Typed Criterion Graphs

• Typed criterion graphs are a categorical and logical framework that uses type graphs, logic, and multiplicity annotations to specify graph constraints.
• They integrate pure type graph languages, restriction graphs, and Boolean operations to enforce both inclusion and exclusion of graph patterns.
• They enable type-safe, mechanized rewriting in open-graph models, supporting applications in verification, architectural modeling, and automated reasoning.

Typed criterion graphs provide a categorical and logical framework for specifying and analyzing structural constraints in graph-based models via typing disciplines, logic over type-graphs, and multiplicity annotations. This formalism encompasses a spectrum of approaches—pure type graphs, type graph logic, and annotated type graphs—enabling the specification of graph languages that range from simple homomorphism tests to expressive constraints involving counting and Boolean combinations. Typed criterion graphs also arise in the context of open-graph rewriting, providing type-safety and mechanizability in rule-based graph transformations. This synthesis of graph-theoretic, categorical, and logic-based viewpoints situates typed criterion graphs as a foundational tool in the verification, synthesis, and mechanized reasoning over models with graph-like structure.

1. Fundamentals of Type Graphs and Typing

A type graph is a finite, edge-labeled directed graph $T$ over a finite label set $\Lambda$. An ordinary $\Lambda$-graph $$G=(V_G, E_G, \operatorname{src}_G, \operatorname{tgt}_G, \operatorname{lab}_G)$$ is assigned a type by a graph homomorphism $h: G \to T$ preserving source, target, and labels for each edge. Typed criterion graphs encode graph constraints by requiring the existence (or absence) of such homomorphisms, and by enriching the type graph with logical operations and annotations.

The pure type graph language associated to a type graph $T$ is

$$L(T) = \{ G \mid \exists\; h: G \to T \ \text{graph homomorphism} \}.$$

This basic mechanism allows the restriction of admissible structure in $G$ to what is "permitted" by $T$ (Corradini et al., 2017).

2. Restriction Graphs, Duality, and Boolean Operations

Restriction graphs represent the dual notion: given a finite graph $R$, the restriction-graph language $\Lambda$0 consists of all graphs $\Lambda$1 where no homomorphism $\Lambda$2 exists. This formalism specifies forbidden patterns.

Closure properties differ: pure type-graph languages are closed under intersection (via categorical products) but not union or complement; restriction-graph languages are closed under union. A "duality pair" $\Lambda$3 satisfies $\Lambda$4 if and only if $\Lambda$5, which holds when $\Lambda$6 is a tree, reflecting results of Nešetřil–Tardif (Corradini et al., 2017).

Type graph logic generalizes these ideas by allowing formulas built from type graphs using $\Lambda$7. The language $\Lambda$8 associated to a formula $\Lambda$9 is closed under all Boolean operations. Deciding membership, emptiness, and inclusion for type graph logic remains decidable, as all reduce to finitely many homomorphism checks to finite graphs.

3. Annotated Type Graphs and Multiplicity Constraints

To model quantitative constraints, type graphs can be annotated with lower and upper bounds (multiplicities) for each node or edge, formulated via an $\Lambda$0-monoid $\Lambda$1. An annotated type graph $\Lambda$2 carries for each annotation $\Lambda$3 a requirement that, for some homomorphism $\Lambda$4, the number of pre-images of each item in $\Lambda$5 falls within $\Lambda$6.

A legal morphism from $\Lambda$7 to $\Lambda$8 ensures that, for some $\Lambda$9, the pushforward count of items mapped to $$G=(V_G, E_G, \operatorname{src}_G, \operatorname{tgt}_G, \operatorname{lab}_G)$$0 obeys $$G=(V_G, E_G, \operatorname{src}_G, \operatorname{tgt}_G, \operatorname{lab}_G)$$1 count $$G=(V_G, E_G, \operatorname{src}_G, \operatorname{tgt}_G, \operatorname{lab}_G)$$2. This enables expressing constraints such as "exactly one server" or "at most one editing edge" in graph models (Corradini et al., 2017).

Inclusion and closure for annotated type graph languages are more involved: intersection corresponds to product graphs and combined annotations, while union to coproducts, but closure under complement is open. Inclusion checking for annotated type graphs is tractable on graphs of bounded pathwidth via automaton-functor methods.

4. Typed Open Graphs and Selective Adhesivity

Typed criterion graphs arise categorically as open-graphs typed over a graphical signature $$G=(V_G, E_G, \operatorname{src}_G, \operatorname{tgt}_G, \operatorname{lab}_G)$$3, distinguishing vertices (boxes) and edge (wire) types. The slice category $$G=(V_G, E_G, \operatorname{src}_G, \operatorname{tgt}_G, \operatorname{lab}_G)$$4–Graph (graphs over the type graph $$G=(V_G, E_G, \operatorname{src}_G, \operatorname{tgt}_G, \operatorname{lab}_G)$$5) structures objects via explicit types for each point and edge (Dixon et al., 2010).

Open-graphs (in OG($$G=(V_G, E_G, \operatorname{src}_G, \operatorname{tgt}_G, \operatorname{lab}_G)$$6)) are those where wire-points (edges) form directed chains or cycles, possibly with unconnected ends, and box-points (vertices from $$G=(V_G, E_G, \operatorname{src}_G, \operatorname{tgt}_G, \operatorname{lab}_G)$$7) have in- and out-arities prescribed by $$G=(V_G, E_G, \operatorname{src}_G, \operatorname{tgt}_G, \operatorname{lab}_G)$$8. Morphisms are those graph maps preserving typing, local arity, and input/output structure. The inclusion functor $$G=(V_G, E_G, \operatorname{src}_G, \operatorname{tgt}_G, \operatorname{lab}_G)$$9 is "selective adhesive": it preserves monos and strong epis, reflects pushouts when all corners remain in $h: G \to T$0, guaranteeing well-behaved, type-safe double-pushout (DPO) rewriting (Dixon et al., 2010).

Rewrite rules are represented as spans $h: G \to T$1 in $h: G \to T$2, where $h: G \to T$3 is the boundary of dangling wires; rewriting is conducted entirely within $h: G \to T$4, ensuring type-safety (no mismatched arities, dangling connections only at allowed typed inputs/outputs).

5. Expressiveness, Algorithmic Properties, and Examples

Typed criterion graphs span a spectrum of expressiveness and closure properties:

Formalism	Expressiveness	Closure Properties	Decidability
Pure type graphs	Structural constraints via homomorphism	$h: G \to T$5; not $h: G \to T$6, complement	Membership, inclusion
Restriction graphs	Forbidden substructures	$h: G \to T$7; not $h: G \to T$8, complement	Membership, inclusion
Type graph logic	Full Boolean closure	$h: G \to T$9, $T$0, complement	Membership, inclusion
Annotated type graphs	Counting, quantitative existence constraints	$T$1, $T$2; complement open	Membership, inclusion*

*For annotated type graphs, inclusion is decidable for graphs of bounded pathwidth via automaton-functor constructions.

Illustrative examples include the enforcement of structural conditions in models, exclusion of particular subgraphs (e.g., acyclicity or absence of forbidden paths), and the specification of component presence with cardinality requirements in architectural models (Corradini et al., 2017).

In the open-graph rewriting context, typed criterion graphs provide a modular representation of compositional rewrite systems, supporting mechanized and type-safe transformations for circuit diagrams, tensor networks, and category-theoretic models (Dixon et al., 2010).

6. Synthesis and Applications

Typed criterion graphs unify logic, combinatorics, and category theory in the specification and analysis of graph constraints. By parameterizing type graphs with logical and quantitative enhancements, they facilitate fine-grained expressiveness and modular reasoning over possibly infinite families of graphs.

In practical contexts, typed criterion graphs underlie:

• Model checking and verification in software and systems engineering (specification of invariants, absence of bad states)
• Structural verification in hardware and architectural models
• Mechanized reasoning in monoidal categories and graphical calculi (via open-graph rewriting)
• Structural shape constraints in heap and memory analysis

Their categorical treatment via slice categories and selective adhesive embeddings ensures that composition, subtraction, and rewriting operations preserve key invariants and constraint satisfaction, supporting robust and extensible mechanized frameworks for graph-based modeling and transformation (Dixon et al., 2010, Corradini et al., 2017).