Finite Automata
An FA accepts input string if final state is ac- cept state; otherwise it rejects. Goddard 1: 4. Page 5. An Example FA. A.
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finite automaton the automaton we use is a McCulloch-Pitts nerve net. Thus their neurons are one example of a kind of "universal elements" for.
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The concatenation of two strings u v ? ?? is the string uv obtained by joining the strings end-to-end. Examples: If u = ab
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For a Deterministic Finite Automaton ?(sa) is a unique state for all s ? S and for all a ? ?. For a Nondeterministic Finite Automaton the transition
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Deterministic Finite Automata A formalism for defining languages consisting of: 1 A finite set of states (Q typically) 2 An input alphabet (? typically) 3 A transition function (? typically) 4 A start state (q 0 in Q typically) 5 A set of final states (F ? Q typically) “Final” and “accepting” are synonyms
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Example: An automaton’s standard transition function takes two parameters: a state and a symbol The “extended transition function” ? takes a state and a string ?can be defined in terms of : Assume that is a string is a symbol in ? and is a state Recursively ? = ? Examples from the previous automaton: ?0 = 0 ?0111 = 1
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Finite Automata Informally a state machine that comprehensively captures all possible states and transitions that a machine can take while responding to a streammachine can take while responding to a stream (or sequence) of input symbols Recognizer for “Regular Languages” Deterministic Finite Automata (DFA)
Searches related to examples of finite automata filetype:pdf
Finite automata (next two weeks) are an abstraction of computers with fnite resource constraints Provide upper bounds for the computing machines that we can actually build Turing machines (later) are an abstraction of computers with unbounded resources Provide upper bounds for what we could ever hope to accomplish
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Finite automata (next two weeks) are an abstraction of computers with finite resource constraints ? Provide upper bounds for the computing machines
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The notes are designed to accompany six lectures on regular languages and finite automata for Part IA of the Cambridge University Computer Science Tripos
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A finite automaton (FA) is a device that recog- nizes a language (set of strings) It has finite memory and an input tape; each input symbol that is read causes
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Labels on arcs tell what causes the transition Page 4 4 Example: Recognizing Strings Ending in “ing
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Since ??(A0100)=A and A is a final state the string 0100 is accepted by this DFA Extended Delta Function – Delta Hat ? ? Example BBM401 Automata Theory
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Example: Detect Even Number of 1s Jim Anderson (modified by Nathan Otterness) 2 This is a “transition diagram” for a deterministic finite automaton
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What is the meaning of automata?
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Are finite automata and hybrid automata labeled transition systems?
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Lecture Notes on
Regular Languages
and Finite Automata for Part IA of the Computer Science TriposMarcelo Fiore
Cambridge University Computer Laboratory
First Edition 1998.Revised 1999, 2000, 2001, 2002, 2003, 2005, 2006, 2007, 2008, 2009, 2010. c2010 A. M. Pitts
ContentsLearning Guideii
1 Regular Expressions1
1.1 Alphabets, strings, and languages . . . . . . . . . . . . . . . . . .. . . . . . 1
1.2 Pattern matching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 Some questions about languages . . . . . . . . . . . . . . . . . . . . .. . . 6
1.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2 Finite State Machines11
2.1 Finite automata . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2 Determinism, non-determinism, and-transitions . . . . . . . . . . . . . . . 14
2.3 A subset construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 17
2.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3 Regular Languages, I23
3.1 Finite automata from regular expressions . . . . . . . . . . . .. . . . . . . . 23
3.2 Decidability of matching . . . . . . . . . . . . . . . . . . . . . . . . . .. . 28
3.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
4 Regular Languages, II31
4.1 Regular expressions from finite automata . . . . . . . . . . . . .. . . . . . 31
4.2 An example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
4.3 Complement and intersection of regular languages . . . . .. . . . . . . . . . 34
4.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
5 The Pumping Lemma39
5.1 Proving the Pumping Lemma . . . . . . . . . . . . . . . . . . . . . . . . . .40
5.2 Using the Pumping Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
5.3 Decidability of language equivalence . . . . . . . . . . . . . . .. . . . . . . 44
5.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
6 Grammars47
6.1 Context-free grammars . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 47
6.2 Backus-Naur Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
6.3 Regular grammars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
6.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
iiLearning GuideThe 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 finite state machines, regular expressions and grammars, and to explain their applications to computer languages. As such, it covers some basic theoretical material which Every Computer Scientist Should Know. Direct applications of the course material occur in the various CST courses on compilers. Further and related developments will be found in the CST Part IB coursesComputation TheoryandSemantics of Programming Languagesand the CST Part II courseTopics inConcurrency.
This course contains the kind of material that is best learned through practice. The books mentioned below contain a large number of problems of varying degrees of difficulty, and some contain solutions to selected problems. A few exercises are given at the end of each section of these notes and relevant past Tripos questions are indicated there. At the end of the course students should be able to explain how to convert between the three ways of representing regular sets of strings introduced in the course; and be able to carry out such conversions by hand for simple cases. They should also be able to prove whether or not a given set of strings is regular. Recommended booksTextbooks which cover the material in this course also tend to cover the material you will meet in the CST Part IB courses onComputation Theoryand Complexity Theory, and the theory underlying parsing in various courses on compilers. There is a large number of such books. Three recommended onesare listed below. J. E. Hopcroft, R. Motwani and J. D. Ullman,Introduction to Automata Theory, Languages, and Computation, Second Edition(Addison-Wesley, 2001). D. C. Kozen,Automata and Computability(Springer-Verlag, New York, 1997). T. A. Sudkamp,Languages and Machines(Addison-Wesley Publishing Company,Inc., 1988).
NoteThe material in these notes has been drawn from several different sources, including the books mentioned above and previous versions of this course by the author and by others. Any errors are of course all the author's own work. A list of corrections will be available from the course web page. Please take time to fill out the on-line lecture feedback form.Marcelo Fiore
Marcelo.Fiore@cl.cam.ac.uk
11 Regular Expressions
Doubtless you have used pattern matching in the command-line shells of various operating systems (Slide 1) and in the search facilities of text editors. Another important example of the same kind is the `lexical analysis' phase in a compiler during which the text of a program is divided up into the allowed tokens of the programming language. The algorithms which implement such pattern-matching operations make use of thenotion of afinite automaton (which is Greeklish forfinite state machine). This course reveals (some of!) the beautiful theory of finite automata (yes, that is the plural of `automaton') and their use for recognising when a particular string matches a particular pattern.Pattern matching
What happens if, at a Unix/Linux shell prompt, you type ls and press return? Suppose the current directory contains files calledregflatex, regflaaux,regflalog,regfladvi, and (strangely)aux. What happens if you type lsaux and press return?Slide 1
1.1 Alphabets, strings, and languages
The purpose of Section 1 is to introduce a particular language for patterns, calledregular expressions, and to formulate some important problems to do with pattern-matching which will be solved in the subsequent sections. But first, here is some notation and terminology to do with character strings that we will be using throughout the course.21 REGULAR EXPRESSIONS
Alphabets
Analphabetis specified by giving a finite set,Σ, whose elements are calledsymbols. For us, any set qualifies as a possible alphabet, so long as it is finite.Examples:
1=0123456789-10-element set of decimal digits.
2=-26-element set of lower-case characters of the
English language.
3=Σ1-210-element set of all subsets of the alphabet of
decimal digits.Non-example:
N=0123- set of all non-negative whole numbers is not an alphabet, because it is infinite.Slide 2
Strings over an alphabet
Astring of length(0) over an alphabetΣis just an ordered-tuple of elements ofΣ, written without punctuation. Example:ifΣ =, then,,, andare strings overΣof lengths one, two, three and four respectively. def=set of all strings overΣof any finite length. N.B. there is a unique string of length zero overΣ, called thenull string (orempty string) and denoted (no matter whichΣwe are talking about).Slide 3
1.1 Alphabets, strings, and languages3
Concatenation of strings
Theconcatenationof two stringsΣis the stringobtained by joining the strings end-to-end.Examples:If=,=and=, then=,=
and=. This generalises to the concatenation of three or more strings.E.g.=.
Slide 4
Slides 2 and 3 define the notions of analphabetΣ, and the setΣof finitestringsover an alphabet. The length of a stringwill be denoted bylength(). Slide 4 defines the operation ofconcatenationof strings. We make no notational distinction between a symbolΣand the corresponding string of length one overΣ: soΣcan be regarded as a subset ofΣ. Note thatΣis never empty-it always contains thenull string,, the unique string of length zero.Note also that for anyΣ
==and()==() andlength() =length() +length().Example 1.1.1.Examples ofΣfor differentΣ:
(i) IfΣ =, thenΣcontains (ii) IfΣ =, thenΣcontains (iii) IfΣ =(theempty set- the unique set with no elements), thenΣ=, the set just containing the null string.41 REGULAR EXPRESSIONS
1.2 Pattern matching
Slide 5 defines the patterns, orregular expressions, over an alphabetΣthat we will use. Each such regular expression,, represents a whole set (possibly an infinite set) of strings inΣthatmatch. The precise definition of this matching relation is given onSlide 6. It might seem odd to include a regular expressionthat is matched by no strings at all-but it is technically convenient to do so. Note that the regular expressionis in fact equivalent to , in the sense that a stringmatchesiff it matches(iff=).Regular expressions over an alphabetΣ
each symbolΣis a regular expression is a regular expression is a regular expression ifandare regular expressions, then so is() ifandare regular expressions, then so is ifis a regular expression, then so is()quotesdbs_dbs3.pdfusesText_6[PDF] examples of hegemony in education
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