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Active Learning of Deterministic Transducers with Outputs in Arbitrary Monoids

Quentin Aristote

TL;DR

This paper addresses the problem of learning and minimizing subsequential-style transducers whose outputs live in an arbitrary monoid $M$, extending classical automata/minimization results to the monoidal setting. It leverages the categorical framework of Colcombet, Petrișan and Stabile to obtain existence and uniqueness conditions for a minimal monoidal transducer and to formulate a general learning algorithm via membership and equivalence queries. The approach models monoidal transducers as functors into the Kleisli category $\\mathbf{Kl}(\\mathcal{T}_M)$, uses a quaternary factorization system to decompose learning/minimization steps, and proves termination under right-noetherianity assumptions on $M$; it also provides concrete guidance for implementing the abstract FunL* learning procedure. The work suggests practical applications, notably learning transducers with outputs in trace monoids for scheduling tasks, and points to generalizations via other monads and subcategories to refine termination and complexity guarantees.

Abstract

We study monoidal transducers, transition systems arising as deterministic automata whose transitions also produce outputs in an arbitrary monoid, for instance allowing outputs to commute or to cancel out. We use the categorical framework for minimization and learning of Colcombet, Petrişan and Stabile to recover the notion of minimal transducer recognizing a language, and give necessary and sufficient conditions on the output monoid for this minimal transducer to exist and be unique (up to isomorphism). The categorical framework then provides an abstract algorithm for learning it using membership and equivalence queries, and we discuss practical aspects of this algorithm's implementation.

Active Learning of Deterministic Transducers with Outputs in Arbitrary Monoids

TL;DR

This paper addresses the problem of learning and minimizing subsequential-style transducers whose outputs live in an arbitrary monoid , extending classical automata/minimization results to the monoidal setting. It leverages the categorical framework of Colcombet, Petrișan and Stabile to obtain existence and uniqueness conditions for a minimal monoidal transducer and to formulate a general learning algorithm via membership and equivalence queries. The approach models monoidal transducers as functors into the Kleisli category , uses a quaternary factorization system to decompose learning/minimization steps, and proves termination under right-noetherianity assumptions on ; it also provides concrete guidance for implementing the abstract FunL* learning procedure. The work suggests practical applications, notably learning transducers with outputs in trace monoids for scheduling tasks, and points to generalizations via other monads and subcategories to refine termination and complexity guarantees.

Abstract

We study monoidal transducers, transition systems arising as deterministic automata whose transitions also produce outputs in an arbitrary monoid, for instance allowing outputs to commute or to cancel out. We use the categorical framework for minimization and learning of Colcombet, Petrişan and Stabile to recover the notion of minimal transducer recognizing a language, and give necessary and sufficient conditions on the output monoid for this minimal transducer to exist and be unique (up to isomorphism). The categorical framework then provides an abstract algorithm for learning it using membership and equivalence queries, and we discuss practical aspects of this algorithm's implementation.
Paper Structure (12 sections, 20 theorems, 3 equations, 3 figures, 1 algorithm)

This paper contains 12 sections, 20 theorems, 3 equations, 3 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Categorical approach to learning minimal automata
  3. Automata and languages as functors
  4. Factorization systems and the minimal automaton recognizing a language
  5. Learning
  6. The category of monoidal transducers
  7. Monoids
  8. Monoidal transducers as functors
  9. The initial and final monoidal transducers recognizing a function
  10. Factorization systems
  11. Active learning of minimal monoidal transducers
  12. Summary and future work

Key Result

Lemma 1.1

Let $A = \{ a \}$, $\Sigma = \{ \alpha, \beta, \gamma \}$, let $\Sigma^*/{\sim}$ be the monoid given by the presentation $\langle \Sigma \mid \alpha \beta = \beta \alpha \rangle$ and let $\pi: \Sigma^* \rightarrow \Sigma^*/{\sim}$ be the corresponding quotient. Consider the function $f: A^* \rightar

Figures (3)

  • Figure 1: Two transducers: unlike automata, the transitions are also labelled with output words.
  • Figure 2: A monoidal transducer $\mathcal{B}$
  • Figure 3: Increasingly small transducers recognizing the same function as the transducer of \ref{['fig:monoidal_transducer']} when $M = \Sigma^{\oast}$

Theorems & Definitions (55)

  • Lemma 1.1
  • proof
  • Definition 2.1: $(\mathcal{C},X,Y)$-automaton
  • Example 2.2: (non-)deterministic automata
  • Definition 2.3: $(\mathcal{C},X,Y)$-language
  • Example 2.4: languages
  • Definition 2.5: category of automata recognizing a language
  • Definition 2.6: factorization system
  • Example 2.7
  • Lemma 2.8: factorization system on $\mathbf{Auto}(\mathcal{L})$ colcombetAutomataMinimizationFunctorial2020
  • ...and 45 more