Part VIII. These languages can be defined using regular expressions. Not every language is regular. A powerful technique, known as Pumping Lemma can be used to show that certain languages are not regular. A Simple Definition Let L be a language and w be a string in the language.
Break w into three parts. Then xy i z must also be a string in the language. Then L is a context free language. For example. This language is regular. Consider one string in this language, aaaa. Break w into three parts, x, y and z. This language is not regular. Since some of the strings of the form xy i zare not in the above language, the language a n b n is not regular.
Proof Let L is regular. Then a DFA exists for L. Let that DFA has n states. That is x takes us to P i once, y takes us from P i back to P i since P i is also P jand z is balance of w. This is shown in the following diagram. Thus xz is accepted by the automation.This Free Python Video Tutorial course has been designed to ensure it will help a beginner Learn Python and become a master programmer of Python.
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These cookies will be stored in your browser only with your consent. You also have the option to opt-out of these cookies.We will discuss the important key points useful for GATE exams in summarized form. For details you may refer this. Finite Automata : It is used to recognize patterns of specific type input.
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Identities of Regular Expression :. Moore Machine : Moore machines are finite state machines with output value and its output depends only on present state. Mealy Machine : Mealy machines are also finite state machines with output value and its output depends on present state and current input symbol.
Push Down Automata : Pushdown Automata has extra memory called stack which gives more power than Finite automata. It is used to recognize context free languages.
Turing Machine : Turing machine has infinite size tape and it is used to accept Recursive Enumerable Languages. Deterministic and Non-Deterministic Turing Machines: In deterministic turing machine, there is only one move from every state on every input symbol but in Non-Deterministic turing machine, there can be more than one move from one state for an input symbol.
Chomsky Classification of Languages:. Relationship between these can be represented as:. Decidable and Undecidable Problems:. A language is Decidable or Recursive if a Turing machine can be constructed which accepts the strings which are part of language and rejects others. A language is Semi — Decidable or Recursive Enumerable if a turing machine can be constructed which accepts the strings which are part of language and it may loop forever for strings which are not part of language.
Countability :. Attention reader! Writing code in comment? Please use ide. See Last Minute Notes on all subjects here. Load Comments.There are two Pumping Lemmas, which are defined for 1. Regular Languages, and 2. Pumping Lemma is used as a proof for irregularity of a language. Thus, if a language is regular, it always satisfies pumping lemma.
If there exists at least one string made from pumping which is not in L, then L is surely not regular. The opposite of this may not always be true. That is, if Pumping Lemma holds, it does not mean that the language is regular.
Let us assume that L is regular, then by Pumping Lemma the above given rules follow. So, by Pumping Lemma, there exists u, v, w such that 1 — 3 hold. We show that for all u, v, w, 1 — 3 does not hold.
For any language L, we break its strings into five parts and pump second and fourth substring. Pumping Lemma, here also, is used as a tool to prove that a language is not CFL. Because, if any one string does not satisfy its conditions, then the language is not CFL. For above example, 0 n 1 n is CFL, as any string can be the result of pumping at two places, one for 0 and other for 1. Let us assume that L is Context-free, then by Pumping Lemma, the above given rules follow.
So, by Pumping Lemma, there exists u, v, w, x, y such that 1 — 3 hold. We show that for all u, v, w, x, y 1 — 3 do not hold. Thus, we have two cases to consider.
Thus L is not context-free. Source : John E. Hopcroft, Rajeev Motwani, Jeffrey D. Ullman Introduction to Automata Theory, Languages, and Computation. This article has been contributed by Nirupam Singh. Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above. Attention reader! Writing code in comment? Please use ide.
Regular Expressions Regular Expressions are used to denote regular languages. An expression is regular if:. Regular Languages : A language is regular if it can be expressed in terms of regular expression. L2 will also be regular. Complement of a language can be found by subtracting strings which are in L G from all possible strings.
D The set of all strings that begin and end with either 0 or 1. Solution : Option A says that it must have substring But is also a part of language but it does not contain 00 as substring. So it is not correct option. Option C says that it must contain atleast two 0. In regular expression, two 0 are present. So this is correct option. Option D says that it contains all strings that begin and end with either 0 or 1.
But it can generate strings which start with 0 and end with 1 or vice versa as well. So it is not correct. Question 2 : Which of the following languages is generated by given grammar? Solution : Option A says that it will have 0 or more a followed by 0 or more b. So A is not correct. Option B says that it will have equal no. So B is not correct. But as shown in option Aba is also part of language.
So C is not correct. So D is correct. So they are equivalent. So they are not equivalent. Solution: Option C stating both both options A and B is the correct regular expression for the stated question.
This article has been contributed by Sonal Tuteja. Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above.In computer sciencethe expressive power also called expressiveness or expressivity of a language is the breadth of ideas that can be represented and communicated in that language.
The more expressive a language is, the greater the variety and quantity of ideas it can be used to represent. So OWL2 EL trades some expressive power for more efficient reasoning processing of the knowledge representation language. The term expressive power may be used with a range of meaning. It may mean a measure of the ideas expressible in that language: . The first sense dominates in areas of mathematics and logic that deal with the formal description of languages and their meaning, such as formal language theorymathematical logic and process algebra.
In informal discussions, the term often refers to the second sense, or to both. This is often the case when discussing programming languages. The notion of expressive power is always relative to a particular kind of thing that the language in question can describe, and the term is normally used when comparing languages that describe the same kind of things, or at least comparable kinds of things.
The design of languages and formalisms involves a trade-off between expressive power and analyzability. The more a formalism can express, the harder it becomes to understand what instances of the formalism say. Decision problems become harder to answer or completely undecidable. Formal language theory mostly studies formalisms to describe sets of strings, such as context-free grammars and regular expressions.
Each instance of a formalism, e. In this context, the expressive power of a formalism is the set of sets of strings its instances describe, and comparing expressive power is a matter of comparing these sets. An important yardstick for describing the relative expressive power of formalisms in this area is the Chomsky hierarchy. It says, for instance, that regular expressionsnondeterministic finite automatons and regular grammars have equal expressive power, while that of context-free grammars is greater; what this means is that the sets of sets of strings described by the first three formalisms are equal, and a proper subset of the set of sets of strings described by context-free grammars.
In this area, the cost of expressive power is a central topic of study. It is known, for instance, that deciding whether two arbitrary regular expressions describe the same set of strings is hard, while doing the same for arbitrary context-free grammars is completely impossible.
However, it can still be efficiently decided whether any given string is in the set. For more expressive formalisms, this problem can be harder, or even undecidable. For a Turing complete formalism, such as arbitrary formal grammarsnot only this problem, but every nontrivial property regarding the set of strings they describe is undecidable, a fact known as Rice's Theorem.
There are some results on conciseness as well; for instance, nondeterministic state machines and regular grammars are more concise than regular expressions, in the sense that the latter can be translated to the former without a blowup in size i.regular expressions in theory of automata - TOC - Lec-40 - Bhanu Priya
Similar considerations apply to formalisms that describe not sets of strings, but sets of trees e. XML schema languagesof graphs, or other structures. Database theory is concerned, among other things, with database queriese. In the predominant relational database paradigm, the contents of a database are described as a finite set of finite mathematical relations; Boolean queries, that always yield true or falseare formulated in first-order logic.
It turns out that first-order logic is lacking in expressive power: it cannot express certain types of Boolean queries, e. Consequently, a literature sprang up in which many query languages and language constructs were compared on expressive power and efficiency, e. Similar considerations apply for query languages on other types of data, e.How to write an Interview Experience? As the placement season is back so are we to help you ace the interview.
There were 3 coding questions to be solved in 90 minutes. How to Practice for the Technical Rounds in Interview? While going to an interview, there can be anything that the interviewer can ask. Hence L has some non, how to write an Interview Experience? Given an array arr of N integers. Given a 2-D array of N rows and M columns and an integer K. Human activity recognition using smartphone sensors like accelerometer is one of the hectic topics of research.
HAR is one of the time series classification problem. It was proposed by researchers at Google Research in Automata Theory is a branch of computer science that deals with designing abstract selfpropelled computing devices that follow a predetermined sequence of operations automatically. An automaton with a finite number of states is called a Finite Automaton. This is a brief and concise tutorial that introduces the fundamental concepts of Finite Automata, Regular Languages, and Pushdown Automata before moving onto Turing machines and Decidability.
Audience This tutorial has been prepared for students pursuing a degree in any information technology or computer science related field. It attempts to help students grasp the essential concepts involved in automata theory. Which regular expression best describes the language accepted by toc non, standard books are recommended but ofcourse they are not the absolute necessity! The atomic fetch, hence either P0 can execute or Gate can execute.
Let us assume that Geeksforgeeks executes, this comment has been removed by the author. Language accepted by NTM, zero value has been returned. Prerequisites This tutorial has a good balance between theory and mathematical rigor. The readers are expected to have a basic understanding of discrete mathematical structures.
Page Not Found Probably, your request is incorrect. A linear bounded automaton is a multi-track non-deterministic Turing machine with a tape of some bounded finite length. The computation is restricted to the constant bounded area.
The input alphabet contains two special symbols which serve as left end markers and right end markers which mean the transitions neither move to the left of the left end marker nor to the right of the right end marker of the tape.