58 research outputs found
Rabin vs. Streett Automata
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 , whereas in the other direction, the translations of both deterministic and nondeterministic Rabin automata to nondeterministic Streett automata involve a state blowup in .
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, size-wise, the results are opposite. That is, the translation of Rabin to Streett involves a size blowup in and of Streett to Rabin in . 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 co-B"uchi and B"uchi automata, respectively. We show that the state blowup involved in these translations, when possible, is in , whereas the size blowup is in
Exact and Approximate Determinization of Discounted-Sum Automata
A discounted-sum automaton (NDA) is a nondeterministic finite automaton with
edge weights, valuing a run by the discounted sum of visited edge weights. More
precisely, the weight in the i-th position of the run is divided by
, where the discount factor is a fixed rational number
greater than 1. The value of a word is the minimal value of the automaton runs
on it. Discounted summation is a common and useful measuring scheme, especially
for infinite sequences, reflecting the assumption that earlier weights are more
important than later weights. Unfortunately, determinization of NDAs, which is
often essential in formal verification, is, in general, not possible. We
provide positive news, showing that every NDA with an integral discount factor
is determinizable. We complete the picture by proving that the integers
characterize exactly the discount factors that guarantee determinizability: for
every nonintegral rational discount factor , there is a
nondeterminizable -NDA. We also prove that the class of NDAs with
integral discount factors enjoys closure under the algebraic operations min,
max, addition, and subtraction, which is not the case for general NDAs nor for
deterministic NDAs. For general NDAs, we look into approximate determinization,
which is always possible as the influence of a word's suffix decays. We show
that the naive approach, of unfolding the automaton computations up to a
sufficient level, is doubly exponential in the discount factor. We provide an
alternative construction for approximate determinization, which is singly
exponential in the discount factor, in the precision, and in the number of
states. We also prove matching lower bounds, showing that the exponential
dependency on each of these three parameters cannot be avoided. All our results
hold equally for automata over finite words and for automata over infinite
words
On the (In)Succinctness of Muller Automata
There are several types of finite automata on infinite words, differing in their acceptance conditions. As each type has its own advantages, there is an extensive research on the size blowup involved in translating one automaton type to another.
Of special interest is the Muller type, providing the most detailed acceptance condition. It turns out that there is inconsistency and incompleteness in the literature results regarding the translations to and from Muller automata. Considering the automaton size, some results take into account, in addition to the number of states, the alphabet length and the number of transitions while ignoring the length of the acceptance condition, whereas other results consider the length of the acceptance condition while ignoring the two other parameters.
We establish a full picture of the translations to and from Muller automata, enhancing known results and adding new ones. Overall, Muller automata can be considered less succinct than parity, Rabin, and Streett automata: translating nondeterministic Muller automata to the other nondeterministic types involves a polynomial size blowup, while the other way round is exponential; translating between the deterministic versions is exponential in both directions; and translating nondeterministic automata of all types to deterministic Muller automata is doubly exponential, as opposed to a single exponent in the translations to the other deterministic types
History Determinism vs. Good for Gameness in Quantitative Automata
Automata models between determinism and nondeterminism/alternations can retain some of the algorithmic properties of deterministic automata while enjoying some of the expressiveness and succinctness of nondeterminism. We study three closely related such models - history determinism, good for gameness and determinisability by pruning - on quantitative automata.
While in the Boolean setting, history determinism and good for gameness coincide, we show that this is no longer the case in the quantitative setting: good for gameness is broader than history determinism, and coincides with a relaxed version of it, defined with respect to thresholds. We further identify criteria in which history determinism, which is generally broader than determinisability by pruning, coincides with it, which we then apply to typical quantitative automata types.
As a key application of good for games and history deterministic automata is synthesis, we clarify the relationship between the two notions and various quantitative synthesis problems. We show that good-for-games automata are central for "global" (classical) synthesis, while "local" (good-enough) synthesis reduces to deciding whether a nondeterministic automaton is history deterministic
REGISTER GAMES
The complexity of parity games is a long standing open problem that saw a
major breakthrough in 2017 when two quasi-polynomial algorithms were published.
This article presents a third, independent approach to solving parity games in
quasi-polynomial time, based on the notion of register game, a parameterised
variant of a parity game. The analysis of register games leads to a
quasi-polynomial algorithm for parity games, a polynomial algorithm for
restricted classes of parity games and a novel measure of complexity, the
register index, which aims to capture the combined complexity of the priority
assignement and the underlying game graph.
We further present a translation of alternating parity word automata into
alternating weak automata with only a quasi-polynomial increase in size, based
on register games; this improves on the previous exponential translation.
We also use register games to investigate the parity index hierarchy: while
for words the index hierarchy of alternating parity automata collapses to the
weak level, and for trees it is strict, for structures between trees and words,
it collapses logarithmically, in the sense that any parity tree automaton of
size n is equivalent, on these particular classes of structures, to an
automaton with a number of priorities logarithmic in n
Discounted-Sum Automata with Multiple Discount Factors
Discounting the influence of future events is a key paradigm in economics and
it is widely used in computer-science models, such as games, Markov decision
processes (MDPs), reinforcement learning, and automata. While a single game or
MDP may allow for several different discount factors, discounted-sum automata
(NDAs) were only studied with respect to a single discount factor. For every
integer , as opposed to every , the class of NDAs with discount factor
(-NDAs) has good computational properties: it is closed
under determinization and under the algebraic operations min, max, addition,
and subtraction, and there are algorithms for its basic decision problems, such
as automata equivalence and containment.
We define and analyze discounted-sum automata in which each transition can
have a different integral discount factor (integral NMDAs). We show that
integral NMDAs with an arbitrary choice of discount factors are not closed
under determinization and under algebraic operations and that their containment
problem is undecidable. We then define and analyze a restricted class of
integral NMDAs, which we call tidy NMDAs, in which the choice of discount
factors depends on the prefix of the word read so far. Some of their special
cases are NMDAs that correlate discount factors to actions (alphabet letters)
or to the elapsed time. We show that for every function that defines
the choice of discount factors, the class of -NMDAs enjoys all of the
above good properties of integral NDAs, as well as the same complexity of the
required decision problems. Tidy NMDAs are also as expressive as deterministic
integral NMDAs with an arbitrary choice of discount factors.
All of our results hold for both automata on finite words and automata on
infinite words.Comment: arXiv admin note: text overlap with arXiv:2301.0408
Good for Games Automata: From Nondeterminism to Alternation
A word automaton recognizing a language is good for games (GFG) if its
composition with any game with winning condition preserves the game's
winner. While all deterministic automata are GFG, some nondeterministic
automata are not. There are various other properties that are used in the
literature for defining that a nondeterministic automaton is GFG, including
"history-deterministic", "compliant with some letter game", "good for trees",
and "good for composition with other automata". The equivalence of these
properties has not been formally shown.
We generalize all of these definitions to alternating automata and show their
equivalence. We further show that alternating GFG automata are as expressive as
deterministic automata with the same acceptance conditions and indices. We then
show that alternating GFG automata over finite words, and weak automata over
infinite words, are not more succinct than deterministic automata, and that
determinizing B\"uchi and co-B\"uchi alternating GFG automata involves a
state blow-up. We leave open the question of whether
alternating GFG automata of stronger acceptance conditions allow for
doubly-exponential succinctness compared to deterministic automata.Comment: Full version of a paper of the same name accepted fr publication at
the 30th International Conference on Concurrency Theor
On the Comparison of Discounted-Sum Automata with Multiple Discount Factors
We look into the problems of comparing nondeterministic discounted-sum
automata on finite and infinite words. That is, the problems of checking for
automata and whether or not it holds that for all words ,
, or .
These problems are known to be decidable when both automata have the same
single integral discount factor, while decidability is open in all other
settings: when the single discount factor is a non-integral rational; when each
automaton can have multiple discount factors; and even when each has a single
integral discount factor, but the two are different.
We show that it is undecidable to compare discounted-sum automata with
multiple discount factors, even if all are integrals, while it is decidable to
compare them if each has a single, possibly different, integral discount
factor. To this end, we also provide algorithms to check for given
nondeterministic automaton and deterministic automaton , each with a
single, possibly different, rational discount factor, whether or not , , or for all words .Comment: This is the full version of a chapter with the same title that
appears in the FoSSaCS 2023 conference proceeding
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