cardinality of regular languages
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Lecture Notes on Regular Languages and Finite Automata
The notes are designed to accompany six lectures on regular languages and finite automata for Part IA of the Cambridge University Computer Science Tripos The aim of this short course will be to introduce the mathematical formalisms of fi nite state machines regular expressions and grammars and to explain their applications to computer |
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Harvard CS 121 and CSCI E-121 Lecture 6: Countability and
A language is regular iff it is accepted by a DFA iff it is accepted by an NFA iff it is represented by a regular expression All these equivalences are CONSTRUCTIVE And we know that this class of languages is closed under union concatenation Kleene * complement intersection So we can mix and match our methods to our problems |
What is a formal language?
We take a very extensional view of language: a formal language is completely determined by the ‘words in the dictionary’, rather than by any grammatical rules. Slide 9 gives some important questions about languages, regular expressions, and the matching relation between strings and regular expressions.
Which connects regular languages with the notion of matching strings?
which connects regular languages with the notion of matching strings to regular expressions introduced in Section 1: the collection of regular languages coincides with the collection of languages determined by matching strings with regular expressions. The aim of this section is to prove part (a) of Kleene’s Theorem.
What is a regular language?
where u is a string of terminals and x and y are non-terminals. Every language generated by a regular grammar is a regular language (i.e. is the set of strings accepted by some DFA). Every regular language can be generated by a regular grammar. right linear), which do generate regular languages.
How to specify an arbitrary language?
To specify an arbitrary language requires an infinite amount of information. For example, an infinite sequence of bits would suffice: Σ∗ has a lexicographic ordering, and the i’th bit of an infinite sequence specifying a language would say whether or not the i’th string is in the language. ⇒ Some language must not be regular. How to formalize?
3 Characterizations Of Regular Language
A language is regular iff it is accepted by a DFA iff it is accepted by an NFA iff it is represented by a regular expression. All these equivalences are CONSTRUCTIVE And we know that this class of languages is closed under union, concatenation, Kleene *, complement, intersection So we can mix and match our methods to our problems scholar.harvard.edu
Goal: Existence of Non-Regular Languages
Intuition: Every regular language can be described by a finite string (namely a regular expression). To specify an arbitrary language requires an infinite amount of information. For example, an infinite sequence of bits would suffice: Σ∗ has a lexicographic ordering, and the i’th bit of an infinite sequence specifying a language would say whether o
• countable if S
is finite or countably infinite • uncountable if it is not countable scholar.harvard.edu
Theorem:
P (N ) is uncountable (The set of all sets of natural numbers) scholar.harvard.edu
Claim:
✪ Pf: D is di Derent is from omitted each row from because the they enumeration, di er at the diagonal. contradicting the assumption that every set of natural numbers is one of the Sis. Pf: D is different from each row because they differ at the diagonal. scholar.harvard.edu
∈ Sk
if and only if • k D ∈ (since D if and only if • k /∈ D = {i : i ∈ Si}) (since D = N − D) if and only if scholar.harvard.edu
Cardinality of Languages
An alphabet Σ is finite by definition Proposition: Σ∗ is countably infinite So every language is either finite or countably infinite P (Σ∗) is uncountable, being the set of subsets of a countable infinite set. i.e. There are uncountably many languages over any alphabet Q: Even if Σ = 1? scholar.harvard.edu
Theorem:
For every alphabet Σ, there exists a non-regular language over Σ. scholar.harvard.edu
Proof:
There are only countably many regular expressions over Σ. ⇒ There are only countably many regular languages over Σ. There are uncountably many languages over Σ. Thus at least one language must be non-regular. ⇒ In fact, “almost all” languages must be non-regular. Q: Could we do this proof using DFAs instead? Q: Can we get our hands on an explicit n
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Regular Languages [Fa14]
You may be tempted to conjecture that all languages are regular but in fact |
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Practice Problems for Final Exam: Solutions CS 341: Foundations of
Regular language. Answer: A regular language is defined by a DFA. iv. Kleene's theorem. Answer: A language is regular if and only if it has a regular expression |
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“CS 374” Fall 2015 — Infinite Fooling Sets Review Guide
1.1 The connection between regular languages and finite automata. Based on the recursive definition It's a regular language with infinite cardinality. |
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Q1 q2 q3 a b b a a b
Definition: A language is regular if it is recognized by some DFA. CS 341: Chapter 1. 1-12. Examples of Deterministic Finite Automata. Example: Consider the |
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Regular Languages
4 L is the concatenation of two regular languages; or. 4 L is the Kleene closure of a regular cardinality argument almost all languages are not regular. |
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Automata Theory and Languages
Automata Theory Languages and Computation - M?rian Halfeld-Ferrari – p. If E and F are regular expressions |
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Harvard CS 121 and CSCI E-121 Lecture 6: Countability and
Sep 19 2013 3 Characterizations Of Regular Language. A language is regular iff it is accepted by a DFA ... ( |
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Theory of Computation Class Notes1
2.2 Regular Expressions . 2.4 Classification of Grammars and Languages . ... The cardinality of a finite set can thus be obtained by counting the ... |
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Weak Cardinality Theorems
Kummer's Cardinality Theorem states that a language A must be recursive if a theorems is prepared in ?2 where the class of regular languages and the ... |
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Solution to Problem Set 0
two sets must be equal. 2. Cardinality. [Category: Proof Points: 40]. (a) Prove that the set of all regular languages (languages accepted by a DFA) on a. |
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Languages & regular expressions notes
Lecture 2: Regular Languages [Fa'14] Caveat lector: This You may be tempted to conjecture that all languages are regular, but in fact, the following cardinality |
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Countability and Uncountability
19 sept 2013 · Goal: Existence of Non-Regular Languages Every regular language can be described by a finite string (S is the size or cardinality of S) |
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Theory of Computation Class Notes1
2 2 Regular Expressions 2 4 Classification of Grammars and Languages The cardinality of a finite set can thus be obtained by counting the elements of |
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Context Free Languages: Cardinality of CFLs • Theorem 131
Context Free Languages: Cardinality of CFLs • Theorem 13 1 – Statement: Regular languages are a proper subset of CFLs – Proof: 1 RLs ⊆ CFLs |
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Non-regular languages and the pumping lemma - MIT
– It turns out that there are many more sets of finite strings than there are DFAs; so just based on cardinality, there must be some non-regular languages – But, |
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Automata Theory and Languages
Automata Theory, Languages and Computation - Mırian Halfeld-Ferrari – p The constants ǫ and ∅ are regular expressions, denoting the language {ǫ} and |
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Regular Languages and Finite Automata
3 Regular Languages, I 23 The notes are designed to accompany six lectures on regular languages and So the answer to (d) is 'no' for cardinality reasons |
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AUTOMATA & LANGUAGES - Distributed Computing
Infinite Cardinality • Question: Are all regular languages finite? • Answer: No Many infinite languages are regular • Question: Give an example of an infinite but |
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Lecture 1
In the last lecture, we defined various operations on regular languages Three of This means that P(\Sigma \ast ) has cardinality larger than that of \Sigma \ast |
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CLASSES OF REGULAR AND CONTEXT-FREE LANGUAGES
For a countably infinite alphabet A, the classes Reg(A) of regular languages For a set S, IS I denotes the cardinality of S Each alphabet 27 can be indexed |
