Graphoid automata and frobenius props unify graph evaluations by matrices
Graphoid Automata, Frobenius PROPs, and Semiring Evaluation
Formal Languages and Automata Theory
Summary
Graphs can be analyzed by breaking them down into smaller parts and looking at how states relate as words. The authors studied the rules that guarantee the results don’t depend on which parts you use or how you break the graph down. They connected these rules to well-known mathematical structures called Frobenius PROPs and explained how changing relations to matrices over semirings affects these rules. Their work helps clarify when two different graph expressions produce the same matrix outcome and what conditions preserve different types of compositions.
What this means in practice
- •For automata theory designers: Decide when different graphical automata representations yield the same evaluated matrix over semirings, aiding correctness in automata design.
- •For programming language theorists: Use the established equivalences to verify semantics of graph-based computation models and optimize their structural transformations.
A theory result. No direct application yet.
Authors
Antonios Kalampakas
Abstract
Graphoid automata evaluate graphs by composing relations between words of states. We examine the equations that make this evaluation independent of the chosen graph expression, and what changes when relations are replaced by matrices over a commutative semiring. We prove that the classical graphoid equations are equivalent to the special commutative Frobenius equations together with the required symmetry laws. The known cospan and hypergraph presentations then describe equality of structural and labelled expressions, with the bone law deleting precisely the isolated internal vertices. We use established partition representations to compare this equality with equality of evaluated matrices. For any nonempty finite state set $Q$ and commutative semiring $K$, we give a uniform criterion for two structural expressions to have the same matrix. The criterion determines which isolated-vertex counts are distinguished and when a vanishing scalar makes matrices from different boundary partitions equal. For the canonical structure, the bone law holds precisely when $|Q|\cdot1_K=1_K$. Preserving all relational compositions is a separate requirement, equivalent to additive idempotency for $|Q|\geq2$. We then examine the characteristic matrices of the known relational structures on $Q$ given by disjoint unions of Abelian groups. Their Frobenius equations hold over every commutative semiring, whereas specialness depends on the order of each group and the bone law depends on the number of groups. These conditions specify which equations remain valid when weights are assigned to a graphoid automaton.