Topic 3: Automata Theory 1

OutlineOutline Finite state machine, Regular expressions, DFA, NDFA, and their equivalence, Grammars and Chomsky hierarchy. 2

What is Automata Theory? Study of abstract computing devices, or “machines” Automaton = an abstract computing device Note: A “device” need not even be a physical hardware! A fundamental question in computer science: Find out what different models of machines can do and cannot do The theory of computation Computability vs. Complexity 3

4 Alan Turing ( )  Father of Modern Computer Science  English mathematician  Studied abstract machines called Turing machines even before computers existed  Heard of the Turing test? (A pioneer of automata theory)

Languages & Grammars Languages: “A language is a collection of sentences of finite length all constructed from a finite alphabet of symbols” Grammars: “A grammar can be regarded as a device that enumerates the sentences of a language” - nothing more, nothing less N. Chomsky, Information and Control, Vol 2, Or “words” Image source: Nowak et al. Nature, vol 417, 2002

The Chomsky Hierachy 6 Regular (DFA) Context- free (PDA) Context- sensitive (LBA) Recursively- enumerable (TM) A containment hierarchy of classes of formal languages

The Central Concepts of Automata Theory 7

AlphabetAlphabet An alphabet is a finite, non-empty set of symbols We use the symbol ∑ (sigma) to denote an alphabet Examples: Binary: ∑ = {0,1} All lower case letters: ∑ = {a,b,c,..z} Alphanumeric: ∑ = {a-z, A-Z, 0-9} DNA molecule letters: ∑ = {a,c,g,t} … 8

StringsStrings A string or word is a finite sequence of symbols chosen from ∑ Empty string is  (or “epsilon”) Length of a string w, denoted by “|w|”, is equal to the number of (non-  ) characters in the string E.g., x = |x| = 6 x = 01  0  1  00  |x| = ? xy = concatentation of two strings x and y 9

LanguagesLanguages 10

The Membership Problem 11

12 LanguagesLanguages  Let  be a set of characters.  is called the alphabet.  A language over  is set of strings of characters drawn from 

13 Example of Languages Alphabet = English characters Language = English sentences Alphabet = ASCII Language = C++ programs, Java, C#

14 NotationNotation  Languages are sets of strings (finite sequence of characters)  Need some notation for specifying which sets we want

15 Regular Languages  Each regular expression is a notation for a regular language (a set of words).  If A is a regular expression, we write L(A) to refer to language denoted by A.

16 Regular Expression  A regular expression (RE) is defined inductively aordinary character from   the empty string

17 Regular Expression R|S= either R or S RS= R followed by S (concatenation) R*= concatenation of R zero or more times (R*=  |R|RR|RRR...)

18 RE Extentions R?=  | R (zero or one R) R + = RR* (one or more R) (R)= R (grouping)

19 RE Extentions [abc]= a|b|c (any of listed) [a-z]= a|b|....|z (range) [^ab]= c|d|... (anything but ‘a’‘b’)

20 Regular Expression REStrings in L(R) a “a” ab “ab” a|b “a” “b” (ab)* “” “ab” “abab”... (a|  )b “ab” “b”

21 Example: integers  integer: a non-empty string of digits  digit = ‘0’|’1’|’2’|’3’|’4’| ’5’|’6’|’7’|’8’|’9’  integer= digit digit*

22 Example: identifiers  identifier: string or letters or digits starting with a letter  C identifier: [a-zA-Z_][a-zA-Z0-9_]*

23 RecapRecap Language L(R): set of strings represented by a regular expression R. L(R) is the language denoted by regular expression R.

24 How to Use REs  We need mechanism to determine if an input string w belongs to L(R), the language denoted by regular expression R.

25 AcceptorAcceptor  Such a mechanism is called an acceptor. input string language w L acceptor yes, if w  L no, if w  L

26 Finite Automata (FA)  Specification: Regular Expressions  Implementation: Finite Automata

27 Finite Automata Finite Automaton consists of  An input alphabet (   A set of states  A start (initial) state  A set of transitions  A set of accepting (final) states

28 Finite Automaton State Graphs A state The start state An accepting state

29 Finite Automaton State Graphs a A transition

30 Finite Automata  A finite automaton accepts a string if we can follow transitions labelled with characters in the string from start state to some accepting state.

31 FA Example A FA that accepts only “1” 1

32 FA Example  A FA that accepts any number of 1’s followed by a single 0 0 1

33 FA Example  A FA that accepts ab*a  Alphabet: {a,b} a b a

34 Table Encoding of FA  Transition table a b a ab 01err

35 RE → Finite Automata  Can we build a finite automaton for every regular expression?  Yes, – build FA inductively based on the definition of Regular Expression

36 NFANFA Nondeterministic Finite Automaton (NFA)  Can have multiple transitions for one input in a given state  Can have  - moves

37 Epsilon Moves  ε – moves machine can move from state A to state B without consuming input  A B

38 NFANFA operation of the automaton is not completely defined by input On input “11”, automaton could be in either state ABC

39 Execution of FA A NFA can choose  Whether to make  -moves.  Which of multiple transitions to take for a single input.

40 Acceptance of NFA  NFA can get into multiple states  Rule: NFA accepts if it can get in a final state ABC 0

41 DFA and NFA Deterministic Finite Automata (DFA)  One transition per input per state.  No  - moves

42 Execution of FA A DFA  can take only one path through the state graph.  Completely determined by input.

43 NFA vs DFA  NFAs and DFAs recognize the same set of languages (regular languages)  DFAs are easier to implement – table driven.

44 NFA vs DFA  For a given language, the NFA can be simpler than the DFA.  DFA can be exponentially larger than NFA.

45 NFA vs DFA  NFAs are the key to automating RE → DFA construction.

46 RE → NFA Construction Thompson’s construction ( CACM 1968 )  Build an NFA for each RE term.  Combine NFAs with  -moves.

47 RE → NFA Construction Subset construction NFA → DFA  Build the simulation.  Minimize number of states in DFA (Hopcroft’s algorithm)

48 RE → NFA Construction Key idea:  NFA pattern for each symbol and each operator.  Join them with  -moves in precedence order.

49 RE → NFA Construction  s0s0 a s1s1 NFA for a s0s0 a s1s1 NFA for ab s3s3 b s4s4

50 RE → NFA Construction s0s0 a s1s1 NFA for a

51 RE → NFA Construction s0s0 a s1s1 NFA for a s3s3 b s4s4 NFA for b

52 RE → NFA Construction s0s0 a s1s1 NFA for a s0s0 a s1s1 s3s3 b s4s4 s3s3 b s4s4 NFA for b

53 RE → NFA Construction  s0s0 a s1s1 NFA for a s0s0 a s1s1 NFA for ab s3s3 b s4s4 s3s3 b s4s4 NFA for b

54 RE → NFA Construction  s0s0 s5s5 s1s1 a s2s2 NFA for a | b s3s3 b s4s4   

55 RE → NFA Construction s1s1 a s2s2 NFA for a

56 RE → NFA Construction s1s1 a s2s2 s3s3 b s4s4 NFA for a and b

57 RE → NFA Construction  s0s0 s5s5 s1s1 a s2s2 NFA for a | b s3s3 b s4s4   

58 RE → NFA Construction  s0s0 s4s4 s1s1 a s2s2 NFA for a*   

59 RE → NFA Construction s1s1 a s2s2 NFA for a

60 RE → NFA Construction  s0s0 s4s4 s1s1 a s2s2 NFA for a*   

61 Example RE → NFA NFA for a ( b|c )*  s3s3 s9s9 s4s4 s5s5 s6s6 s7s7 s8s8 s0s0 s1s1 s2s2 a       b c 

62 Example RE → NFA building NFA for a ( b|c )* s0s0 s1s1 a

63 Example RE → NFA NFA for a, b and c s4s4 s5s5 s6s6 s7s7 s0s0 s1s1 a b c

64 Example RE → NFA NFA for a and b|c s3s3 s4s4 s5s5 s6s6 s7s7 s8s8 s0s0 s1s1 a     b c

65 Example RE → NFA NFA for a and ( b|c )*  s3s3 s9s9 s4s4 s5s5 s6s6 s7s7 s8s8 s0s0 s1s1 s2s2 a       b c 

66 Example RE → NFA NFA for a ( b|c )*  s3s3 s9s9 s4s4 s5s5 s6s6 s7s7 s8s8 s0s0 s1s1 s2s2 a       b c 