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Factoring through monomial representations: arithmetic characterizations and ambiguity of weighted automata

Antoni Puch, Daniel Smertnig

TL;DR

A full correspondence between arithmetic properties and a complexity hierarchy for WFA based on ambiguity is discovered, and it is shown that the $M-ambiguity, finite ambiguity, and polynomial ambiguity properties are algorithmically decidable for invertible WFA.

Abstract

We characterize group representations that factor through monomial representations, respectively, block-triangular representations with monomial diagonal blocks, by arithmetic properties. Similar results are obtained for semigroup representations by invertible transformations. The characterizations use results on unit equations from Diophantine number theory (by Evertse, van der Poorten, and Schlickewei in characteristic zero, and by Derksen and Masser in positive characteristic). Specialized to finitely generated groups in characteristic zero, one of our main theorems recovers an improvement of a very recent similar characterization by Corvaja, Demeio, Rapinchuk, Ren, and Zannier that was motivated by the study of the bounded generation (BG) property. In positive characteristic, we get a characterization of linear BG groups, recovering a theorem of Abért, Lubotzky, and Pyber from 2003. Our motivation comes from weighted finite automata (WFA) over a field. For invertible WFA we show that $M$-ambiguity, finite ambiguity, and polynomial ambiguity are characterized by arithmetic properties. We discover a full correspondence between arithmetic properties and a complexity hierarchy for WFA based on ambiguity. In the invertible case, this is a far-reaching generalization of a recent result by Bell and the second author, characterizing unambiguous WFA, that resolved a 1979 conjecture of Reutenauer. As a consequence, using the computability of the (linear) Zariski closure of a finitely generated matrix semigroup, the $M$-ambiguity, finite ambiguity, and polynomial ambiguity properties are algorithmically decidable for invertible WFA.

Factoring through monomial representations: arithmetic characterizations and ambiguity of weighted automata

TL;DR

A full correspondence between arithmetic properties and a complexity hierarchy for WFA based on ambiguity is discovered, and it is shown that the $M-ambiguity, finite ambiguity, and polynomial ambiguity properties are algorithmically decidable for invertible WFA.

Abstract

We characterize group representations that factor through monomial representations, respectively, block-triangular representations with monomial diagonal blocks, by arithmetic properties. Similar results are obtained for semigroup representations by invertible transformations. The characterizations use results on unit equations from Diophantine number theory (by Evertse, van der Poorten, and Schlickewei in characteristic zero, and by Derksen and Masser in positive characteristic). Specialized to finitely generated groups in characteristic zero, one of our main theorems recovers an improvement of a very recent similar characterization by Corvaja, Demeio, Rapinchuk, Ren, and Zannier that was motivated by the study of the bounded generation (BG) property. In positive characteristic, we get a characterization of linear BG groups, recovering a theorem of Abért, Lubotzky, and Pyber from 2003. Our motivation comes from weighted finite automata (WFA) over a field. For invertible WFA we show that -ambiguity, finite ambiguity, and polynomial ambiguity are characterized by arithmetic properties. We discover a full correspondence between arithmetic properties and a complexity hierarchy for WFA based on ambiguity. In the invertible case, this is a far-reaching generalization of a recent result by Bell and the second author, characterizing unambiguous WFA, that resolved a 1979 conjecture of Reutenauer. As a consequence, using the computability of the (linear) Zariski closure of a finitely generated matrix semigroup, the -ambiguity, finite ambiguity, and polynomial ambiguity properties are algorithmically decidable for invertible WFA.
Paper Structure (19 sections, 52 theorems, 56 equations, 1 figure)

This paper contains 19 sections, 52 theorems, 56 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Notions and Preliminaries
  3. Statements of Main Results
  4. Guiding Examples and Special Cases
  5. One-letter WFA: Linear Recurrence Sequences (LRS)
  6. Non-finitely generated groups
  7. $k$-regular and $k$-automatic sequences
  8. Bounded generation
  9. Unit Equations: Linear Equations over Finitely Generated Groups
  10. Locally Bézivin implies Weakly Epimonomial
  11. Characterization of Locally Bézivin Groups
  12. Maximal separating elements
  13. Lifting to monomial representations
  14. Not necessarily finitely generated groups
  15. Proofs of \ref{['t:main-loc-bezivin', 'cor:locbezivin-fg', 'cor:locbezivin-char0']}
  16. ...and 4 more sections

Key Result

Theorem 1.1

Let $K$ be a field and $G \le \mathop{\mathrm{GL}}\nolimits_d(K)$. Suppose that $\mathop{\mathrm{char}}\nolimits{K}=0$ or that $G$ is finitely generated. Then the following statements are equivalent.

Figures (1)

  • Figure 1: The natural complexity hierarchy for rational series (functions computable by WFA) based on ambiguity is reflected in arithmetic properties of the outputs, respectively the semigroup of a minimal linear representation. Full arrows represent unconditional implications, the implications of the dashed arrows are currently known in the invertible case (\ref{['t:wfa-ambiguity']}, \ref{['p:wfa-necessary']}, and bell-smertnig21). The picture represents the case of algebraically closed fields, for simplicity.

Theorems & Definitions (108)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Corollary 1.4: bernik05
  • Corollary 1.5
  • Theorem 1.6
  • Corollary 1.7
  • Lemma 2.1
  • proof
  • Definition 2.2
  • ...and 98 more