1,786 research outputs found
Two iteration theorems for the LL(k) languages
AbstractThe structure of derivation trees over an LL(k) grammar is explored and a property of these trees obtained which is shown to characterize the LL(k) grammars. This characterization, called the LL(k) Left Part Theorem, makes it possible to establish a pair of iteration theorems for the LL(k) languages. These theorems provide a general and powerful method of showing that a language is not LL(k) when that is the case. They thus provide for the first time a flexible tool with which to explore the structure of the LL(k) languages and with which to discriminate between the LL(k) and LR(k) language classes.Examples are given of LR(k) languages which, for various reasons, fail to be LL(k). Easy and rigorous proofs to this effect are given using our LL(k) iteration theorems. In particular, it is proven that the dangling-ELSE construct allowed in PL/I and Pascal cannot be generated by any LL(k) grammar. We also give a new and straightforward proof based on the LL(k) Left Part Theorem that every LL(k) grammar is LR(k)
Commutative Languages and their Composition by Consensual Methods
Commutative languages with the semilinear property (SLIP) can be naturally
recognized by real-time NLOG-SPACE multi-counter machines. We show that unions
and concatenations of such languages can be similarly recognized, relying on --
and further developing, our recent results on the family of consensually
regular (CREG) languages. A CREG language is defined by a regular language on
the alphabet that includes the terminal alphabet and its marked copy. New
conditions, for ensuring that the union or concatenation of CREG languages is
closed, are presented and applied to the commutative SLIP languages. The paper
contributes to the knowledge of the CREG family, and introduces novel
techniques for language composition, based on arithmetic congruences that act
as language signatures. Open problems are listed.Comment: In Proceedings AFL 2014, arXiv:1405.527
Proper Functors and Fixed Points for Finite Behaviour
The rational fixed point of a set functor is well-known to capture the
behaviour of finite coalgebras. In this paper we consider functors on algebraic
categories. For them the rational fixed point may no longer be fully abstract,
i.e. a subcoalgebra of the final coalgebra. Inspired by \'Esik and Maletti's
notion of a proper semiring, we introduce the notion of a proper functor. We
show that for proper functors the rational fixed point is determined as the
colimit of all coalgebras with a free finitely generated algebra as carrier and
it is a subcoalgebra of the final coalgebra. Moreover, we prove that a functor
is proper if and only if that colimit is a subcoalgebra of the final coalgebra.
These results serve as technical tools for soundness and completeness proofs
for coalgebraic regular expression calculi, e.g. for weighted automata
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
Complexity Hierarchies Beyond Elementary
We introduce a hierarchy of fast-growing complexity classes and show its
suitability for completeness statements of many non elementary problems. This
hierarchy allows the classification of many decision problems with a
non-elementary complexity, which occur naturally in logic, combinatorics,
formal languages, verification, etc., with complexities ranging from simple
towers of exponentials to Ackermannian and beyond.Comment: Version 3 is the published version in TOCT 8(1:3), 2016. I will keep
updating the catalogue of problems from Section 6 in future revision
Deterministic Real-Time Tree-Walking-Storage Automata
We study deterministic tree-walking-storage automata, which are finite-state
devices equipped with a tree-like storage. These automata are generalized stack
automata, where the linear stack storage is replaced by a non-linear tree-like
stack. Therefore, tree-walking-storage automata have the ability to explore the
interior of the tree storage without altering the contents, with the possible
moves of the tree pointer corresponding to those of tree-walking automata. In
addition, a tree-walking-storage automaton can append (push) non-existent
descendants to a tree node and remove (pop) leaves from the tree. Here we are
particularly considering the capacities of deterministic tree-walking-storage
automata working in real time. It is shown that even the non-erasing variant
can accept rather complicated unary languages as, for example, the language of
words whose lengths are powers of two, or the language of words whose lengths
are Fibonacci numbers. Comparing the computational capacities with automata
from the classical automata hierarchy, we derive that the families of languages
accepted by real-time deterministic (non-erasing) tree-walking-storage automata
is located between the regular and the deterministic context-sensitive
languages. There is a context-free language that is not accepted by any
real-time deterministic tree-walking-storage automaton. On the other hand,
these devices accept a unary language in non-erasing mode that cannot be
accepted by any classical stack automaton, even in erasing mode and arbitrary
time. Basic closure properties of the induced families of languages are shown.
In particular, we consider Boolean operations (complementation, union,
intersection) and AFL operations (union, intersection with regular languages,
homomorphism, inverse homomorphism, concatenation, iteration). It turns out
that the two families in question have the same properties and, in particular,
share all but one of these closure properties with the important family of
deterministic context-free languages.Comment: In Proceedings NCMA 2023, arXiv:2309.0733
New Bounds for the Garden-Hose Model
We show new results about the garden-hose model. Our main results include
improved lower bounds based on non-deterministic communication complexity
(leading to the previously unknown bounds for Inner Product mod 2
and Disjointness), as well as an upper bound for the
Distributed Majority function (previously conjectured to have quadratic
complexity). We show an efficient simulation of formulae made of AND, OR, XOR
gates in the garden-hose model, which implies that lower bounds on the
garden-hose complexity of the order will be
hard to obtain for explicit functions. Furthermore we study a time-bounded
variant of the model, in which even modest savings in time can lead to
exponential lower bounds on the size of garden-hose protocols.Comment: In FSTTCS 201
- …