Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Localising Stochasticity in Weighted Automata

Smayan Agarwal, Aalok Thakkar

TL;DR

It is proved that every weighted automaton whose transition matrix has spectral radius strictly less than one can be normalised, by a semantics-preserving rescaling of transition weights, into an equivalent locally stochastic probabilistic automaton.

Abstract

Weighted automata over the nonnegative reals form a fundamental model for quantitative languages. We show that, up to scaling, this model collapses to probabilistic automata. Concretely, we prove that every weighted automaton whose transition matrix has spectral radius strictly less than one can be normalised, by a semantics-preserving rescaling of transition weights, into an equivalent locally stochastic probabilistic automaton. Thus, finite-mass weighted automata and probabilistic automata coincide up to normalisation. The construction is effective and relies on Perron-Frobenius theory. We further characterise probabilistic automata by stochastic regular expressions equipped with a geometrically weighted star. Beyond the finite-mass setting, we show that the behaviour of an arbitrary weighted automaton admits a decomposition into an exponential growth rate and a normalised probabilistic component, separating quantitative growth from stochastic structure.

Localising Stochasticity in Weighted Automata

TL;DR

It is proved that every weighted automaton whose transition matrix has spectral radius strictly less than one can be normalised, by a semantics-preserving rescaling of transition weights, into an equivalent locally stochastic probabilistic automaton.

Abstract

Weighted automata over the nonnegative reals form a fundamental model for quantitative languages. We show that, up to scaling, this model collapses to probabilistic automata. Concretely, we prove that every weighted automaton whose transition matrix has spectral radius strictly less than one can be normalised, by a semantics-preserving rescaling of transition weights, into an equivalent locally stochastic probabilistic automaton. Thus, finite-mass weighted automata and probabilistic automata coincide up to normalisation. The construction is effective and relies on Perron-Frobenius theory. We further characterise probabilistic automata by stochastic regular expressions equipped with a geometrically weighted star. Beyond the finite-mass setting, we show that the behaviour of an arbitrary weighted automaton admits a decomposition into an exponential growth rate and a normalised probabilistic component, separating quantitative growth from stochastic structure.
Paper Structure (26 sections, 18 theorems, 24 equations, 4 figures, 1 algorithm)

This paper contains 26 sections, 18 theorems, 24 equations, 4 figures, 1 algorithm.

Table of Contents

  1. Introduction and Overview
  2. From Global Properties to Local Properties
  3. Stochastic Regular Expressions
  4. Beyond Probabilistic Automata
  5. Probabilistic Normal Forms for Finite-Mass Automata
  6. Finite Mass and Spectral Radius
  7. Strongly Connected Components and Block Structure
  8. Normalizing Strongly Connected Components
  9. Normalizing the Acyclic Structure
  10. Main Theorem
  11. Stochastic Regular Expressions
  12. Syntax and Semantics
  13. From Probabilistic Automata to SREs
  14. Main Equivalence Theorem
  15. Normal Forms Beyond Finite Mass
  16. ...and 11 more sections

Key Result

Proposition 2

Let $A$ be a weighted automaton over $\mathbb{R}_{\geq 0}$ with joint transition matrix $M$. Then $\|f_{A}\|_1 < \infty$ if and only if $\rho(M) < 1$, where $\rho(M)$ denotes the spectral radius of $M$.

Figures (4)

  • Figure 1: A weighted automaton $A$ with alphabet $\Sigma = \{a, b\}$ and total mass $\|f_{A}\|_1 = 28$. The automaton is globally stochastic after normalisation by $1/28$, but not locally stochastic: state $q_0$ has outgoing weight $\frac{2}{5} + \frac{3}{5}+ 2 + 1 > 1$.
  • Figure 2: The weighted automaton from Figure \ref{['fig:running-example']} can be written down as acyclic components according to its topological sorting. The machine in this figure corresponds to the partial order $\{q_0\} \prec \{q_1,q_2\} \prec \{q_3\} \prec \{q_4\}$.
  • Figure 3: This is the machine we obtain after applying Algorithm \ref{['alg:acyclic-normalisation']} to the machine in Figure \ref{['fig:topologically-sorted']}
  • Figure 4: This automaton represents the normalised form of the automaton from Figure \ref{['fig:running-example']}

Theorems & Definitions (34)

  • Definition 1: Probabilistic Automaton
  • Proposition 2: Spectral characterisation of Finite Mass
  • Lemma 3: Spectral Radius of SCCs
  • Lemma 4: Well-definedness of $d$
  • Theorem 5: Local Stochasticity of normalised SCC
  • Lemma 6: Acyclic normalisation Preserves Semantics
  • Theorem 7: Local normalisation After Acyclic normalisation
  • Theorem 8: Finite-Mass Normal Form
  • Definition 9: Syntax of Stochastic Regular Expressions
  • Theorem 10: State Elimination for Probabilistic Automata
  • ...and 24 more