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Full Definability in a Profunctorial Model

This paper establishes that all logical families of stable and total profunctors in a proof-relevant relational model based on groupoids are fully definable by proof-nets of multiplicative linear logic with MIX, demonstrating that stability serves as a crucial correctness criterion for this characterization.

Original authors: Takeshi Tsukada, Kazuyuki Asada, Kengo Hirata

Published 2026-04-30

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CC BY 4.0

Original authors: Takeshi Tsukada, Kazuyuki Asada, Kengo Hirata

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to build a perfect dictionary that translates between two languages: the language of computer programs (proofs) and the language of mathematical meaning (semantics).

Usually, when we translate a program into math, we lose some details. It's like taking a high-resolution photo and shrinking it to a thumbnail; you can still recognize the face, but you've lost the texture of the skin or the individual strands of hair. In computer science, a model is called "fully definable" only if it's a perfect, lossless translation. This means every single piece of math in the model corresponds to an actual, existing program. If there's a math piece with no program behind it, the dictionary is "broken" or incomplete.

This paper, by Tsukada, Asada, and Hirata, builds a new, incredibly detailed dictionary. They use a complex mathematical structure called Profunctors to do this.

Here is the breakdown of their work using simple analogies:

1. The Problem: From "Yes/No" to "How Many Ways"

Think of the old way of modeling programs as a checklist.

The Old Way (Relations): You ask, "Is there a connection between Program A and Data B?" The answer is a simple "Yes" or "No." It's like a light switch: on or off.
The New Way (Profunctors): The authors use Profunctors, which are like a multi-lane highway. Instead of just asking "Is there a road?", they ask, "How many different roads connect A to B? Are there bridges? Are there tunnels? Do the roads merge?"

Profunctors carry much richer information. However, because they are so complex, it's very hard to know which ones actually correspond to real programs. It's like having a map of every possible path in a city; you need a rule to tell you which paths are actual, drivable roads and which are just imaginary lines on the map.

2. The Solution: Two Special Filters

To find the "real" roads (definable profunctors) among the imaginary ones, the authors use two special filters, or "rules of the road":

Filter 1: Stability (The "Rigid Structure" Rule)
Imagine a building made of blocks. If you push one block, the whole thing shouldn't wobble unpredictably. In math, this is called Stability. The authors show that if a profunctor is "stable," it behaves like a well-constructed proof.
- The Analogy: Think of a stability check like a quality control test for a bridge. If the bridge sways too much when a car drives over it, it's "unstable" and doesn't count as a real bridge. The authors prove that this stability check is actually a correctness test for computer proofs. If a proof structure passes this test, it's a valid proof.
Filter 2: Totality (The "No Duplicates" Rule)
Imagine you are organizing a library. If you have two books that are identical copies, you only want one on the shelf. Totality ensures that for every piece of data, there is exactly one "canonical" way to represent it.
- The Analogy: In the old "checklist" models, you could have a list that said "Yes" to a connection, but it didn't matter how you got there. In this new model, Totality ensures that if you have a connection, it's the only connection. It prevents the model from having "ghost" connections that don't correspond to a unique program.

3. The Big Discovery: The "Strict Factorization" Secret

When the authors combined these two filters (Stability + Totality), something surprising happened. They discovered that the resulting structure naturally organizes itself into Strict Factorization Systems.

The Analogy: Imagine you have a complex puzzle piece. You want to know if it fits. The authors found that these pieces can always be broken down into two specific, non-overlapping parts: a "left" part and a "right" part, and there is only one way to snap them together.
This is significant because, in previous research, mathematicians had to force this "one-way snapping" rule onto their models. Here, the authors show that this rule emerges naturally just by applying the Stability and Totality filters. It's as if they found a law of physics that explains why the puzzle pieces fit the way they do, rather than just gluing them together.

4. The Result: A Perfect Dictionary

The paper proves that if you take any "Logical Family" of these profunctors that passes both the Stability and Totality tests, it is guaranteed to be the mathematical meaning of a real computer program (specifically, a proof in Multiplicative Linear Logic with MIX).

In short: They built a model where:
1. Every math object is a real program (Full Definability).
2. They found a new way to check if a proof is correct (using Stability).
3. They discovered that the complex math of these models naturally organizes itself into neat, unique patterns (Strict Factorization Systems).

Why This Matters (According to the Paper)

The authors don't claim this will immediately fix bugs in your phone or cure diseases. Instead, they are solving a deep theoretical puzzle in computer science. They are showing that even though "Profunctors" are much more complicated than simple "Relations," we can still understand them perfectly if we use the right combination of rules (Stability and Totality).

They also highlight that their method of checking for "correctness" (Stability) is a new, independent discovery that works just as well as older methods, but in a more detailed, "high-definition" setting.

Summary Metaphor:
If the old models were a black-and-white sketch of a city, this paper creates a 3D, high-definition simulation. The authors figured out the specific "physics" (Stability and Totality) that make the simulation real, proving that every building in this 3D city corresponds to a real blueprint (a program), and that the city naturally organizes itself into perfect, non-redundant blocks.

1. Problem Statement

The paper addresses the problem of full definability (or full completeness) in semantic models for Multiplicative Linear Logic with MIX (MLL + Mix).

Context: A model is fully definable if every semantic element in the model corresponds to the denotation of some proof (or $\lambda$ -term). While fully definable models exist for the relational model (e.g., Loader's totality spaces) and game semantics, extending these results to more expressive models is challenging.
The Gap: The authors focus on the (bi)category of profunctors on groupoids. Profunctors are a "categorification" of relations; unlike relations which only indicate if two elements are related, profunctors encode how they are related (including the number of ways and relationships between those ways).
The Challenge: While profunctor models capture rich quantitative and combinatorial information (useful for Taylor expansions and intersection types), they are significantly more complex than relational models. The central question is: What intrinsic properties characterize the profunctors that are denotations of valid MLL + Mix proofs? The authors aim to construct a fully definable model based on profunctors and characterize the definable elements.

2. Methodology

The authors construct a new semantic model by combining two established concepts from the literature: Totality (from Loader's relational model) and Stability (from domain theory and Fiore et al.'s work on profunctors).

A. Theoretical Foundations

Profunctors on Groupoids: The model is built on the category Prof, where objects are groupoids and morphisms are isomorphism classes of profunctors ( $P: A^{op} \times B \to \mathbf{Set}$ ). This category is compact closed, making it a model for MLL where tensor ( $\otimes$ ) and par ( $\parr$ ) coincide.
Orthogonality: The authors utilize the framework of orthogonality (Hyland and Schalk) to define subcategories of "good" profunctors.
Creeds and Kits: They employ creeds (families of subsets of endomorphisms) and kits (families of subgroups) to constrain the structure of profunctors. These concepts, originally introduced by Taylor and Fiore et al., serve as the mechanism for defining stability.

B. Core Construction: Stably Total Orthogonality

The authors define a new orthogonality relation, stably total orthogonality ( $\between$ ), which combines:

Stability: Ensures the profunctor behaves well with respect to the logical structure (related to the "creed-kit" criterion).
Totality: Ensures a form of "orbit uniqueness" or maximality, preventing the sum of two distinct proofs from being a single definable element (a common issue in relational models).

The resulting category, STProf (Stably Total Profunctors), is a full subcategory of the tight orthogonality category derived from this new orthogonality.

C. Proof Techniques

Correctness Criterion: They prove that stability serves as a correctness criterion for proof structures. A proof structure is correct (a proof-net) if and only if its interpretation is stable with respect to all Boolean creeds/kits.
Strict Factorization Systems: A key technical insight is that the interaction between stability and totality on single-object groupoids (groups) naturally yields strict factorization systems. This provides a new perspective on why factorization systems appear in profunctor semantics.
Logical Families: To handle the dependency on variable assignments, the authors define logical families of profunctors, ensuring that the interpretation respects the functorial structure of the logic (using double categories).

3. Key Contributions

A Fully Definable Profunctor Model:
The paper constructs a model STProf based on profunctors that is fully definable with respect to MLL + Mix. This is the first such result for a profunctor-based model that captures the full expressive power of the logic.
Characterization of Definable Profunctors:
The authors provide a complete characterization: A family of profunctors is definable by an MLL + Mix proof if and only if it is logical (respects the functorial structure of the logic) and stably total.
- Logicality ensures the family behaves correctly under substitution and variable renaming.
- Stably Total ensures the profunctor corresponds to a unique proof structure without "spurious" elements (like sums of disjoint proofs).
Stability as a Correctness Criterion:
They prove Theorem 17: A cut-free proof structure is correct (a proof-net) if and only if it satisfies the creed-kit criterion (a stability condition). This establishes stability as a semantic correctness criterion analogous to Retoré's coherence criterion for the relational model, but adapted for the richer structure of profunctors.
Emergence of Strict Factorization Systems:
The paper demonstrates that strict factorization systems arise naturally from the combination of stability and totality on groupoids. Specifically, for a group $G$ , a profunctor is stably total if and only if its associated subgroups form a strict factorization system. This offers a new theoretical justification for the role of factorization systems in profunctor semantics.

4. Results

Theorem 38 (Main Result): A family of profunctors is definable by an MLL + Mix proof (up to natural isomorphism) iff it is a logical and stably total family.
Theorem 17: The creed-kit criterion is necessary and sufficient for the correctness of MLL + Mix proof structures.
Proposition 29: On single-object groupoids, the stably total orthogonality coincides with the existence of a strict factorization system.
Lemma 35 (Single Orbit Property): Every stably total profunctor has a "single orbit" property, meaning any two elements in the same isomorphism class of objects are related by a unique action of the groupoid morphisms. This prevents the "sum of proofs" problem found in simpler models.

5. Significance

Bridging Game Semantics and Profunctors: Game semantics is known for full definability but is complex. Profunctor models are often viewed as a simplified variant of game semantics (by discarding certain structures). This paper demonstrates that game-semantic reasoning (specifically full definability) can be successfully carried out in the simplified setting of profunctors, provided the correct orthogonality (stably total) is used.
Quantitative Semantics: By using profunctors, the model retains "quantitative" or "combinatorial" information (e.g., the number of ways a proof can be constructed) that is lost in standard relational models. This makes the result highly relevant for the study of Taylor expansions and resource-sensitive computation.
New Perspective on Factorization Systems: The derivation of strict factorization systems from stability and totality provides a fresh theoretical link between domain theory concepts (stability) and combinatorial structures (factorization systems) in the context of linear logic.
Foundational for Future Extensions: The authors note that while this work covers MLL + Mix, the approach offers a pathway to extending full definability to logics with exponentials (which model resource management), a major open problem in categorical logic.

In summary, the paper successfully lifts the concept of full definability from the relational world to the richer world of profunctors, using a novel combination of stability and totality to characterize exactly which semantic elements correspond to valid proofs.