1,078 research outputs found
Piecewise Testable Languages and Nondeterministic Automata
A regular language is -piecewise testable if it is a finite boolean
combination of languages of the form , where and . Given a DFA and , it is an NL-complete problem to decide whether the language is
piecewise testable and, for , it is coNP-complete to decide whether the
language is -piecewise testable. It is known that the depth of the
minimal DFA serves as an upper bound on . Namely, if is piecewise
testable, then it is -piecewise testable for equal to the depth of .
In this paper, we show that some form of nondeterminism does not violate this
upper bound result. Specifically, we define a class of NFAs, called ptNFAs,
that recognize piecewise testable languages and show that the depth of a ptNFA
provides an (up to exponentially better) upper bound on than the minimal
DFA. We provide an application of our result, discuss the relationship between
-piecewise testability and the depth of NFAs, and study the complexity of
-piecewise testability for ptNFAs.Comment: Corrections in section 4.
Complexity of Deciding Detectability in Discrete Event Systems
Detectability of discrete event systems (DESs) is a question whether the
current and subsequent states can be determined based on observations. Shu and
Lin designed a polynomial-time algorithm to check strong (periodic)
detectability and an exponential-time (polynomial-space) algorithm to check
weak (periodic) detectability. Zhang showed that checking weak (periodic)
detectability is PSpace-complete. This intractable complexity opens a question
whether there are structurally simpler DESs for which the problem is tractable.
In this paper, we show that it is not the case by considering DESs represented
as deterministic finite automata without non-trivial cycles, which are
structurally the simplest deadlock-free DESs. We show that even for such very
simple DESs, checking weak (periodic) detectability remains intractable. On the
contrary, we show that strong (periodic) detectability of DESs can be
efficiently verified on a parallel computer
A Note on Limited Pushdown Alphabets in Stateless Deterministic Pushdown Automata
Recently, an infinite hierarchy of languages accepted by stateless
deterministic pushdown automata has been established based on the number of
pushdown symbols. However, the witness language for the n-th level of the
hierarchy is over an input alphabet with 2(n-1) elements. In this paper, we
improve this result by showing that a binary alphabet is sufficient to
establish this hierarchy. As a consequence of our construction, we solve the
open problem formulated by Meduna et al. Then we extend these results to
m-state realtime deterministic pushdown automata, for all m at least 1. The
existence of such a hierarchy for m-state deterministic pushdown automata is
left open
Complexity of Infimal Observable Superlanguages
The infimal prefix-closed, controllable and observable superlanguage plays an
essential role in the relationship between controllability, observability and
co-observability -- the central notions of supervisory control theory. Existing
algorithms for its computation are exponential and it is not known whether a
polynomial algorithm exists. In this paper, we study the state complexity of
this language. State complexity of a language is the number of states of the
minimal DFA for the language. For a language of state complexity , we show
that the upper-bound state complexity on the infimal prefix-closed and
observable superlanguage is and that this bound is asymptotically
tight. It proves that there is no algorithm computing a DFA of the infimal
prefix-closed and observable superlanguage in polynomial time. Our construction
further shows that such a DFA can be computed in time . The
construction involves NFAs and a computation of the supremal prefix-closed
sublanguage. We study the computation of the supremal prefix-closed sublanguage
and show that there is no polynomial-time algorithm that computes an NFA of the
supremal prefix-closed sublanguage of a language given as an NFA even if the
language is unary
Comparison of Two Context-Free Rewriting Systems with Simple Context-Checking Mechanisms
This paper solves an open problem concerning the generative power of
nonerasing context-free rewriting systems using a simple mechanism for checking
for context dependencies, in the literature known as semi-conditional grammars
of degree (1,1). In these grammars, two nonterminal symbols are attached to
each context-free production, and such a production is applicable if one of the
two attached symbols occurs in the current sentential form, while the other
does not. Specifically, this paper demonstrates that the family of languages
generated by semi-conditional grammars of degree (1,1) coincides with the
family of random context languages. In addition, it shows that the normal form
proved by Mayer for random context grammars with erasing productions holds for
random context grammars without erasing productions, too. It also discusses two
possible definitions of the relation of the direct derivation step used in the
literature.Comment: Unpublished as a stronger result is under consideration
Complexity of Verifying Nonblockingness in Modular Supervisory Control
Complexity analysis becomes a common task in supervisory control. However,
many results of interest are spread across different topics. The aim of this
paper is to bring several interesting results from complexity theory and to
illustrate their relevance to supervisory control by proving new nontrivial
results concerning nonblockingness in modular supervisory control of discrete
event systems modeled by finite automata
A Note on Undecidability of Observation Consistency for Non-Regular Languages
One of the most interesting questions concerning hierarchical control of
discrete-event systems with partial observations is a condition under which the
language observability is preserved between the original and the abstracted
plant. Recently, we have characterized two such sufficient
conditions---observation consistency and local observation consistency. In this
paper, we prove that the condition of observation consistency is undecidable
for non-regular (linear, deterministic context-free) languages. The question
whether the condition is decidable for regular languages is open
Separability by Piecewise Testable Languages is PTime-Complete
Piecewise testable languages form the first level of the Straubing-Th\'erien
hierarchy. The membership problem for this level is decidable and testing if
the language of a DFA is piecewise testable is NL-complete. The question has
not yet been addressed for NFAs. We fill in this gap by showing that it is
PSpace-complete. The main result is then the lower-bound complexity of
separability of regular languages by piecewise testable languages. Two regular
languages are separable by a piecewise testable language if the piecewise
testable language includes one of them and is disjoint from the other. For
languages represented by NFAs, separability by piecewise testable languages is
known to be decidable in PTime. We show that it is PTime-hard and that it
remains PTime-hard even for minimal DFAs.Comment: Revised version of related results separated from arXiv:1409.394
On the State Complexity of the Reverse of R- and J-trivial Regular Languages
The tight upper bound on the state complexity of the reverse of R-trivial and
J-trivial regular languages of the state complexity n is 2^{n-1}. The witness
is ternary for R-trivial regular languages and (n-1)-ary for J-trivial regular
languages. In this paper, we prove that the bound can be met neither by a
binary R-trivial regular language nor by a J-trivial regular language over an
(n-2)-element alphabet. We provide a characterization of tight bounds for
R-trivial regular languages depending on the state complexity of the language
and the size of its alphabet. We show the tight bound for J-trivial regular
languages over an (n-2)-element alphabet and a few tight bounds for binary
J-trivial regular languages. The case of J-trivial regular languages over an
(n-k)-element alphabet, for 2 <= k <= n-3, is open.Comment: Full version of the paper accepted for DCFS 201
On Verification of D-Detectability for Discrete Event Systems
Detectability has been introduced as a generalization of state-estimation
properties of discrete event systems studied in the literature. It asks whether
the current and subsequent states of a system can be determined based on
observations. Since, in some applications, to exactly determine the current and
subsequent states may be too strict, a relaxed notion of D-detectability has
been introduced, distinguishing only certain pairs of states rather than all
states. Four variants of D-detectability have been defined: strong (periodic)
D-detectability and weak (periodic) D-detectability. Deciding weak (periodic)
D-detectability is PSpace-complete, while deciding strong (periodic)
detectability or strong D-detectability is polynomial (and we show that it is
actually NL-complete). However, to the best of our knowledge, it is an open
problem whether there exists a polynomial-time algorithm deciding strong
periodic D-detectability. We solve this problem by showing that deciding strong
periodic D-detectability is a PSpace-complete problem, and hence there is no
polynomial-time algorithm unless PSpace = P. We further show that there is no
polynomial-time algorithm deciding strong periodic D-detectability even for
systems with a single observable event, unless P = NP. Finally, we propose a
class of systems for which the problem is tractable.Comment: Extended version of a paper accepted for WODES 202
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