across every negative; use near-miss pairs to isolate which relation rather than object identity controls the split. As a collapse test, the case exits when the answer is a panel position, a list of exceptions, or a rule that fails a displayed case. A fourth check is to prefer the simplest rule that accounts for all panels without panel-specific exceptions. A final check is to test the rule on imagined or held-out diagrams to expose accidental correlations.

Knowledge Transfer

The task format transfers literally to new visual domains when positive and negative panels support an all-versus-none rule. Calling any difficult classification a ‘Bongard problem’ is loose analogy unless sparse contrastive examples and explicit concept formulation are central. No canonical parent prime is currently asserted; broader structural comparisons remain related-prime analogies until separately adjudicated in the DAG. Negative cases make the decision boundary informative. The solver generalizes a rule from sparse examples.

Relationships to Other Abstractions

Current abstraction Bongard Problem Domain-specific

Parents (1) — more general patterns this builds on

• Bongard Problem is a kind of Logic Puzzle Domain-specific

  Bongard Problem satisfies the defining boundary of Logic Puzzle: A logic puzzle is a deliberately constructed problem that presents entities, states, clues, and explicit or inferable constraints and asks a solver to derive a required configuration, classification, quantity, or explanation primarily through valid deduction and exhaustive consistency rather than hidden factual knowledge.

Hierarchy path (1) — routes to 1 parentless root

• Bongard Problem → Logic Puzzle

Neighborhood in Abstraction Space

Bongard Problem sits in a crowded region of the domain-specific corpus (39^th percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.

Family — Visual & Cinematic Composition Techniques (24 abstractions)

Nearest neighbors

• Semiorder — 0.88
• Logical graph — 0.87
• Constructional System — 0.87
• Suppressed Correlative — 0.87
• Writing system — 0.87

Computed from structural-signature embeddings · 2026-10-08