Automata_and_Complexity_Theommmmmmmry-RevisionforExitExam.ppt

Automata and
Complexity Theory

 Alphabets and Strings: An alphabet is a finite set of symbols, and a string is a
finite sequence of symbols from an alphabet.
 Languages and Grammars: A language is a set of strings, and a grammar is a set
of rules that generate a language.
 Automata:
 Finite Automata: A computational model that recognizes regular languages. It
consists of a set of states, transitions between states, and an input alphabet.
 Deterministic Finite Automata (DFA): A finite automaton where each state and input
symbol uniquely determines the next state.
 Non-deterministic Finite Automata (NFA): A finite automaton where the next state
may not be uniquely determined by the current state and input symbol.
Introduction

 Regular Expressions: A formal notation for describing regular languages.
 Connection between Regular Expression and Regular Languages: Regular expressions
and regular grammars generate the same class of languages, known as regular languages.
 Regular Grammar: A type of formal grammar that generates regular languages.
 Pumping Lemma and Non-Regular Language Grammars: The pumping lemma is a
tool used to prove that certain languages are not regular.
Regular Expression and Regular languages

 Context-Free Languages: A class of formal languages that can be generated by context-
free grammars.
 Parsing and Ambiguity: Parsing is the process of determining the syntactic structure of
a string according to the rules of a grammar. Ambiguity arises when a string has
multiple valid parse trees.
 Sentential Forms and Derivation Tree or Parse Tree: Sentential forms are intermediate
steps in deriving a string from a grammar's start symbol. Derivation trees or parse trees
represent the hierarchical structure of how a string is derived from a grammar.
Context-Free Languages

 Simplification of Context-Free Grammars:
 Methods for Transforming Grammars: Techniques to transform a grammar into an
equivalent but simpler form.
 Chomsky's Hierarchy of Grammars: A classification of formal grammars based on
the restrictions placed on the production rules.
…Context-Free Languages

PDA: A computational model that recognizes context-free languages,
with a stack to store and retrieve information.
Pushdown Automata and Context-Free Languages: Pushdown
automata and context-free grammars recognize the same class of
languages.
Deterministic Pushdown Automata: A special case of pushdown
automata where the next state and stack operation are uniquely
determined.
Deterministic Context-Free Languages: The class of languages
recognized by deterministic pushdown automata.
Push Down Automata

A Turing machine is a mathematical model of computation that
consists of a finite-state control, a read/write head, and an infinite
tape divided into cells.
The tape represents the input, output, and working memory of the
machine.
The finite-state control determines the next action based on the current
state and the symbol under the read/write head.
Turing Machine (TM)

Turing Decidable and Turing Acceptable Languages:
A language is Turing-decidable (or recursive) if there exists a Turing machine
that halts and accepts all strings in the language and rejects all strings not in
the language.
A language is Turing-acceptable (or recursively enumerable) if there exists a
Turing machine that halts and accepts all strings in the language, but may not
halt on strings not in the language.
Undecidable Problems:
•Some problems are not Turing-decidable, meaning there is no Turing machine
that can solve them. These are called undecidable problems.
•The most famous undecidable problem is the Halting Problem,
…Turing Machine (TM)

Turing machines are the fundamental model of computation in
computability theory, and they define the class of Turing-decidable
(recursive) and Turing-acceptable (recursively enumerable)
languages.
Computability

The relationship among formal languages, formal grammars, and automata is a fundamental
topic in computer science and the theory of computation.
Formal Languages:
A formal language is a set of strings (sequences of symbols) that are defined by a set of
rules.
Formal languages can be classified into different types, such as regular languages, context-free
languages, and context-sensitive languages, based on the complexity of their underlying
structures.
Acquire insights into the relationship among
formal languages, formal grammars, and
automata.

Formal Grammars:
A formal grammar is a set of rules that define the syntax of a formal language.
Formal grammars can be classified into different types, such as regular grammars, context-
free grammars, and context-sensitive grammars, based on the complexity of the rules they use
to generate the language.
…Acquire insights into the relationship among
automata.

Automata:
Automata are mathematical models of computation that can recognize and accept formal
languages.
The different types of automata, such as finite-state automata, pushdown automata, and
Turing machines, correspond to the different types of formal languages and grammars.
In summary,
Formal languages, formal grammars, and automata are interconnected concepts in the theory of
computation. Formal grammars provide the rules for generating formal languages, and
automata are the computational models that can recognize and accept those formal languages.
…Acquire insights into the relationship among
automata.

1.Regular Languages:
1. Formal Grammar: Regular Grammars
2. Automata: Finite-State Automata (FSA)
3. Characteristics: Regular languages are the simplest and most restrictive class of formal
languages. They can be generated by regular grammars and recognized by finite-state
automata.
2.Context-Free Languages:
1. Formal Grammar: Context-Free Grammars (CFG)
2. Automata: Pushdown Automata (PDA)
3. Characteristics: Context-free languages are more expressive than regular languages. They
can be generated by context-free grammars and recognized by pushdown automata.
Identify different formal language classes and
their relationships

3. Context-Sensitive Languages:
1. Formal Grammar: Context-Sensitive Grammars (CSG)
2. Automata: Linear-Bounded Automata (LBA)
3. Characteristics: Context-sensitive languages are more expressive than context-free
languages. They can be generated by context-sensitive grammars and recognized by linear-
bounded automata.
4. Recursively Enumerable Languages:
1. Formal Grammar: Type-0 Grammars (Unrestricted Grammars)
2. Automata: Turing Machines (TM)
3. Characteristics: Recursively enumerable languages are the most expressive class of formal
languages. They can be generated by unrestricted grammars (type-0 grammars) and
recognized by Turing machines.
…Identify different formal language classes and
their relationships

•The relationships among these formal language classes can be summarized as
follows:
– Regular Languages Context-Free Languages Context-Sensitive
⊆ ⊆
Languages Recursively Enumerable Languages
⊆
– This means that every regular language is also a context-free language, every
context-free language is also a context-sensitive language, and every context-
sensitive language is also a recursively enumerable language.
…Identify different formal language classes and
their relationships

Refer Your Note!
Design grammars and recognizer for different
formal languages

Understand Complexity classes, P/NP/PSPACE,
reductions, hardness, completeness, hierarchy,
relationships between complexity classes. complexity

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