Pumping lemma for regular language
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This document discusses the properties of regular languages including closure properties like union, intersection, concatenation and complementation as well as star and reversal operations. It also covers decision properties such as membership, emptiness, finiteness, infiniteness and equivalence that can be tested for regular languages.
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This document summarizes a lecture on automata theory, specifically discussing non-regular languages, the pumping lemma, and regular expressions. It introduces the language B={0n1n | n ≥ 0} as a non-regular language that cannot be recognized by a DFA. It then proves the pumping lemma theorem and uses it to show that B and the language of strings with equal numbers of 0s and 1s are non-regular. Finally, it defines regular expressions as a way to describe languages and provides examples of regular expressions and their meanings.
Pumping lemma
1) The pumping lemma can be used to prove that a language is not regular or context-free.
2) For regular languages, the pumping lemma states that for any string x longer than n, x can be broken into uvw such that pumping v any number of times results in strings still in the language.
3) For context-free languages, the pumping lemma similarly states that any string x longer than n can be broken into five parts such that pumping the second and fourth parts any number of times results in strings still in the language.
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Pumping lemma
The document discusses the pumping lemma and how it can be used to prove that certain languages are not regular. It provides examples of applying the pumping lemma to the languages L1={0n1n:n≥0}, L2={0n10n1:n≥0}, L3={x: x has the same number of 01s and 10s}, L4={x: x has the same number of 0s and 1s}, and L5={1p: p is prime}. In each case, it shows how to derive a contradiction by "pumping" a string in the language to produce a string not in the language, proving the original language cannot be regular.
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Pumping lemma Theory Of Automata
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This document summarizes a lecture on automata theory, specifically discussing non-regular languages, the pumping lemma, and regular expressions. It introduces the language B={0n1n | n ≥ 0} as a non-regular language that cannot be recognized by a DFA. It then proves the pumping lemma theorem and uses it to show that B and the language of strings with equal numbers of 0s and 1s are non-regular. Finally, it defines regular expressions as a way to describe languages and provides examples of regular expressions and their meanings.
Pumping lemma
1) The pumping lemma can be used to prove that a language is not regular or context-free.
2) For regular languages, the pumping lemma states that for any string x longer than n, x can be broken into uvw such that pumping v any number of times results in strings still in the language.
3) For context-free languages, the pumping lemma similarly states that any string x longer than n can be broken into five parts such that pumping the second and fourth parts any number of times results in strings still in the language.
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The document discusses the pumping lemma and how it can be used to prove that certain languages are not regular. It provides examples of applying the pumping lemma to the languages L1={0n1n:n≥0}, L2={0n10n1:n≥0}, L3={x: x has the same number of 01s and 10s}, L4={x: x has the same number of 0s and 1s}, and L5={1p: p is prime}. In each case, it shows how to derive a contradiction by "pumping" a string in the language to produce a string not in the language, proving the original language cannot be regular.
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This document provides an introduction to automata theory and finite automata. It defines an automaton as an abstract computing device that follows a predetermined sequence of operations automatically. A finite automaton has a finite number of states and can be deterministic or non-deterministic. The document outlines the formal definitions and representations of finite automata. It also discusses related concepts like alphabets, strings, languages, and the conversions between non-deterministic and deterministic finite automata. Methods for minimizing deterministic finite automata using Myhill-Nerode theorem and equivalence theorem are also introduced.
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This document provides an overview of deterministic finite automata (DFA) through examples and practice problems. It begins with defining the components of a DFA, including states, alphabet, transition function, start state, and accepting states. An example DFA is given to recognize strings ending in "00". Additional practice problems involve drawing minimal DFAs, determining the minimum number of states for a language, and completing partially drawn DFAs. The document aims to help students learn and practice working with DFA models.
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FellowBuddy.com is an innovative platform that brings students together to share notes, exam papers, study guides, project reports and presentation for upcoming exams.
We connect Students who have an understanding of course material with Students who need help.
Benefits:-
# Students can catch up on notes they missed because of an absence.
# Underachievers can find peer developed notes that break down lecture and study material in a way that they can understand
# Students can earn better grades, save time and study effectively
Our Vision & Mission – Simplifying Students Life
Our Belief – “The great breakthrough in your life comes when you realize it, that you can learn anything you need to learn; to accomplish any goal that you have set for yourself. This means there are no limits on what you can be, have or do.”
Like Us - https://www.facebook.com/FellowBuddycom
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This document provides an overview of the pumping lemma for regular languages. It introduces the pumping lemma, which states that if a language L is regular, then there exists a pumping length p such that any string s in L of length at least p can be broken down into sections x, y, z such that xyiz is also in L for all i >= 0. The document then analyzes several example languages to determine whether they are regular or not based on whether the pumping lemma applies to them. It explores languages with patterns like an, anbn, and w1wR.
2. context free langauages
The document discusses context-free languages and pushdown automata. It defines context-free grammars and languages, and provides examples of grammars and the strings they generate. It also defines pushdown automata formally as a 6-tuple with states, input alphabet, stack alphabet, transition function, start state, and accept states. Pushdown automata are similar to finite automata but have an additional stack which allows them to recognize some non-regular languages.
Push Down Automata (PDA)
This document contains information about pushdown automata (PDA) and examples of using PDA to represent certain formal languages. It provides examples of drawing PDA for languages of strings over an alphabet where the number of as equals the number of bs and for strings that represent legal arithmetic expressions over an alphabet of integer, operators and parentheses. Links to external resources on the topic are also included.
Turing machines
Turing Machines are a simple mathematical model of a general purpose computer invented by Alan Turing in 1936. A Turing Machine consists of an infinite tape divided into cells, a head that reads and writes symbols on the tape, a finite set of states, and transition rules determining the behavior of the machine. The machine operates by reading a symbol on the tape, updating the symbol according to its transition rules, moving the head left or right, and transitioning to a new state. Turing Machines can simulate any algorithm and are capable of performing any calculation that can be performed by any computing machine.
Context free grammars
Context free grammars (CFGs) are formal systems that describe the structure of languages. A CFG consists of variables, terminals, production rules, and a start variable. Production rules take the form of a single variable producing a string of terminals and/or variables. CFGs can capture the recursive structure of natural languages while ignoring agreement and reference. They are used to define context-free languages and generate parse trees. Ambiguous grammars have sentences with multiple parse trees, and disambiguation aims to impose an ordering on derivations. While ambiguity cannot always be eliminated, simplifying and restricting grammars has theoretical and practical benefits.
Dfa vs nfa
The document describes the conversion of a nondeterministic finite automaton (NFA) to a deterministic finite automaton (DFA). It involves three steps: 1) The initial state of the DFA is a set containing the initial state of the NFA. 2) Transition functions for the DFA are determined by considering all possible transitions from states in the NFA. 3) A state in the DFA is marked as final if it contains any final states from the NFA. The procedure guarantees that the languages accepted by the original NFA and resulting DFA are equivalent. This proves that NFAs and DFAs have equal computational power in accepting regular languages.
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- 2. Properties of Regular Languages • Closure Properties – Union – Intersection – Concatenation – Complementation – Star – Reversal • Decision Properties – Membership – Emptiness – Finiteness – Infiniteness – Equivalence
- 18. Thank You!