256,146 research outputs found
Formal Languages in Dynamical Systems
We treat here the interrelation between formal languages and those dynamical
systems that can be described by cellular automata (CA). There is a well-known
injective map which identifies any CA-invariant subshift with a central formal
language. However, in the special case of a symbolic dynamics, i.e. where the
CA is just the shift map, one gets a stronger result: the identification map
can be extended to a functor between the categories of symbolic dynamics and
formal languages. This functor additionally maps topological conjugacies
between subshifts to empty-string-limited generalized sequential machines
between languages. If the periodic points form a dense set, a case which arises
in a commonly used notion of chaotic dynamics, then an even more natural map to
assign a formal language to a subshift is offered. This map extends to a
functor, too. The Chomsky hierarchy measuring the complexity of formal
languages can be transferred via either of these functors from formal languages
to symbolic dynamics and proves to be a conjugacy invariant there. In this way
it acquires a dynamical meaning. After reviewing some results of the complexity
of CA-invariant subshifts, special attention is given to a new kind of
invariant subshift: the trapped set, which originates from the theory of
chaotic scattering and for which one can study complexity transitions.Comment: 23 pages, LaTe
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Information-theoretic Characterization of the Sub-regular Hierarchy
Our goal is to link two different formal notions of complexity: the complexity classes defined by Formal Language Theory (FLT)—in particular, the Sub-regular Hierarchy (Rogers et al., 2013; Lai, 2015; Heinz, 2018)—and Statistical Com- plexity Theory (Feldman and Crutchfield, 1998; Crutchfield and Marzen, 2015). The motivation for exploring this connection is that factors involving memory resources have been hypothesized to explain why phonological processes seem to inhabit the Sub-regular Hierarchy, and Statistical Complexity Theory gives an information-theoretic characterization of memory use. It is currently not known whether statistical complexity and FLT define equivalent complexity classes, or whether statistical complexity cross-cuts the usual FLT hierarchies. Our work begins to bridge the gap between FLT and Information Theory by presenting characterizations of certain Sub-regular languages in terms of statistical complexit
Non-Deterministic Communication Complexity of Regular Languages
In this thesis, we study the place of regular languages within the
communication complexity setting. In particular, we are interested in the
non-deterministic communication complexity of regular languages.
We show that a regular language has either O(1) or Omega(log n)
non-deterministic complexity. We obtain several linear lower bound results
which cover a wide range of regular languages having linear non-deterministic
complexity. These lower bound results also imply a result in semigroup theory:
we obtain sufficient conditions for not being in the positive variety Pol(Com).
To obtain our results, we use algebraic techniques. In the study of regular
languages, the algebraic point of view pioneered by Eilenberg (\cite{Eil74})
has led to many interesting results. Viewing a semigroup as a computational
device that recognizes languages has proven to be prolific from both semigroup
theory and formal languages perspectives. In this thesis, we provide further
instances of such mutualism.Comment: Master's thesis, 93 page
State Complexity of Combined Operations on Finite Languages
State complexity is a descriptive complexity measure for regular languages. It is a fundamental topic in automata and formal language theory. The state complexity of a regular language is the number of states in the minimal complete deterministic finite automaton accepting the language. During the last few decades, many publications have focused and studied the state complexity of many individual as well as combined operations on regular languages. Also, the state complexity of some basic operations on finite languages has been studied. But until now there has been no study on the state complexity of combined operations on finite languages.
In this thesis, we will first study the state complexity of the combined operation, star of union, on finite languages and give an exact bound. Then we will investigate the state complexity of star of catenation and show its approximation with a good ratio bound and finally, we will prove an upper bound for star of intersection
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
Adaptive Intelligent Tutoring System for learning Computer Theory
In this paper, we present an intelligent tutoring system developed to help students in learning Computer Theory. The Intelligent tutoring system was built using ITSB authoring tool. The system helps students to learn finite automata, pushdown automata, Turing machines and examines the relationship between these automata and formal languages, deterministic and nondeterministic machines, regular expressions, context free grammars, undecidability, and complexity. During the process the intelligent tutoring system gives assistance and feedback of many types in an intelligent manner according to the behavior of the student. An evaluation of the intelligent tutoring system has revealed reasonably acceptable results in terms of its usability and learning abilities are concerned
Advanced Topics on State Complexity of Combined Operations
State complexity is a fundamental topic in formal languages and automata theory. The study of state complexity is also strongly motivated by applications of finite automata in software engineering, programming languages, natural language and speech processing and other practical areas. Since many of these applications use automata of large sizes, it is important to know the number of states of the automata. In this thesis, we firstly discuss the state complexities of individual operations on regular languages, including union, intersection, star, catenation, reversal and so on. The state complexity of an operation on unary languages is usually different from that of the same operation on languages over a larger alphabet. Both kinds of state complexities are reviewed in the thesis. Secondly, we study the exact state complexities of twelve combined operations on regular languages. The state complexities of most of these combined operations are not equal to the compositions of the state complexities of the individual operations which make up these combined operations. We also explore the reason for this difference. Finally, we introduce the concept of estimation and approximation of state complexity. We show close estimates and approximations of the state complexities of six combined operations on regular languages which are good enough to use in practice
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