7,478 research outputs found
Tightening the Complexity of Equivalence Problems for Commutative Grammars
We show that the language equivalence problem for regular and context-free
commutative grammars is coNEXP-complete. In addition, our lower bound
immediately yields further coNEXP-completeness results for equivalence problems
for communication-free Petri nets and reversal-bounded counter automata.
Moreover, we improve both lower and upper bounds for language equivalence for
exponent-sensitive commutative grammars.Comment: 21 page
On the equivalence, containment, and covering problems for the regular and context-free languages
We consider the complexity of the equivalence and containment problems for regular expressions and context-free grammars, concentrating on the relationship between complexity and various language properties. Finiteness and boundedness of languages are shown to play important roles in the complexity of these problems. An encoding into grammars of Turing machine computations exponential in the size of the grammar is used to prove several exponential lower bounds. These lower bounds include exponential time for testing equivalence of grammars generating finite sets, and exponential space for testing equivalence of non-self-embedding grammars. Several problems which might be complex because of this encoding are shown to simplify for linear grammars. Other problems considered include grammatical covering and structural equivalence for right-linear, linear, and arbitrary grammars
A new variant of Petri net controlled grammars
A Petri net controlled grammar is a Petri net with respect to a context-free grammar where the successful derivations of the grammar can be simulated using the occurrence sequences of the net. In this paper, we introduce a new variant of Petri net controlled grammars, called a place-labeled Petri net controlled grammar, which is a context-free grammar equipped with a Petri net and a function which maps places of the net to productions of the grammar. The language consists of all terminal strings that can be obtained by parallelly applying multisets of the rules which are the images of the sets of the input places of transitions in a successful occurrence sequence of the Petri net. We study the effect of the different labeling strategies to the computational power and establish lower and upper bounds for the generative capacity of place- labeled Petri net controlled grammars
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Packing and Counting Permutations
A permutation class is a set of permutations closed under taking subpermutations. We study two aspects of permutation classes: enumeration and packing.
Our work on enumeration consists of two campaigns. First, we enumerate all juxtaposition classes of the form “Av(abc) next to Av(xy)”, where abc and xy are permutations of lengths three and two, respectively. We represent elements from such a juxtaposition class by Dyck paths decorated with sequences of points. Context-free grammars are then used to enumerate these decorated Dyck paths. Second, we classify as algebraic the generating functions of 1×m permutation grid classes where one cell is context-free and the remaining cells are monotone. We rely on properties of combinatorial specifications of context-free classes and use operators to express juxtapositions. Repeated application of operators resolves cases for m > 2. We provide examples to re-prove known results and give new ones. Our methods are algorithmic and could be implemented on a PC.
Our work on packing consolidates current knowledge about packing densities of 4-point permutations. We also improve the lower bounds for the packing densities of 1324 and 1342 and provide rigorous upper bounds for the packing densities of 1324, 1342, and 2413. All our bounds are within 10-4 of the true packing densities. Together with the known bounds, we have a fairly complete picture of 4-point packing densities. Additionally, we obtain several bounds (lower and upper) for permutations of length at least five. Our main tool for the upper bounds is the framework of flag algebras introduced by Razborov in 2007. We also present Permpack — a flag algebra package for permutations
Grammar Boosting: A New Technique for Proving Lower Bounds for Computation over Compressed Data
Grammar compression is a general compression framework in which a string
of length is represented as a context-free grammar of size whose
language contains only . In this paper, we focus on studying the limitations
of algorithms and data structures operating on strings in grammar-compressed
form. Previous work focused on proving lower bounds for grammars constructed
using algorithms that achieve the approximation ratio
. Unfortunately, for the majority of
grammar compressors, is either unknown or satisfies
. In their seminal paper, Charikar et al. [IEEE
Trans. Inf. Theory 2005] studied seven popular grammar compression algorithms:
RePair, Greedy, LongestMatch, Sequential, Bisection, LZ78, and
-Balanced. Only one of them (-Balanced) is known to achieve
.
We develop the first technique for proving lower bounds for data structures
and algorithms on grammars that is fully general and does not depend on the
approximation ratio of the used grammar compressor. Using this
technique, we first prove that time is required
for random access on RePair, Greedy, LongestMatch, Sequential, and Bisection,
while time is required for random access to LZ78. All
these lower bounds hold within space and
match the existing upper bounds. We also generalize this technique to prove
several conditional lower bounds for compressed computation. For example, we
prove that unless the Combinatorial -Clique Conjecture fails, there is no
combinatorial algorithm for CFG parsing on Bisection (for which it holds
) that runs in time for all constants and . Previously,
this was known only for
On vocabulary size of grammar-based codes
We discuss inequalities holding between the vocabulary size, i.e., the number
of distinct nonterminal symbols in a grammar-based compression for a string,
and the excess length of the respective universal code, i.e., the code-based
analog of algorithmic mutual information. The aim is to strengthen inequalities
which were discussed in a weaker form in linguistics but shed some light on
redundancy of efficiently computable codes. The main contribution of the paper
is a construction of universal grammar-based codes for which the excess lengths
can be bounded easily.Comment: 5 pages, accepted to ISIT 2007 and correcte
On Measuring Non-Recursive Trade-Offs
We investigate the phenomenon of non-recursive trade-offs between
descriptional systems in an abstract fashion. We aim at categorizing
non-recursive trade-offs by bounds on their growth rate, and show how to deduce
such bounds in general. We also identify criteria which, in the spirit of
abstract language theory, allow us to deduce non-recursive tradeoffs from
effective closure properties of language families on the one hand, and
differences in the decidability status of basic decision problems on the other.
We develop a qualitative classification of non-recursive trade-offs in order to
obtain a better understanding of this very fundamental behaviour of
descriptional systems
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