5.10 Boden’s modes revisited

The theory/model distinction sharpens Boden’s three modes.

Combinational creativity.

Construct new terms, diagrams, or composites from registered generators while remaining in the same presented theory. In the double setting, the factors \(\mathbb A,\mathbb B\) and their interchange law remain fixed; novelty lies in a composite model or artifact. For PROPs, this includes serial, parallel, and symmetric composition.

Exploratory creativity.

Search the model category of a fixed theory: find models, countermodels, invariants, conjectures, parameter regimes, or unexpected consequences while preserving the doctrine. A claimed trajectory must also pass the declared integrability or finite-persistence observer.

Transformational creativity.

Change the presentation or doctrine: introduce a sort, operation, equation, limit, tensor generator, topology, or class of admissible models. In DIAL this may change \(\mathbb A\), \(\mathbb B\), their tensor-product interchange, the domain sketch \(\mathbb S\), or several of these, together with explicit transport from the old theory.

Not every sketch edit is transformational. A definitional extension may make implicit structure explicit without changing the theory’s model content. Conversely, a small edit—one new equation or one forbidden diagonal—can radically change the model category. Creativity is assessed semantically, not by textual edit distance.

Chapter 4’s Markdown programs have a clear target: their skill sections propose and test controlled edits of \(\mathbb S\), its doctrine, and its models. Bracket nonclosure can identify where the current skill or theory declaration is inadequate. It cannot certify that the proposed completion is coherent or scientifically valuable; that is the role of functorial semantics and independent admission.

Chapter 6 next places this target alongside the major traditions in computational creativity. That comparison is essential: novelty, artifact generation, heuristic discovery, symbolic regression, and theory construction are related achievements, but they do not create the same object and cannot be admitted by the same certificate.

. A synthetic theory is not admitted because its presentation is novel. It must have a declared preservation doctrine, nonempty and interpretable model semantics, explicit transport of prior results, resistance to countermodels, and consequences not used to propose it. Equivalent presentations are counted as one theoretical contribution unless the new presentation itself yields a separately demonstrated computational or explanatory advantage.