161 research outputs found
State Space Reduction For Parity Automata
Exact minimization of ?-automata is a difficult problem and heuristic algorithms are a subject of current research. We propose several new approaches to reduce the state space of deterministic parity automata. These are based on extracting information from structures within the automaton, such as strongly connected components, coloring of the states, and equivalence classes of given relations, to determine states that can safely be merged. We also establish a framework to generalize the notion of quotient automata and uniformly describe such algorithms. The description of these procedures consists of a theoretical analysis as well as data collected from experiments
A Rice-style theorem for parallel automata
AbstractWe present a general result, similar to Rice’s theorem, concerning the complexity of detecting properties on finite automata enriched by bounded cooperative concurrency, such as statecharts and abstract parallel automata, which we denote by CFAs (Concurrent Finite Automata). On one extreme, the complexity of detecting non-trivial properties that preserve equivalence of machines, i.e. properties of the accepted language, on finite automata, can be as little as O(1). On the other extreme, Rice’s theorem states that all such properties on Turing machines are undecidable. We state that all the non-trivial properties of the regular (or ω-regular) languages, are PSPACE-hard on CFAs with ϵ-moves and on CFAs without ϵ-moves accepting infinite words. We also extend this result to CFAs without ϵ-moves accepting finite words that satisfy a condition that holds for many properties
Index problems for game automata
For a given regular language of infinite trees, one can ask about the minimal
number of priorities needed to recognize this language with a
non-deterministic, alternating, or weak alternating parity automaton. These
questions are known as, respectively, the non-deterministic, alternating, and
weak Rabin-Mostowski index problems. Whether they can be answered effectively
is a long-standing open problem, solved so far only for languages recognizable
by deterministic automata (the alternating variant trivializes).
We investigate a wider class of regular languages, recognizable by so-called
game automata, which can be seen as the closure of deterministic ones under
complementation and composition. Game automata are known to recognize languages
arbitrarily high in the alternating Rabin-Mostowski index hierarchy; that is,
the alternating index problem does not trivialize any more.
Our main contribution is that all three index problems are decidable for
languages recognizable by game automata. Additionally, we show that it is
decidable whether a given regular language can be recognized by a game
automaton
Prompt Delay
Delay games are two-player games of infinite duration in which one player may
delay her moves to obtain a lookahead on her opponent's moves. Recently, such
games with quantitative winning conditions in weak MSO with the unbounding
quantifier were studied, but their properties turned out to be unsatisfactory.
In particular, unbounded lookahead is in general necessary. Here, we study
delay games with winning conditions given by Prompt-LTL, Linear Temporal Logic
equipped with a parameterized eventually operator whose scope is bounded. Our
main result shows that solving Prompt-LTL delay games is complete for
triply-exponential time. Furthermore, we give tight triply-exponential bounds
on the necessary lookahead and on the scope of the parameterized eventually
operator. Thus, we identify Prompt-LTL as the first known class of well-behaved
quantitative winning conditions for delay games. Finally, we show that applying
our techniques to delay games with \omega-regular winning conditions answers
open questions in the cases where the winning conditions are given by
non-deterministic, universal, or alternating automata
VLDL Satisfiability and Model Checking via Tree Automata
We present novel algorithms solving the satisfiability problem and the model
checking problem for Visibly Linear Dynamic Logic (VLDL) in asymptotically
optimal time via a reduction to the emptiness problem for tree automata with
B\"uchi acceptance. Since VLDL allows for the specification of important
properties of recursive systems, this reduction enables the efficient analysis
of such systems.
Furthermore, as the problem of tree automata emptiness is well-studied, this
reduction enables leveraging the mature algorithms and tools for that problem
in order to solve the satisfiability problem and the model checking problem for
VLDL.Comment: 14 page
Wadge Degrees of -Languages of Petri Nets
We prove that -languages of (non-deterministic) Petri nets and
-languages of (non-deterministic) Turing machines have the same
topological complexity: the Borel and Wadge hierarchies of the class of
-languages of (non-deterministic) Petri nets are equal to the Borel and
Wadge hierarchies of the class of -languages of (non-deterministic)
Turing machines which also form the class of effective analytic sets. In
particular, for each non-null recursive ordinal there exist some -complete and some -complete -languages of Petri nets, and the supremum of
the set of Borel ranks of -languages of Petri nets is the ordinal
, which is strictly greater than the first non-recursive ordinal
. We also prove that there are some -complete, hence non-Borel, -languages of Petri nets, and
that it is consistent with ZFC that there exist some -languages of
Petri nets which are neither Borel nor -complete. This
answers the question of the topological complexity of -languages of
(non-deterministic) Petri nets which was left open in [DFR14,FS14].Comment: arXiv admin note: text overlap with arXiv:0712.1359, arXiv:0804.326
The Complexity of Flat Freeze LTL
We consider the model-checking problem for freeze LTL on one-counter automata (OCAs). Freeze LTL extends LTL with the freeze quantifier, which allows one to store different counter values of a run in registers so that they can be compared with one another. As the model-checking problem is undecidable in general, we focus on the flat fragment of freeze LTL, in which the usage of the freeze quantifier is restricted. Recently, Lechner et al. showed that model checking for flat freeze LTL on OCAs with binary encoding of counter updates is decidable and in 2NEXPTIME. In this paper, we prove that the problem is, in fact, NEXPTIME-complete no matter whether counter updates are encoded in unary or binary. Like Lechner et al., we rely on a reduction to the reachability problem in OCAs with parameterized tests (OCAPs). The new aspect is that we simulate OCAPs by alternating two-way automata over words. This implies an exponential upper bound on the parameter values that we exploit towards an NP algorithm for reachability in OCAPs with unary updates. We obtain our main result as a corollary
- …