200,535 research outputs found
Boolean Circuit Complexity of Regular Languages
In this paper we define a new descriptional complexity measure for
Deterministic Finite Automata, BC-complexity, as an alternative to the state
complexity. We prove that for two DFAs with the same number of states
BC-complexity can differ exponentially. In some cases minimization of DFA can
lead to an exponential increase in BC-complexity, on the other hand
BC-complexity of DFAs with a large state space which are obtained by some
standard constructions (determinization of NFA, language operations), is
reasonably small. But our main result is the analogue of the "Shannon effect"
for finite automata: almost all DFAs with a fixed number of states have
BC-complexity that is close to the maximum.Comment: In Proceedings AFL 2014, arXiv:1405.527
Digraph Complexity Measures and Applications in Formal Language Theory
We investigate structural complexity measures on digraphs, in particular the
cycle rank. This concept is intimately related to a classical topic in formal
language theory, namely the star height of regular languages. We explore this
connection, and obtain several new algorithmic insights regarding both cycle
rank and star height. Among other results, we show that computing the cycle
rank is NP-complete, even for sparse digraphs of maximum outdegree 2.
Notwithstanding, we provide both a polynomial-time approximation algorithm and
an exponential-time exact algorithm for this problem. The former algorithm
yields an O((log n)^(3/2))- approximation in polynomial time, whereas the
latter yields the optimum solution, and runs in time and space O*(1.9129^n) on
digraphs of maximum outdegree at most two. Regarding the star height problem,
we identify a subclass of the regular languages for which we can precisely
determine the computational complexity of the star height problem. Namely, the
star height problem for bideterministic languages is NP-complete, and this
holds already for binary alphabets. Then we translate the algorithmic results
concerning cycle rank to the bideterministic star height problem, thus giving a
polynomial-time approximation as well as a reasonably fast exact exponential
algorithm for bideterministic star height.Comment: 19 pages, 1 figur
Bounded Languages Meet Cellular Automata with Sparse Communication
Cellular automata are one-dimensional arrays of interconnected interacting
finite automata. We investigate one of the weakest classes, the real-time
one-way cellular automata, and impose an additional restriction on their
inter-cell communication by bounding the number of allowed uses of the links
between cells. Moreover, we consider the devices as acceptors for bounded
languages in order to explore the borderline at which non-trivial decidability
problems of cellular automata classes become decidable. It is shown that even
devices with drastically reduced communication, that is, each two neighboring
cells may communicate only constantly often, accept bounded languages that are
not semilinear. If the number of communications is at least logarithmic in the
length of the input, several problems are undecidable. The same result is
obtained for classes where the total number of communications during a
computation is linearly bounded
Quantum Arthur-Merlin Games
This paper studies quantum Arthur-Merlin games, which are Arthur-Merlin games
in which Arthur and Merlin can perform quantum computations and Merlin can send
Arthur quantum information. As in the classical case, messages from Arthur to
Merlin are restricted to be strings of uniformly generated random bits. It is
proved that for one-message quantum Arthur-Merlin games, which correspond to
the complexity class QMA, completeness and soundness errors can be reduced
exponentially without increasing the length of Merlin's message. Previous
constructions for reducing error required a polynomial increase in the length
of Merlin's message. Applications of this fact include a proof that logarithmic
length quantum certificates yield no increase in power over BQP and a simple
proof that QMA is contained in PP. Other facts that are proved include the
equivalence of three (or more) message quantum Arthur-Merlin games with
ordinary quantum interactive proof systems and some basic properties concerning
two-message quantum Arthur-Merlin games.Comment: 22 page
On two-way communication in cellular automata with a fixed number of cells
The effect of adding two-way communication to k cells one-way cellular automata (kC-OCAs) on their size of description is studied. kC-OCAs are a parallel model for the regular languages that consists of an array of k identical deterministic finite automata (DFAs), called cells, operating in parallel. Each cell gets information from its right neighbor only. In this paper, two models with different amounts of two-way communication are investigated. Both models always achieve quadratic savings when compared to DFAs. When compared to a one-way cellular model, the result is that minimum two-way communication can achieve at most quadratic savings whereas maximum two-way communication may provide savings bounded by a polynomial of degree k
Uniform Random Sampling of Traces in Very Large Models
This paper presents some first results on how to perform uniform random walks
(where every trace has the same probability to occur) in very large models. The
models considered here are described in a succinct way as a set of
communicating reactive modules. The method relies upon techniques for counting
and drawing uniformly at random words in regular languages. Each module is
considered as an automaton defining such a language. It is shown how it is
possible to combine local uniform drawings of traces, and to obtain some global
uniform random sampling, without construction of the global model
- …