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Time complexity for deterministic string machines

Ali Cataltepe, Vanessa Kosoy

TL;DR

This work formulates deterministic string machines as category-theoretic transducers with a filtered-set enrichment to bound description and runtime complexity. It introduces filtered transducers, a state-category framework, and a meta-vertex hierarchy that allows string machines to build and run other transducers, enabling polynomial-time guarantees under finite-state assumptions. The paper analyzes expressivity, conjecturing that non-meta-vertex finite-state machines characterize regular languages while meta-vertices extend to more expressive classes (e.g., palindrome recognition). These results provide a principled, compositional approach to automata-based hypothesis spaces with formal runtime bounds and outline future directions for description-runtime coupling and monadic extensions.

Abstract

Algorithms which learn environments represented by automata in the past have had complexity scaling with the number of states in the automaton, which can be exponentially large even for automata recognizing regular expressions with a small description length. We thus formalize a compositional language that can construct automata as transformations between certain types of category, representable as string diagrams, which better reflects the description complexity of various automata. We define complexity constraints on this framework by having them operate on categories enriched over filtered sets, and using these constraints, we prove elementary results on the runtime and expressivity of a subset of these transformations which operate deterministically on finite state spaces. These string diagrams, or "string machines," are themselves morphisms in a category, so it is possible for string machines to create other string machines in runtime to model computations which take more than constant memory. We prove sufficient conditions for polynomial runtime guarantees on these, which can help develop complexity constraints on string machines which also encapsulate runtime complexity.

Time complexity for deterministic string machines

TL;DR

This work formulates deterministic string machines as category-theoretic transducers with a filtered-set enrichment to bound description and runtime complexity. It introduces filtered transducers, a state-category framework, and a meta-vertex hierarchy that allows string machines to build and run other transducers, enabling polynomial-time guarantees under finite-state assumptions. The paper analyzes expressivity, conjecturing that non-meta-vertex finite-state machines characterize regular languages while meta-vertices extend to more expressive classes (e.g., palindrome recognition). These results provide a principled, compositional approach to automata-based hypothesis spaces with formal runtime bounds and outline future directions for description-runtime coupling and monadic extensions.

Abstract

Algorithms which learn environments represented by automata in the past have had complexity scaling with the number of states in the automaton, which can be exponentially large even for automata recognizing regular expressions with a small description length. We thus formalize a compositional language that can construct automata as transformations between certain types of category, representable as string diagrams, which better reflects the description complexity of various automata. We define complexity constraints on this framework by having them operate on categories enriched over filtered sets, and using these constraints, we prove elementary results on the runtime and expressivity of a subset of these transformations which operate deterministically on finite state spaces. These string diagrams, or "string machines," are themselves morphisms in a category, so it is possible for string machines to create other string machines in runtime to model computations which take more than constant memory. We prove sufficient conditions for polynomial runtime guarantees on these, which can help develop complexity constraints on string machines which also encapsulate runtime complexity.
Paper Structure (17 sections, 5 theorems, 1 equation, 1 figure)

This paper contains 17 sections, 5 theorems, 1 equation, 1 figure.

Table of Contents

  1. Introduction
  2. Motivations
  3. Prior work
  4. Deterministic transducers
  5. Composition of transducers
  6. The meta-vertex and looping
  7. Parameterizing complexity in deterministic string machines
  8. Filtered transducers
  9. Finite-state string machines
  10. String machine description length
  11. The expressivity of finite-state string machines
  12. Without a meta-vertex
  13. With a meta-vertex
  14. Conclusions
  15. Further work
  16. ...and 2 more sections

Key Result

Proposition 3.1

If $S(B)$ is finite, and, for every $x \in S(A)$, the linear term of the degree of every variable in $V_{A}(x)$ is nonzero, then, for every $x \in S(A)$ and every primary input $\mathcal{C}$-morphism $\alpha$, the number of copies of the generator corresponding to each auxiliary input $X_i \to Y_i$

Figures (1)

  • Figure 1: Assuming the runtime of a state-category morphism is proportional to its degree, we must impose either a constant or logarithmic bound on the number of primary-input compositions where the output is multiplied to preserve runtime that is polynomial in the total number of transducers. In the figure, vertical arrows go to a transducer's primary input, while horizontal ones go to its auxiliary input.

Theorems & Definitions (30)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 3.1
  • Definition 3.2
  • Remark
  • Remark
  • Definition 3.3
  • Definition 3.4
  • ...and 20 more