Formal languages | Theoretical computer science | Automata (computation) | Trees (data structures)
In automata theory, a tree is a particular way of representing a tree structure as sequences of natural numbers. For example, each node of the tree is a word over set of natural numbers, which helps this definition to be used in automata theory. A tree is a set T ⊆ * such that if t.c ∈ T, with t ∈ * and c ∈ , then t ∈ T and t.c1 ∈ T for all 0 ≤ c1 < c. The elements of T are known as nodes, and the empty word ε is the (single) root of T. For every t ∈ T, the element t.c ∈ T is a successor of t in direction c. The number of successors of t is called its degree or arity, and represented as d(t). A node is a leaf if it has no successors. If every node of a tree has finitely many successors, then it is called a finitely, otherwise an infinitely branching tree. A path π is a subset of T such that ε ∈ π and for every t ∈ T, either t is a leaf or there exists a unique c ∈ such that t.c ∈ π. A path may be a finite or infinite set. If all paths of a tree are finite then the tree is called finite, otherwise infinite. A tree is called fully infinite if all its paths are infinite. Given an alphabet Σ, a Σ-labeled tree is a pair (T,V), where T is a tree and V: T → Σ maps each node of T to a symbol in Σ. A labeled tree formally defines a commonly used term tree structure. A set of labeled trees is called a tree language. A tree is called ordered if there is an order among the successors of each of its nodes. The above definition of tree naturally suggests an order among the successors, which can be used to make the tree ranked. In the case of ranked alphabets, an extra function Ar: Σ → is defined. This function associates a fixed arity to each symbol of the alphabet. In this case, each t ∈ T has to satisfy Ar(V(t)) = d(t). The trees that satisfy this property are called ranked trees. The trees that do not (necessarily) satisfy that property are called unranked. For example, the above definition is used in the definition of an infinite tree automaton. (Wikipedia).
Graph Theory: 36. Definition of a Tree
In this video I define a tree and a forest in graph theory. I discuss the difference between labelled trees and non-isomorphic trees. I also show why every tree must have at least two leaves. An introduction to Graph Theory by Dr. Sarada Herke. Related Videos: http://youtu.be/zxu0dL436gI
From playlist Graph Theory part-7
Intro to Tree Graphs | Trees in Graph Theory, Equivalent Definitions
What are trees in graph theory? Tree graphs are connected graphs with no cycles. We'll introduce them and some equivalent definitions, with of course examples of tree graphs in today's graph theory video lesson! Some equivalent definitions of tree graphs are as follows. A graph is a tree
From playlist Graph Theory
Tree Graphs - Intro to Algorithms
This video is part of an online course, Intro to Algorithms. Check out the course here: https://www.udacity.com/course/cs215.
From playlist Introduction to Algorithms
Graph Theory FAQs: 02. Graph Automorphisms
An automorphism of a graph G is an isomorphism between G and itself. The set of automorphisms of a graph forms a group under the operation of composition and is denoted Aut(G). The automorphisms of a graph describe the symmetries of the graph. We look at a few examples of graphs and det
From playlist Graph Theory FAQs
Cellular automata: emergence in not-so-complex systems
In this video we explore the concept of emergence through the lens of cellular automata. 00:00 Intro 02:13 Our model system, the cellular automaton 04:56 Visualizing automata through time 05:12 Types of automata, from periodicity to chaos 08:24 Looking for emergence in cellular automata
From playlist Summer of Math Exposition Youtube Videos
Automorphism groups and modular arithmetic
Jacob explains the concept of the automorphism group of a group, as well as how such groups give rise to useful properties of multiplication in modular arithmetic, including Fermat's Little Theorem.
From playlist Basics: Group Theory
Data structures: Introduction to Trees
See complete series on data structures here: http://www.youtube.com/playlist?list=PL2_aWCzGMAwI3W_JlcBbtYTwiQSsOTa6P In this lesson, we have described tree data structure as a logical model in computer science. We have briefly discussed tree as a non-linear hierarchical data structure, i
From playlist Data structures
Make a Tree - Intro to Algorithms
This video is part of an online course, Intro to Algorithms. Check out the course here: https://www.udacity.com/course/cs215.
From playlist Introduction to Algorithms
Diego Figueira: Semistructured data, Logic, and Automata – lecture 2
Semistructured data is an umbrella term encompassing data models which are not logically organized in tables (i.e., the relational data model) but rather in hierarchical structures using markers such as tags to separate semantic elements and data fields in a ‘self-describing’ way. In this
From playlist Logic and Foundations
Introduction to Trees and Properties of Trees
This video introduces defines and gives the properties of tree graphs. mathispower4u.com
From playlist Graph Theory (Discrete Math)
Introduction to a Unified Model of Cellular Automata
This is an introduction to a unified model of Cellular Automata in which a rule is represented not by a single function but by a vector of functions we call genes. These functions can be ordered so that they maintain the same order regardless of the rule space where they are realized. This
From playlist Wolfram Technology Conference 2022
What We've Learned from NKS Chapter 3: The World of Simple Programs
In this episode of "What We've Learned from NKS", Stephen Wolfram is counting down to the 20th anniversary of A New Kind of Science with [another] chapter retrospective. If you'd like to contribute to the discussion in future episodes, you can participate through this YouTube channel or th
From playlist Science and Research Livestreams
Thomas Colcombet : Algebra vs Logic over (generalised) words
CONFERENCE Recording during the thematic meeting : « Discrete mathematics and logic: between mathematics and the computer science » the January 17, 2023 at the Centre International de Rencontres Mathématiques (Marseille, France) Filmmaker: Jean Petit Find this video and other talks give
From playlist Logic and Foundations
Diego Figueira: Semistructured data, Logic, and Automata – lecture 1
Semistructured data is an umbrella term encompassing data models which are not logically organized in tables (i.e., the relational data model) but rather in hierarchical structures using markers such as tags to separate semantic elements and data fields in a ‘self-describing’ way. In this
From playlist Logic and Foundations
4. Pushdown Automata, Conversion of CFG to PDA and Reverse Conversion
MIT 18.404J Theory of Computation, Fall 2020 Instructor: Michael Sipser View the complete course: https://ocw.mit.edu/18-404JF20 YouTube Playlist: https://www.youtube.com/playlist?list=PLUl4u3cNGP60_JNv2MmK3wkOt9syvfQWY Quickly reviewed last lecture. Defined context free grammars (CFGs) a
From playlist MIT 18.404J Theory of Computation, Fall 2020
Bernhard Reinke: Iterated monodromy groups and transcendental dynamics
HYBRID EVENT Recorded during the meeting "Advancing Bridges in Complex Dynamics" the September 21, 2021 by the Centre International de Rencontres Mathématiques (Marseille, France) Filmmaker: Guillaume Hennenfent Find this video and other talks given by worldwide mathematicians on CIRM
From playlist Dynamical Systems and Ordinary Differential Equations
Find this video and other talks given by worldwide mathematicians on CIRM's Audiovisual Mathematics Library: http://library.cirm-math.fr. And discover all its functionalities: - Chapter markers and keywords to watch the parts of your choice in the video - Videos enriched with abstracts, b
From playlist Mathematical Aspects of Computer Science
Learning Automata with Hankel Matrices - Borja Balle, Amazon Research Cambridge
The Hankel matrix is a fundamental tool in the theory of weighted automata. In this talk we will describe a general framework for learning automata with Hankel matrices. Our framework provides a unified view of many classical and recent algorithms for learning automata under different lear
From playlist Logic and learning workshop
What We've Learned from NKS Chapter 5: Two Dimensions and Beyond
In this episode of "What We've Learned from NKS", Stephen Wolfram is counting down to the 20th anniversary of A New Kind of Science with [another] chapter retrospective. If you'd like to contribute to the discussion in future episodes, you can participate through this YouTube channel or th
From playlist Science and Research Livestreams