22 research outputs found
Regular realizability problems and context-free languages
We investigate regular realizability (RR) problems, which are the problems of
verifying whether intersection of a regular language -- the input of the
problem -- and fixed language called filter is non-empty. In this paper we
focus on the case of context-free filters. Algorithmic complexity of the RR
problem is a very coarse measure of context-free languages complexity. This
characteristic is compatible with rational dominance. We present examples of
P-complete RR problems as well as examples of RR problems in the class NL. Also
we discuss RR problems with context-free filters that might have intermediate
complexity. Possible candidates are the languages with polynomially bounded
rational indices.Comment: conference DCFS 201
Space complexity in on-line computation
AbstractA technique is developed for determining space complexity in on-line computation. It is shown that each of the following functions requires linear space: (i) the conversion of binary numbers into ternary numbers, (ii) the multiplication of integers and (iii) the translation of arithmetic expressions in infix notation into Polish notation
One-Tape Turing Machine Variants and Language Recognition
We present two restricted versions of one-tape Turing machines. Both
characterize the class of context-free languages. In the first version,
proposed by Hibbard in 1967 and called limited automata, each tape cell can be
rewritten only in the first visits, for a fixed constant .
Furthermore, for deterministic limited automata are equivalent to
deterministic pushdown automata, namely they characterize deterministic
context-free languages. Further restricting the possible operations, we
consider strongly limited automata. These models still characterize
context-free languages. However, the deterministic version is less powerful
than the deterministic version of limited automata. In fact, there exist
deterministic context-free languages that are not accepted by any deterministic
strongly limited automaton.Comment: 20 pages. This article will appear in the Complexity Theory Column of
the September 2015 issue of SIGACT New
On Monotone Formulae with Restricted Depth
We prove a hierarchy theorem for the representation of monotone Boolean functions by monotone Boolean functions by monotone formulae with restricted depth. Specifically, we show that there are functions with Πk-formulae of size n for which every Σk-formula has size exp Ω(n1/(k-1)). A similar lower bound applies to concrete functions such as transitive closure and clique. We also show that any function with a formula of size n (and any depth) has a Σk-formula of size exp O(n1/(k-1)). Thus our hierarchy theorem is the best possible
Space bounds for processing contentless inputs
The space and time bounds of Turing machines which process contentless inputs, i.e., inputs of the form an are investigated. There is such a Turing machine which uses space bounded by log log n but not space bounded by any constant. Properties of this processor are given. The general properties of Turing machines processing contentless inputs are discussed. Any nontrivial processor can be transformed into a recognizer of a nonregular language in the same input alphabet and using exactly the same space. Finally, a theorem which establishes a hierarchy of contentless languages whose recognizers require at least log n space is given
Constructing Small Tree Grammars and Small Circuits for Formulas
It is shown that every tree of size n over a fixed set of sigma different ranked symbols can be decomposed into O(n/log_sigma(n)) = O((n * log(sigma))/ log(n)) many hierarchically defined pieces. Formally, such a hierarchical decomposition has the form of a straight-line linear context-free tree grammar of size O(n/log_sigma(n)), which can be used as a compressed representation of the input tree. This generalizes an analogous result for strings. Previous grammar-based tree compressors were not analyzed for the worst-case size of the computed grammar, except for the top dag of Bille et al., for which only the weaker upper bound of O(n/log^{0.19}(n)) for unranked and unlabelled trees has been derived. The main result is used to show that every arithmetical formula of size n, in which only m <= n different variables occur, can be transformed (in time O(n * log(n)) into an arithmetical circuit of size O((n * log(m))/log(n)) and depth O(log(n)). This refines a classical result of Brent, according to which an arithmetical formula of size n can be transformed into a logarithmic depth circuit of size O(n)
Time and tape complexity of pushdown automaton languages
An algorithm is presented which will determine whether any string w in Σ*, of length n, is contained in a language L ⊆ Σ* defined by a two-way nondeterministic pushdown automation. This algorithm requires time n3 when implemented on a random access computer. It requires n4 time and n2 tape when implemented on a multitape Turing machine.If the pushdown automaton is deterministic, the algorithm requires n2 time on a random access computer and n2 log n time on a multitape Turing machine
Tree-size bounded alternation
AbstractThe size of an accepting computation tree of an alternating Turing machine (ATM) is introduced as a complexity measure. We present a number of applications of tree-size to the study of more traditional complexity classes. Tree-size on ATMs is shown to closely correspond to time on nondeterministic TMs and on nondeterministic auxiliary pushdown automata. One application of the later is a useful new characterization of the class of languages log-space-reducible to context-free languages. Surprising relationships with parallel-time complexity are also demonstrated. ATM computations using at most space S(n) and tree-size Z(n) (simultaneously) can be simulated in alternating space S(n) and time S(n) · log Z(n) (simultaneously). Several well-known simulations, e.g., Savitch's theorem, are special cases of this result. It also leads to improved parallel complexity bounds for many problems in terms of both time and number of “processors.” As one example we show that context-free language recognition in time O(log2 n) is possible on several parallel models. Further, this bound is achievable with only a polynomial number of processors, in contrast to all previously known sub-linear time CFL recognizers
Nondeterministic one-tape off-line Turing machines and their time complexity
In this paper we consider the time and the crossing sequence complexities of
one-tape off-line Turing machines. We show that the running time of each
nondeterministic machine accepting a nonregular language must grow at least as
n\log n, in the case all accepting computations are considered (accept
measure). We also prove that the maximal length of the crossing sequences used
in accepting computations must grow at least as \log n. On the other hand, it
is known that if the time is measured considering, for each accepted string,
only the faster accepting computation (weak measure), then there exist
nonregular languages accepted in linear time. We prove that under this measure,
each accepting computation should exhibit a crossing sequence of length at
least \log\log n. We also present efficient implementations of algorithms
accepting some unary nonregular languages.Comment: 18 pages. The paper will appear on the Journal of Automata, Languages
and Combinatoric
How to Walk Your Dog in the Mountains with No Magic Leash
We describe a -approximation algorithm for computing the
homotopic \Frechet distance between two polygonal curves that lie on the
boundary of a triangulated topological disk. Prior to this work, algorithms
were known only for curves on the Euclidean plane with polygonal obstacles.
A key technical ingredient in our analysis is a -approximation
algorithm for computing the minimum height of a homotopy between two curves. No
algorithms were previously known for approximating this parameter.
Surprisingly, it is not even known if computing either the homotopic \Frechet
distance, or the minimum height of a homotopy, is in NP