Boker, Udi
Rabin vs. Streett Automata
Abstract
The Rabin and Streett acceptance conditions are dual. Accordingly, deterministic Rabin and Streett automata are dual. Yet, when adding nondeterminsim, the picture changes dramatically. In fact, the state blowup involved in translations between Rabin and Streett automata is a longstanding open problem, having an exponential gap between the known lower and upper bounds.
We resolve the problem, showing that the translation of Streett to Rabin automata involves a state blowup in $\Theta(n^2)$, whereas in the other direction, the translations of both deterministic and nondeterministic Rabin automata to nondeterministic Streett automata involve a state blowup in $2^{\Theta(n)}$.
Analyzing this substantial difference between the two directions, we get to the conclusion that when studying translations between automata, one should not only consider the state blowup, but also the \emph{size} blowup, where the latter takes into account all of the automaton elements. More precisely, the size of an automaton is defined to be the maximum of the alphabet length, the number of states, the number of transitions, and the acceptance condition length (index).
Indeed, sizewise, the results are opposite. That is, the translation of Rabin to Streett involves a size blowup in $\Theta(n^2)$ and of Streett to Rabin in $2^{\Theta(n)}$. The core difference between state blowup and size blowup stems from the tradeoff between the index and the number of states. (Recall that the index of Rabin and Streett automata might be exponential in the number of states.)
We continue with resolving the open problem of translating deterministic Rabin and Streett automata to the weaker types of deterministic coB\"uchi and B\"uchi automata, respectively. We show that the state blowup involved in these translations, when possible, is in $2^{\Theta(n)}$, whereas the size blowup is in $\Theta(n^2)$.
BibTeX  Entry
@InProceedings{boker:LIPIcs:2018:8378,
author = {Udi Boker},
title = {{Rabin vs. Streett Automata}},
booktitle = {37th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2017)},
pages = {17:117:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959770552},
ISSN = {18688969},
year = {2018},
volume = {93},
editor = {Satya Lokam and R. Ramanujam},
publisher = {Schloss DagstuhlLeibnizZentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2018/8378},
URN = {urn:nbn:de:0030drops83782},
doi = {10.4230/LIPIcs.FSTTCS.2017.17},
annote = {Keywords: Finite automata on infinite words, translations, automata size, state space}
}
12.02.2018
Keywords: 

Finite automata on infinite words, translations, automata size, state space 
Seminar: 

37th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2017)

Issue date: 

2018 
Date of publication: 

12.02.2018 