Continuous Uniformization of Rational Relations and Synthesis of Computable Functions

03/09/2021
by   Emmanuel Filiot, et al.
0

A uniformizer of a binary relation is a function whose graph is contained in the relation and which is defined on the same domain as the relation. It is known that any rational relation of infinite words, i.e. a relation given as a transducer, admits a rational uniformizer. Although rational, those uniformizers are not necessarily well-behaved, in the sense that the ith letter of the output word may depend on the whole infinite input word. In other words, those uniformizers might not be continuous (for the Cantor topology). This paper addresses the question of whether rational relations of infinite words can be uniformized by continuous functions. On the negative side, continuous uniformizers might not exist in general and we prove that deciding their existence is algorithmically impossible. On the positive side, we exhibit a large class of rational relations of infinite words, called weakly deterministic rational relations, for which deciding whether a relation in this class admits a continuous uniformizer is an ExpTime-c problem. This class includes the known classes of deterministic rational relations and automatic relations of infinite words. As an application of the previous result, and by exploiting a connection between computability and continuity for rational functions of infinite words, we show a result on the synthesis of computable functions from specifications given as weakly deterministic rational relations. In particular, we show that deciding the existence of a computable uniformizer is ExpTime-c and if there is one, it is possible to effectively synthesize a deterministic two-way transducer computing it. This generalizes the classical setting of Church synthesis to asynchronous implementations which can arbitrarily delay the production of their output signals.

READ FULL TEXT

page 1

page 2

page 3

page 4

research
05/15/2019

Deciding the Computability of Regular Functions over Infinite Words

The class of regular functions from infinite words to infinite words is ...
research
04/21/2023

Revisiting Membership Problems in Subclasses of Rational Relations

We revisit the membership problem for subclasses of rational relations o...
research
09/10/2018

On Computing the Measures of First-Order Definable Sets of Trees

We consider the problem of computing the measure of a regular language o...
research
01/18/2021

Computability of Data-Word Transductions over Different Data Domains

In this paper, we investigate the problem of synthesizing computable fun...
research
01/28/2018

Characterisation of (Sub)sequential Rational Functions over a General Class Monoids

In this technical report we describe a general class of monoids for whic...
research
02/19/2020

On Computability of Data Word Functions Defined by Transducers

In this paper, we investigate the problem of synthesizing computable fun...
research
02/28/2018

Continuity and Rational Functions

A word-to-word function is continuous for a class of languages V if its ...

Please sign up or login with your details

Forgot password? Click here to reset