256 research outputs found
Multi-Head Finite Automata: Characterizations, Concepts and Open Problems
Multi-head finite automata were introduced in (Rabin, 1964) and (Rosenberg,
1966). Since that time, a vast literature on computational and descriptional
complexity issues on multi-head finite automata documenting the importance of
these devices has been developed. Although multi-head finite automata are a
simple concept, their computational behavior can be already very complex and
leads to undecidable or even non-semi-decidable problems on these devices such
as, for example, emptiness, finiteness, universality, equivalence, etc. These
strong negative results trigger the study of subclasses and alternative
characterizations of multi-head finite automata for a better understanding of
the nature of non-recursive trade-offs and, thus, the borderline between
decidable and undecidable problems. In the present paper, we tour a fragment of
this literature
Descriptional Complexity of Finite Automata -- Selected Highlights
The state complexity, respectively, nondeterministic state complexity of a
regular language is the number of states of the minimal deterministic,
respectively, of a minimal nondeterministic finite automaton for . Some of
the most studied state complexity questions deal with size comparisons of
nondeterministic finite automata of differing degree of ambiguity. More
generally, if for a regular language we compare the size of description by a
finite automaton and by a more powerful language definition mechanism, such as
a context-free grammar, we encounter non-recursive trade-offs. Operational
state complexity studies the state complexity of the language resulting from a
regularity preserving operation as a function of the complexity of the argument
languages. Determining the state complexity of combined operations is generally
challenging and for general combinations of operations that include
intersection and marked concatenation it is uncomputable
On the Succinctness of Alternating Parity Good-For-Games Automata
We study alternating parity good-for-games (GFG) automata, i.e., alternating parity automata where both conjunctive and disjunctive choices can be resolved in an online manner, without knowledge of the suffix of the input word still to be read.
We show that they can be exponentially more succinct than both their nondeterministic and universal counterparts. Furthermore, we present a single exponential determinisation procedure and an Exptime upper bound to the problem of recognising whether an alternating automaton is GFG.
We also study the complexity of deciding "half-GFGness", a property specific to alternating automata that only requires nondeterministic choices to be resolved in an online manner. We show that this problem is PSpace-hard already for alternating automata on finite words
Alternation and Bounded Concurrency Are Reverse Equivalent
AbstractNumerous models of concurrency have been considered in the framework of automata. Among the more interesting concurrency models are classical nondeterminism and pure concurrency, the two facets of alternation, and the bounded concurrency model. Bounded concurrency was previously considered to be similar to nondeterminism and pure concurrency in the sense of the succinctness of automata augmented with these features. In this paper we show that, when viewed more broadly, the power (of succinctness) of bounded concurrency is in fact most similar to the power of alternation. Our contribution is that, just like nondeterminism and pure concurrency are “complement equivalent,” bounded concurrency and alternation are “reverse equivalent” over finite automata. The reverse equivalence is expressed by the existence of polynomial transformations, in both directions, between bounded concurrency and alternation for the reverse of the language accepted by the other. It follows, that bounded concurrency is double-exponentially more succinct than DFAs with respect to reverse, while alternation only saves one exponent. This is as opposed to the direct case where alternation saves two exponents and bounded concurrency saves only one. An immediate corollary is that for languages over a one-letter alphabet, bounded concurrency and alternation are equivalent. We complete the picture of succinctness results for these languages by considering the different combinations of the concurrency models using additional lower bounds
From Finite Automata to Regular Expressions and Back--A Summary on Descriptional Complexity
The equivalence of finite automata and regular expressions dates back to the
seminal paper of Kleene on events in nerve nets and finite automata from 1956.
In the present paper we tour a fragment of the literature and summarize results
on upper and lower bounds on the conversion of finite automata to regular
expressions and vice versa. We also briefly recall the known bounds for the
removal of spontaneous transitions (epsilon-transitions) on non-epsilon-free
nondeterministic devices. Moreover, we report on recent results on the average
case descriptional complexity bounds for the conversion of regular expressions
to finite automata and brand new developments on the state elimination
algorithm that converts finite automata to regular expressions.Comment: In Proceedings AFL 2014, arXiv:1405.527
How Deterministic are Good-For-Games Automata?
In GFG automata, it is possible to resolve nondeterminism in a way that only
depends on the past and still accepts all the words in the language. The
motivation for GFG automata comes from their adequacy for games and synthesis,
wherein general nondeterminism is inappropriate. We continue the ongoing effort
of studying the power of nondeterminism in GFG automata. Initial indications
have hinted that every GFG automaton embodies a deterministic one. Today we
know that this is not the case, and in fact GFG automata may be exponentially
more succinct than deterministic ones.
We focus on the typeness question, namely the question of whether a GFG
automaton with a certain acceptance condition has an equivalent GFG automaton
with a weaker acceptance condition on the same structure. Beyond the
theoretical interest in studying typeness, its existence implies efficient
translations among different acceptance conditions. This practical issue is of
special interest in the context of games, where the Buchi and co-Buchi
conditions admit memoryless strategies for both players. Typeness is known to
hold for deterministic automata and not to hold for general nondeterministic
automata.
We show that GFG automata enjoy the benefits of typeness, similarly to the
case of deterministic automata. In particular, when Rabin or Streett GFG
automata have equivalent Buchi or co-Buchi GFG automata, respectively, then
such equivalent automata can be defined on a substructure of the original
automata. Using our typeness results, we further study the place of GFG
automata in between deterministic and nondeterministic ones. Specifically,
considering automata complementation, we show that GFG automata lean toward
nondeterministic ones, admitting an exponential state blow-up in the
complementation of a Streett automaton into a Rabin automaton, as opposed to
the constant blow-up in the deterministic case
On the Succinctness of Good-for-MDPs Automata
Good-for-MDPs and good-for-games automata are two recent classes of
nondeterministic automata that reside between general nondeterministic and
deterministic automata. Deterministic automata are good-for-games, and
good-for-games automata are good-for-MDPs, but not vice versa. One of the
question this raises is how these classes relate in terms of succinctness.
Good-for-games automata are known to be exponentially more succinct than
deterministic automata, but the gap between good-for-MDPs and good-for-games
automata as well as the gap between ordinary nondeterministic automata and
those that are good-for-MDPs have been open. We establish that these gaps are
exponential, and sharpen this result by showing that the latter gap remains
exponential when restricting the nondeterministic automata to separating safety
or unambiguous reachability automata.Comment: 18 page
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