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# Formal languages and automata theory

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1. The Chinese University of Hong KongFall 2011 Formal languagesand automata theory Andrej Bogdanov http://www.cse.cuhk.edu.hk/~andrejb/csc3130

3. What else? Search Fly Play chess Is there anything a computer cannot do?

4. Impossibilities Why do we care about the impossible?

5. Perpetual motion In the middle ages, people wanted a machine that does not use any energy Later, discoveries in physics showed that energy cannot be created out of thin air Perpetual motion is a futile endeavor Understanding the impossible helps us channel our energies towards the more useful.

6. The laws of computation Just like the laws of physics tell uswhat is (im)possible for nature to do... ...the laws of computation tell us what is (im)possible for computers.

7. Automata theory Automata theory studies the laws of computation. In reality, the laws of computation are not understood, but automata theory is a good start.

8. A gumball machine machine takes \$5 and \$10 coins a gumball costs \$15 actions: +5,+10,Release 一五 +10 +10 +5 +5 +5, +10 \$0 \$5 \$10 R R R +5, +10 R

9. What is the difference? • Such questions are difficult to reason about, because devices can be designed in infinitely many ways • Automata theory will help answer (some of) them

10. What can automata do • They can describe the operation of a small device • They can be used to verify (simple) software* • They are used in lexical analyzers to recognize expressions in programming languages: ab1 is a legal name of a variable in java 5u= is not

11. Different kinds of machines • This was only one example of a machine • We will look at different kinds of machines and ask: • What kinds of problems can this kind of machine solve? • What things are impossible for this kind of machine? • Is machine A more powerful than machine B? +10 +10 +5 +5 +5, +10 \$0 \$5 \$10 R R R +5, +10 R

12. Some kinds of machines

13. Some highlights of the course • Finite automata • Automata are closely related to the task of searching for patterns in text • Grammars • Grammars describe the meaning of sentences in English, and the meaning of programs in Java • We will see how to extract the meaning out of a program find (ab)*(ab) in abracadabra

14. Some highlights of the course • Turing Machines • This is a general model of a computer, capturing anything we could ever hope to compute • But there are many things that computers cannot do: Given the code of a computer program, can you tell if the program prints the string “banana”? #include <stdio.h> main(t,_,a)char *a;{return!0<t?t<3?main(-79,-13,a+main(-87,1-_, main(-86,0,a+1)+a)):1,t<_?main(t+1,_,a):3,main(-94,-27+t,a)&&t==2?_<13? main(2,_+1,"%s %d %d\n"):9:16:t<0?t<-72?main(_,t, "@n'+,#'/*{}w+/w#cdnr/+,{}r/*de}+,/*{*+,/w{%+,/w#q#n+,/#{l,+,/n{n+,/+#n+,/#\ ;#q#n+,/+k#;*+,/'r :'d*'3,}{w+K w'K:'+}e#';dq#'l \ q#'+d'K#!/+k#;q#'r}eKK#}w'r}eKK{nl]'/#;#q#n'){)#}w'){){nl]'/+#n';d}rw' i;# \ ){nl]!/n{n#'; r{#w'r nc{nl]'/#{l,+'K {rw' iK{;[{nl]'/w#q#n'wk nw' \ iwk{KK{nl]!/w{%'l##w#' i; :{nl]'/*{q#'ld;r'}{nlwb!/*de}'c \ ;;{nl'-{}rw]'/+,}##'*}#nc,',#nw]'/+kd'+e}+;#'rdq#w! nr'/ ') }+}{rl#'{n' ')# \ }'+}##(!!/") :t<-50?_==*a?putchar(31[a]):main(-65,_,a+1):main((*a=='/')+t,_,a+1) :0<t?main(2,2,"%s"):*a=='/'||main(0,main(-61,*a, "!ek;dc i@bK'(q)-[w]*%n+r3#l,{}:\nuwloca-O;m .vpbks,fxntdCeghiry"),a+1);} banana ?

15. Some highlights of the course • Time-bounded Turing Machines • Many problems are possible to solve on a computer in principle, but take too much time in practice • Traveling salesman: Given a list of cities, find the shortest way to visit them and come back home • Hard in practice: For 100 cities, this would take 100+ years even on the fastest computer! Beijing Xian Chengdu Shanghai Guangzhou Hong Kong

16. Preliminaries of automata theory • How do we ask the question • First, we need a way of describing the problems that we are interested in solving Can machine A solve problem B?

17. Problems • Examples of problems we will consider • Given a words, does it contain “to” as a subword? • Given a numbern, is it divisible by 7? • Given two wordss and t, are they the same? • All of these have “yes/no” answers. • There are other types of problems, like “Find this” or “How many of that” but we wont look at them.

18. Alphabets and strings • A common way to talk about words, numbers, pairs of words, etc. is by representing them as strings • To define strings, we start with an alphabet • Examples An alphabet is a finite set of symbols. S1 = {a, b, c, d, …, z}: the set of letters in English S2 = {0, 1, …, 9}: the set of (base 10) digits S3= {a, b, …, z, #}: the set of letters plus the special symbol #

19. Strings • The empty string will be denoted by e • We write S* for the set of all strings over S A string over alphabet S is a finite sequenceof symbols in S. abfbz is a string over S1 = {a, b, c, d, …, z} 9021 is a string over S2 = {0, 1, …, 9} ab#bc is a string over S3= {a, b, …, z, #}

20. Languages • Languages describe problems with “yes/no” answers: A language is a set of strings (over the same alphabet). L1 = All strings that contain the substring “to” S1 = {a, b, …, z} stop, to, toe are inL1 e,oysterare not inL1 L1 = {x Î S1*: xcontains the substring “to”}

21. Examples of languages L2 = {x Î S2*: xis divisible by 7} S2 = {0, 1, …, 9} = {7, 14, 21, …} S3= {a, b, …, z, #} L3 = {s#s: s Î {a, b, …, z}*} inL3 ab#ab ab#ba not inL3 not inL3 a##a#

22. Finite Automata

23. Example of a finite automaton • There are states\$0, \$5, \$10, go, the start state is \$0 • The automaton takes inputs from {+5, +10, R} • The state go is an accepting state • There are transitions saying what to do for every state and every alphabet symbol go +10 +10 +5 +5 +5, +10 \$0 \$5 \$10 R R R +5, +10 R

24. Deterministic finite automata • A finite automaton (DFA) is a 5-tuple (Q, S, d, q0, F) where • Q is a finite set of states • S is an alphabet • d: Q×S → Q is a transition function • q0ÎQ is the initial state • F ÍQ is a set of accepting states (or final states). • In diagrams, the accepting states will be denoted by double loops

25. Example 0 1 0,1 1 0 q0 q1 q2 table of transition function d: alphabet S= {0, 1} statesQ = {q0, q1, q2} initial stateq0 accepting statesF = {q0, q1} inputs 0 1 q0 q0 q1 q1 q2 q1 states q2 q2 q2

26. Language of a DFA The language of a DFA (Q, S, d, q0, F) is the set of all strings over S that, starting from q0 and following the transitions as the string is read leftto right, will reach some accepting state. go +10 +10 +5 +5 +5, +10 M: \$0 \$5 \$10 R R R +5, +10 R +5 +10, +5 R R +5 +10 are in the language +5, +5 +10 R are not

27. Examples S= {a, b} S= {a, b} q0 b a b a q1 q3 b a a q0 q1 b a a b b q2 q4 b a 0 1 0, 1 0 1 q2 q0 q1 S= {0, 1} What are the languages of these DFAs?

28. Examples • Construct a DFA over alphabet {0, 1} that accepts all strings with at most three 1s

29. Examples • Construct a DFA over alphabet {0, 1} that accepts all strings with at most three 1s • Answer 0 0 0, 1 0 0 1 1 1 1 q4+ q2 q3 q0 q1

30. Examples • Construct a DFA for the language L = {010, 1} ( S = {0, 1} )

31. Examples • Construct a DFA that accepts the language • Answer L = {010, 1} ( S = {0, 1} ) 1 0 q0 q01 q010 0 0 1 qe 0, 1 1 0, 1 q1 qdie 0, 1

32. Examples • Construct a DFA over alphabet {0, 1} that accepts all strings that end in 01

33. Examples • Construct a DFA over alphabet {0, 1} that accepts all strings that end in 01 • Hint: The DFA must “remember” the last 2 bits of the string it is reading

34. Examples • Construct a DFA over alphabet {0, 1} that accepts all strings that end in 01 • Answer: 0 0 q00 q0 1 1 0 q01 0 qe 1 0 1 q10 1 0 0 q1 1 q11 1

35. Examples • Construct a DFA over alphabet {0, 1} that accepts all strings that end in 101 • Answer: