Representation Independent Decompositions of Computation

TL;DR

This work addresses the challenge of robustly decomposing computation by modeling state-transition systems as typed semigroupoids, enabling representation-independent, category-theoretic decomposition. It extends the Covering Lemma from transformation semigroups to semigroupoids via relational functors, formalizing a three-step collapse–copy–compress algorithm and introducing the tracing product, kernel compression, and pinhole cascade product to emulate the original dynamics. The holonomy decomposition is shown to be compatible with this framework, linking image-set based decompositions to the cascade approach. The results open avenues for generalizing Krohn–Rhodes style decompositions to broader algebraic structures and improving toolchains for computer algebra and formal reasoning about computation.

Abstract

Constructing complex computation from simpler building blocks is a defining problem of computer science. In algebraic automata theory, we represent computing devices as semigroups. Accordingly, we use mathematical tools like products and homomorphisms to understand computation through hierarchical decompositions. To address the shortcomings of some of the existing decomposition methods, we generalize semigroup representations to semigroupoids by introducing types. On the abstraction level of category theory, we describe a flexible, iterative and representation independent algorithm. Moving from the specific state transition model to the abstract composition of arrows unifies seemingly different decomposition methods and clarifies the three algorithmic stages: collapse, copy and compress. We collapse some dynamics through a morphism to the top level; copy the forgotten details into the bottom level; and finally we apply compression there. The hierarchical connections are solely for locating the repeating patterns in the compression. These theoretical findings pave the way for more precise computer algebra tools and allow for understanding computation with other algebraic structures.

Figures (9)

• Figure 1: Representations of the flip-flop monoid.
• Figure 2: The camera obscura (pinhole camera) metaphor for hierarchical decompositions of state transition systems. The surjective morphism $\varphi$ defines the top level of the decomposition. All the information lost in this map is 'projected' down to the second, lower level component. However, the resulting bottom level component is not a well-defined semigroup in the case $\varphi$ is a morphism of semigroups.
• Figure 3: Decomposing $\mathbb{Z}_4$ counter. On the left: the original permutation group. On the right: a two-level decomposition. The bottom level is defined by the pinhole projections. To avoid clutter, the $^+3$ permutation is not shown (the arrows can be obtained by reversing the $^+1$ arrows). The issue is the bottom level does not form a transformation semigroup.
• Figure 4: A semigroupoid with two objects and six arrows. Diagram of objects and arrows on the left, the corresponding composition table in the middle, and the simplified composition table with the types only on the right.
• Figure 5: Transformation semigroupoid with stabilizers $a,b$ and $f$, and with transporters $c,d,e$.

Theorems & Definitions (27)

• Definition 2.1
• Definition 2.2: Transformation Semigroups
• Example 2.3: The flip-flop monoid
• Definition 2.4: Relational Morphism
• Example 2.5: Building a modulo four counter $\mathbb{Z}_4$ from two $\mathbb{Z}_2$ counters.
• Definition 3.1
• Example 3.2: Two-object semigroupoid
• Definition 3.3: Transformation semigroupoid
• Example 3.4
• Definition 3.5: Relational Functor