Σ1 = Σ where xy denotes the concatenation of x and y If Σ = {01} then Σ2 = {00011011} If Σ = {abc z} then Σ3 = {act bad cat den } Note that fre aat lkj ∈ Σ3 Empty string λ is the string containing no symbols λ is not the blank symbol and Σ o = { λ} λ cannot be a symbol in an alphabet More Powers
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In Discrete Maths you inductively defined subsets of (languages) S using axioms and rules Below are some example axioms and rules for generating a language L over the alphabet = fa bg S that contains strings of an a followed by zero or more b’s i e L = fa ab abb abbb g Axioms Axioms specify elements of S that exist in L
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phrase-structure grammar G = (V; T; S; P ) consists of a vocabulary V a subset T of V con-sisting of terminal symbols a start symbol S from V and a finite set of productions P The set T is denoted by N Elements of N are called nonterminal symbols Every production in P must contain at least one nonterminal on its left side Derivability
What is the language of all strings of even length?
Let A = {aa,ab,ba,bb}. Then is the language of all strings of even length. An atomic language is a language that contains only one string, and this string has length 1. {a} For short we denote such a language simply by a Every language that contains only one string can be represented as a concatenation of atomic languages.
Which alphabet is a finite set?
An alphabet is any finite set. Σ Its elements are called symbols or letters {0,1} a binary alphabet {0,1,2,3,4,5,6,7,8,9} {a,b,...,x,y,z} Latin alphabet {а, б, ..., э, ю, я} Cyrillic ..... the dancing men alphabet Strings that are obtained by concatenation of the same number of symbols are grouped into powers of the alphabet.
What are examples of strings over a language over the alphabet?
If = fa, bg then e, ba, bab, aab are examples of strings over . language over alphabet . using axioms and rules. Below are some example axioms and rules for generating a language, L, over the alphabet = fa, bg, L = fa, ab, abb, abbb, ...g. respectively). The following is a unary rule where u indicates some string in :
Are two grammars weakly equivalent if they derive the same set of strings?
to denote the reflexive, transitive closure of derivation steps, consequently L(G) = fw 2 jS = ) wg. Two grammars are weakly equivalent if they derive the same set of strings. They are strongly equivalent if they derive the same set of strings with the same tree structures.
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[Discrete Mathematics] Formal Languages
[Discrete Mathematics] Formal Languages Examples
Grammars and Languages in Discrete Mathematics.
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Page 1 To understand a program, you must become both the Languages and Grammars Vocabulary(V ) a finite set of symbols, eg, V = {0, 1} a finite set of productions P By convention, S is the start symbol, and nonterminal symbols are in uppercase italics This grammar generates odd binary numbers S (
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A word (or sentence) over V is a string of finite length elements of V The empty string or null string, denoted by λ, is the string containing no symbols The set of all words over V is denoted by V ∗ A language over V is a subset of V ∗
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Noam Chomsky, Three models for the description of language, IRE Transactions on Information Theory, (1956) N2, 113–124 4 Noam Chomsky, On certain
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