66 research outputs found
Revisiting Membership Problems in Subclasses of Rational Relations
We revisit the membership problem for subclasses of rational relations over
finite and infinite words: Given a relation R in a class C_2, does R belong to
a smaller class C_1? The subclasses of rational relations that we consider are
formed by the deterministic rational relations, synchronous (also called
automatic or regular) relations, and recognizable relations. For almost all
versions of the membership problem, determining the precise complexity or even
decidability has remained an open problem for almost two decades. In this
paper, we provide improved complexity and new decidability results. (i) Testing
whether a synchronous relation over infinite words is recognizable is
NL-complete (PSPACE-complete) if the relation is given by a deterministic
(nondeterministic) omega-automaton. This fully settles the complexity of this
recognizability problem, matching the complexity of the same problem over
finite words. (ii) Testing whether a deterministic rational binary relation is
recognizable is decidable in polynomial time, which improves a previously known
double exponential time upper bound. For relations of higher arity, we present
a randomized exponential time algorithm. (iii) We provide the first algorithm
to decide whether a deterministic rational relation is synchronous. For binary
relations the algorithm even runs in polynomial time
Highly Undecidable Problems For Infinite Computations
We show that many classical decision problems about 1-counter
omega-languages, context free omega-languages, or infinitary rational
relations, are -complete, hence located at the second level of the
analytical hierarchy, and "highly undecidable". In particular, the universality
problem, the inclusion problem, the equivalence problem, the determinizability
problem, the complementability problem, and the unambiguity problem are all
-complete for context-free omega-languages or for infinitary rational
relations. Topological and arithmetical properties of 1-counter
omega-languages, context free omega-languages, or infinitary rational
relations, are also highly undecidable. These very surprising results provide
the first examples of highly undecidable problems about the behaviour of very
simple finite machines like 1-counter automata or 2-tape automata.Comment: to appear in RAIRO-Theoretical Informatics and Application
Decidability and syntactic control of interference
AbstractWe investigate the decidability of observational equivalence and approximation in Reynolds’ “Syntactic Control of Interference” (SCI), a prototypical functional-imperative language in which covert interference between functions and their arguments is prevented by the use of an affine typing discipline.By associating denotations of terms in a fully abstract “relational” model of finitary basic SCI (due to Reddy) with multitape finite state automata, we show that observational approximation is not decidable (even at first order), but that observational equivalence is decidable for all terms.We then consider the same problems for basic SCI extended with non-local control in the form of backwards jumps. We show that both observational approximation and observational equivalence are decidable in this “observably sequential” version of the language by describing a fully abstract games model in which strategies are regular languages
A SURVEY OF LIMITED NONDETERMINISM IN COMPUTATIONAL COMPLEXITY THEORY
Nondeterminism is typically used as an inherent part of the computational models used incomputational complexity. However, much work has been done looking at nondeterminism asa separate resource added to deterministic machines. This survey examines several differentapproaches to limiting the amount of nondeterminism, including Kintala and Fischer\u27s βhierarchy, and Cai and Chen\u27s guess-and-check model
Small overlap monoids II: automatic structures and normal forms
We show that any finite monoid or semigroup presentation satisfying the small
overlap condition C(4) has word problem which is a deterministic rational
relation. It follows that the set of lexicographically minimal words forms a
regular language of normal forms, and that these normal forms can be computed
in linear time. We also deduce that C(4) monoids and semigroups are rational
(in the sense of Sakarovitch), asynchronous automatic, and word hyperbolic (in
the sense of Duncan and Gilman). From this it follows that C(4) monoids satisfy
analogues of Kleene's theorem, and admit decision algorithms for the rational
subset and finitely generated submonoid membership problems. We also prove some
automata-theoretic results which may be of independent interest.Comment: 17 page
An Application of the Feferman-Vaught Theorem to Automata and Logics for<br> Words over an Infinite Alphabet
We show that a special case of the Feferman-Vaught composition theorem gives
rise to a natural notion of automata for finite words over an infinite
alphabet, with good closure and decidability properties, as well as several
logical characterizations. We also consider a slight extension of the
Feferman-Vaught formalism which allows to express more relations between
component values (such as equality), and prove related decidability results.
From this result we get new classes of decidable logics for words over an
infinite alphabet.Comment: 24 page
The Computational Complexity of Symbolic Dynamics at the Onset of Chaos
In a variety of studies of dynamical systems, the edge of order and chaos has
been singled out as a region of complexity. It was suggested by Wolfram, on the
basis of qualitative behaviour of cellular automata, that the computational
basis for modelling this region is the Universal Turing Machine. In this paper,
following a suggestion of Crutchfield, we try to show that the Turing machine
model may often be too powerful as a computational model to describe the
boundary of order and chaos. In particular we study the region of the first
accumulation of period doubling in unimodal and bimodal maps of the interval,
from the point of view of language theory. We show that in relation to the
``extended'' Chomsky hierarchy, the relevant computational model in the
unimodal case is the nested stack automaton or the related indexed languages,
while the bimodal case is modeled by the linear bounded automaton or the
related context-sensitive languages.Comment: 1 reference corrected, 1 reference added, minor changes in body of
manuscrip
On Rational Recursive Sequences
We study the class of rational recursive sequences (ratrec) over the rational
numbers. A ratrec sequence is defined via a system of sequences using mutually
recursive equations of depth 1, where the next values are computed as rational
functions of the previous values. An alternative class is that of simple ratrec
sequences, where one uses a single recursive equation, however of depth k: the
next value is defined as a rational function of k previous values.
We conjecture that the classes ratrec and simple ratrec coincide. The main
contribution of this paper is a proof of a variant of this conjecture where the
initial conditions are treated symbolically, using a formal variable per
sequence, while the sequences themselves consist of rational functions over
those variables. While the initial conjecture does not follow from this
variant, we hope that the introduced algebraic techniques may eventually be
helpful in resolving the problem.
The class ratrec strictly generalises a well-known class of polynomial
recursive sequences (polyrec). These are defined like ratrec, but using
polynomial functions instead of rational ones. One can observe that if our
conjecture is true and effective, then we can improve the complexities of the
zeroness and the equivalence problems for polyrec sequences. Currently, the
only known upper bound is Ackermanian, which follows from results on polynomial
automata. We complement this observation by proving a PSPACE lower bound for
both problems for polyrec. Our lower bound construction also implies that the
Skolem problem is PSPACE-hard for the polyrec class
Finite state verifiers with constant randomness
We give a new characterization of as the class of languages
whose members have certificates that can be verified with small error in
polynomial time by finite state machines that use a constant number of random
bits, as opposed to its conventional description in terms of deterministic
logarithmic-space verifiers. It turns out that allowing two-way interaction
with the prover does not change the class of verifiable languages, and that no
polynomially bounded amount of randomness is useful for constant-memory
computers when used as language recognizers, or public-coin verifiers. A
corollary of our main result is that the class of outcome problems
corresponding to O(log n)-space bounded games of incomplete information where
the universal player is allowed a constant number of moves equals NL.Comment: 17 pages. An improved versio
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Verifying and comparing finite state machines for systems that have distributed interfaces
This paper concerns state-based systems that interact with their environment at physically distributed interfaces, called ports. When such a system is used a projection of the global trace, a local trace, is observed at each port. As a result the environment has reduced observational power: the set of local traces observed need not define the global trace that occurred. We consider the previously defined implementation relation ⊆s and prove that it is undecidable whether N ⊆s M and so it is also undecidable whether testing can distinguishing two states or FSMs. We also prove that a form of model-checking is undecidable when we have distributed observations and give conditions under which N ⊆s M is decidable. We then consider implementation relation ⊆sk that concerns input sequences of length κ or less. If we place bounds on κ and the number of ports then we can decide N ⊆sk M in polynomial time but otherwise this problem is NP-hard
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