Two-Way Automata over Locally Finite Semirings

Louis-Marie Dando,Sylvain Lombardy

doi:10.1007/978-3-319-94631-3_6

Abstract

Two-way transducers or weighted automata are in general more powerful than one-way ones. We show that two-way automata over locally finite semirings may have undefined behaviour. We prove that it is decidable whether this behaviour is defined, and, if it is, we show that two-way automata over locally finite semirings are equivalent to one-way automata.

Highlights

Weighted two-way automata and transducers have been recently intensively studied for their interest in verification [3]
We consider in this paper two-way automata over locally finite semirings
Finite or locally finite semirings occur in many models, like distributive lattices or fuzzy logic

Summary

Introduction

Weighted two-way automata and transducers have been recently intensively studied for their interest in verification [3]. For every two-way automaton over a locally finite semiring, we build an automaton that describes the potentially infinite family of weights of runs on every input. It leads to a deterministic oneway automaton where each final state describes the (potentially infinite) family of weights of runs of the two-way automaton on the input. It is decidable whether the weight of every input is defined, and, if it is, the deterministic automaton can be turned into a (deterministic) one-way automaton equivalent to the two-way automaton

Locally Finite Semirings

Definition and Behaviour

Characteristic and δ-normalized Two-way Automaton

Counting Paths

Crossing Sequences

Automaton of Crossing Sequences with One Repetition

Gathering Runs with the Same Label

Conclusion