Graphical Regular Logic

• Graphical Regular Logic is a diagrammatic formulation of regular logic that leverages free regular categories and bicategories of relations to represent syntax and semantics.
• It encodes formulas into posets by entailment and employs string-diagram calculus to visually capture proof rules such as conjunction and existential quantification.
• Its framework establishes a precise syntax-semantics correspondence, unifying relational algebra with internal logic through categorical equivalences.

Searching arXiv for recent and foundational papers on Graphical Regular Logic. Graphical regular logic is a diagrammatic presentation of regular logic—the fragment of first-order logic generated by equality, truth, conjunction, and existential quantification—developed by interpreting logical syntax through the free regular category on a set of types and its bicategory of relations. In this formulation, formulas in context are organized by entailment into posets, and regular theories are encoded by certain monoidal $2$-functors from a $2$-category of contexts to $\mathsf{Poset}$. The result is a string-diagram calculus in which shells, wires, and support dots realize the usual proof rules of regular logic while remaining tightly linked to the semantics of regular categories and relations (Fong et al., 2018).

1. Logical and categorical basis

Regular logic is the fragment of first-order logic generated by equality $(=)$, truth $(\top$ or $true)$, conjunction $(\wedge)$, and existential quantification $(\exists)$. In a regular category $R$, an $n$-ary relation on objects $2$0 is a subobject $2$1. For binary relations $2$2 and $2$3, composition is given by

$2$4

and this coincides with image-factorizing the pullback of $2$5 and $2$6 over $2$7 (Fong et al., 2020).

A category is regular when it has all finite limits, every morphism admits an image factorization as a regular epi followed by a mono, and regular epis are stable under pullback. This is precisely the categorical environment in which equality, conjunction, and existential quantification admit canonical interpretations via diagonals, pullbacks, and image factorizations. One of the central observations behind graphical regular logic is that the same fragment is also the minimal fragment required to express composition of binary relations, so the logic is simultaneously the internal language of regular categories and the native logic of relation composition (Fong et al., 2019).

This dual role is fundamental. It means that graphical regular logic is neither merely a visual syntax for relational algebra nor merely a reformulation of sequent calculus. Rather, it sits at the intersection of internal logic and categorical relation theory: the proof theory is rendered diagrammatically, while the semantics is supplied by regular categories and their bicategories of relations.

2. Free regular categories and the category of contexts

Fix a set of types $2$8. The free regular category on $2$9 is

$\mathsf{Poset}$0

where $\mathsf{Poset}$1 is the poset of finite subsets of $\mathsf{Poset}$2. Its objects may be written as contexts

$\mathsf{Poset}$3

with $\mathsf{Poset}$4, $\mathsf{Poset}$5 finite, and $\mathsf{Poset}$6. A morphism $\mathsf{Poset}$7 is a function $\mathsf{Poset}$8 satisfying $\mathsf{Poset}$9 and the stated support condition. The terminal object is $(=)$0, the product is $(=)$1, and the category has pullbacks, monos, and regular epis. The associated adjunction

$(=)$2

expresses $(=)$3 as the free regular category on $(=)$4 (Fong et al., 2020).

Passing from $(=)$5 to relations yields

$(=)$6

a locally-posetal symmetric monoidal $(=)$7-category. Its objects are again contexts, now understood graphically as shells with named ports for variables and a white dot representing extra support. A $(=)$8-cell

$(=)$9

is a wiring diagram: an equivalence relation on the union of the ports and support dots, drawn by placing inner shells inside an outer shell and connecting ports by wires. The order on homs is refinement of wiring, described as breaking wires or removing support (Fong et al., 2020).

This construction is the categorical core of graphical regular logic. The contexts are not merely tuples of variables, but objects of a free regular category, and the structural manipulations of logic become morphisms in a relation bicategory built from that free object.

3. Regular calculi and predicate-valued functors

A regular calculus is a pair $(\top$0 where $(\top$1 is a set and

$(\top$2

is an ajax functor, meaning a lax monoidal $(\top$3-functor whose unit laxator

$(\top$4

and binary laxators

$(\top$5

are right adjoints. Such a functor assigns to each context the poset of formulas in that context, ordered by entailment, and sends wiring diagrams to monotone operations on predicates. Morphisms of regular calculi consist of a function on types together with a strict monoidal natural transformation respecting the laxators on the nose (Fong et al., 2018).

Every regular category produces a regular calculus. If $(\top$6 is regular, then the composite

$(\top$7

is a strong symmetric monoidal $(\top$8-functor whose laxators are right adjoints. In this way, $(\top$9 is the regular calculus of predicates in $true)$0 (Fong et al., 2020).

A closely related presentation appears in "Regular Calculi I: Graphical Regular Logic", where the syntax category $true)$1 is a strict symmetric-monoidal po-category whose objects are finite tuples of sorts and whose hom-posets are equivalence relations on the disjoint union of source and target contexts. In that presentation, a regular calculus is a symmetric-monoidal lax po-functor $true)$2, and the four wiring generators $true)$3 act respectively as forgetting a variable, adding equality, adding a fresh variable, and merging wires for substitution or identification (Clingman et al., 2021).

These formulations are equivalent in purpose. The $true)$4-based account emphasizes freeness and regular categories; the $true)$5 account emphasizes a syntax category generated by wiring. Together they show that graphical regular logic is not an external notation layered on top of logic, but a categorical organization of syntax itself.

4. Diagrammatic proof theory

Given a regular calculus $true)$6, each object $true)$7 is a context and each element $true)$8 is a predicate in that context. A graphical term $true)$9, where $(\wedge)$0, is evaluated by

$(\wedge)$1

Semantically, this is the composite of tensoring the input predicates via the lax monoidal structure and then pushing them through the wiring diagram (Fong et al., 2020).

The proof rules of regular logic become local diagrammatic operations. Conjunction is the merge of two annotated shells into one. Truth is the empty shell in a context and may be erased. Existential quantification is performed by hiding wires: from $(\wedge)$2, one forms $(\wedge)$3, where $(\wedge)$4 is the right adjoint to the binary laxator. Substitution is re-routing along a map $(\wedge)$5. Identity is the unchanged shell, and composition is diagram nesting followed by elimination of interior support dots (Fong et al., 2018).

The same proof theory can be expressed through the four core generators

$(\wedge)$6

together with braiding. These satisfy the axioms of a special commutative Frobenius monoid plus adjunction inequalities $(\wedge)$7 and $(\wedge)$8. In this reading, cups, caps, copy maps, deletions, and merges encode the interactions of equality, weakening, conjunction, and existential elimination; many standard derivations become planar isotopies or local rewrites (Clingman et al., 2021).

A canonical example is the sequent

$(\wedge)$9

Here $(\exists)$0 and $(\exists)$1 are represented by two inner shells wired together along the common $(\exists)$2-port; conjunction merges the shells, and existential quantification discards the internal $(\exists)$3-wire by attaching a white dot. The resulting diagram denotes exactly the relational composite mediated by $(\exists)$4 (Fong et al., 2020).

A common misconception is that these diagrams are heuristic pictures for ordinary sequent proofs. In fact, the graphical language is categorical: relational po-categories automatically carry a hypergraph or self-dual compact closed structure with wires, cups, caps, and spiders, and the logic is presented by the corresponding Frobenius, special, and lax-homomorphism axioms (Fong et al., 2019).

5. Reconstruction, reflectivity, and equivalence

The central structural result is a syntax-semantics correspondence between regular categories and regular calculi. There are mutually adjoint functors

$(\exists)$5

where $(\exists)$6 sends a regular category to its calculus of predicates and $(\exists)$7 sends a calculus $(\exists)$8 to its syntactic category

$(\exists)$9

For every small regular category $R$0, there is a natural equivalence

$R$1

and for every regular calculus $R$2, there is an equivalence

$R$3

in the $R$4-categorical sense (Fong et al., 2020).

In the construction of $R$5, objects are pairs $R$6, and morphisms are predicates $R$7 that act as relations from $R$8 to $R$9 and admit right adjoints; equivalently, they are total and deterministic. The resulting category has finite limits and stable image factorizations, hence is regular. Conversely, starting from a regular category, the subobject calculus $n$0 yields an ajax functor because $n$1 is ajax (Fong et al., 2020).

This is the precise sense in which the category of regular categories is essentially reflective in that of regular calculi, a point emphasized in both the 2018 and 2021 accounts (Fong et al., 2018). In parallel, the equivalence

$n$2

between regular categories and relational po-categories shows that the bicategory-of-relations viewpoint is not ancillary. It is a complete reformulation of regular semantics, and therefore the string-diagram language for relational po-categories is sound and complete for the internal regular logic of any regular category (Fong et al., 2019).

The significance of these equivalences is methodological. Graphical regular logic is not simply compatible with ordinary regular logic; it reconstructs the same semantic content from diagrammatic data, and the diagrammatic proof theory is justified by categorical equivalence rather than by translation alone.

6. Double-categorical extensions and applications

Recent work extends the framework from bicategories of relations to a double-categorical setting. A regular hyperdoctrine over a finite-limit category $n$3 is a pseudofunctor

$n$4

whose substitution maps $n$5 admit left adjoints $n$6, subject to Beck–Chevalley and Frobenius coherence: pullback commutes with $n$7, and $n$8 commutes with $n$9 up to a distributivity law. Siqueira shows that generalized regular hyperdoctrines correspond to normal, lax symmetric monoidal double pseudofunctors

$2$00

whose monoidal laxators are companion-commuting cells in the sense of Paré (Siqueira, 8 Aug 2025).

This yields the notion of a regular double hyperdoctrine, defined as a lax symmetric monoidal double pseudofunctor

$2$01

satisfying the companion-commuter condition on its monoidal laxators. In the associated string-diagram syntax, vertical strings represent tight arrows $2$02, horizontal strings represent existential quantifiers $2$03, caps and cups encode units and counits of the adjunction $2$04, a node labelled $2$05 represents conjunction, and pasting of squares expresses Beck–Chevalley and Frobenius $2$06-cells (Siqueira, 8 Aug 2025).

The same paper discusses how one can recover a form of graphical regular logic suitable for modelling specifications of systems, including port-plugging systems that compose operadically. In this setting, specifications are boxes with typed interface ports; composition glues outputs to inputs; the regular double hyperdoctrine assigns predicates to multimorphisms in the corresponding operad of wiring graphs; and internal wires are quantified out by existential quantification along spans. The stated applications include open-systems and network models, graph rewriting via double-pushout spans enriched in a logic, and categorical databases and constraint satisfaction via $2$07 (Siqueira, 8 Aug 2025).

This suggests a broadening of the role of graphical regular logic. In its original form, it provides a categorical proof theory for regular logic based on free regular categories, bicategories of relations, and ajax functors to posets. In the double-categorical generalization, the same logical content is reorganized so that substitution, quantification, conjunction, Beck–Chevalley, and Frobenius coherence inhabit different dimensions of a single double-categorical object. The underlying theme remains unchanged: regular logic is rendered compositional by wiring, and its semantics is controlled by categorical structure rather than by syntax alone.