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CS 154 Formal Languages and Computability February 11 Class Meeting Department of Computer Science San Jose State University Spring 2016 Instructor: Ron.

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Presentation on theme: "CS 154 Formal Languages and Computability February 11 Class Meeting Department of Computer Science San Jose State University Spring 2016 Instructor: Ron."— Presentation transcript:

1 CS 154 Formal Languages and Computability February 11 Class Meeting Department of Computer Science San Jose State University Spring 2016 Instructor: Ron Mak www.cs.sjsu.edu/~mak

2 Computer Science Dept. Spring 2016: February 11 CS 154: Formal Languages and Computability © R. Mak JFLAP Manual Derivation  Jexample1.13.jff  Input string aaabbbbaabab  The automatic CYK Parser gets it wrong!  Use the User Control Parser. 2 JFLAP demo Jexample1.13.jff

3 Computer Science Dept. Spring 2016: February 11 CS 154: Formal Languages and Computability © R. Mak Distinguishable vs. Indistinguishable States  Consider two states p and q of a DFA and all strings w in Σ*.  If there is path from p to a final state implies there is a path from q to a final state, and  If there is no path from p to a final state implies there is no path from q to a final state,  Then states p and q are indistinguishable.  However, if for any one string w there is a path from p to a final state but no path from q to a final state (or vice versa),  Then the states p and q are distinguishable. 3 Not necessarily the same final state.

4 Computer Science Dept. Spring 2016: February 11 CS 154: Formal Languages and Computability © R. Mak Reducing the Number of States  Given a DFA, how can we simplify it by reducing the number of states? Of course, we want the simplified DFA to be equivalent to the original one.  One way: Find and combine indistinguishable states.  Plan: First eliminate inaccessible states. Then repeatedly partition the states into equivalence classes of indistinguishable states. 4

5 Computer Science Dept. Spring 2016: February 11 CS 154: Formal Languages and Computability © R. Mak State Reduction Example #1  From either q 1 or q 2, input 1 and input 01 lead to a final state, so they’re together in another equivalence class.  We can’t partition any further, so make new states out of each equivalence class. 5 q 0 q 1 q 2 | q 3 q 4  Remove inaccessible state q5.  Final states q 3 and q 4 are in one equivalence class: q 0 | q 1 q 2 | q 3 q 4 Formal Languages and Automata, 5 th ed. Peter Linz Jones & Bartlett, 2012

6 Computer Science Dept. Spring 2016: February 11 CS 154: Formal Languages and Computability © R. Mak State Reduction Example #2  States q 2 and q 4 are final:   From q 1 and q 3, strings 0 and 1 both lead to final states:   δ(q 4, 0) = q 4 but δ(q 2, 0) = q 1 :  No further partitioning is possible. 6 0 1 3 | 2 4 0 | 1 3 | 2 4 0 | 1 3 | 2 | 4 JFLAP demo Jexample2.15.jff Formal Languages and Automata, 5 th ed. Peter Linz Jones & Bartlett, 2012

7 Computer Science Dept. Spring 2016: February 11 CS 154: Formal Languages and Computability © R. Mak Regular Languages and Automata  A language L is called regular if and only if there exists a finite acceptor M such that L = L(M).  The finite acceptor can be a DFA or an NFA.  Is there a more concise way to describe a regular language? 7

8 Computer Science Dept. Spring 2016: February 11 CS 154: Formal Languages and Computability © R. Mak Regular Expressions  A regular expression consists of strings of symbols from an alphabet Σ, parentheses, and the operators: + for union: a + b  for concatenation: a  b which can also be written ab * for star-closure: a*  Example: (a + (b  c))* is the star-closure of, which is the language {λ, a, bc, aa, abc, bca, bcbc, aaa, aabc, … } 8

9 Computer Science Dept. Spring 2016: February 11 CS 154: Formal Languages and Computability © R. Mak Regular Expressions, cont’d Let Σ be an alphabet. Then ① The primitive regular expressions are Ø, λ, and. ② If r 1 and r 2 are regular expressions, then r 1 + r 2, r 1  r 2, r 1 *, and (r 1 ) are also regular expressions. ③ A string is a regular expression if and only if it can be derived from the primitive regular expressions by a finite number of applications of the rules in (2). 9

10 Computer Science Dept. Spring 2016: February 11 CS 154: Formal Languages and Computability © R. Mak Regular Expression Example  Is (a + b  c)*  (c + ϕ ) a regular expression?  Yes, since it is derived from the primitive regular expressions and repeated applications of the rules in (2) on the previous slide.  But (a + b + ) is not. 10

11 Computer Science Dept. Spring 2016: February 11 CS 154: Formal Languages and Computability © R. Mak Regular Expression Languages We can use a regular expression (RE) r to describe an associated language L(r). 1. Ø is a RE denoting the empty set. 2. λ is a RE denoting {λ}. 3. For every, a is a RE denoting {a}. If r 1 and r 2 are regular expressions, then 4. 5. 6. 7. 11 recursive definitions terminating conditions

12 Computer Science Dept. Spring 2016: February 11 CS 154: Formal Languages and Computability © R. Mak Regular Expression Language Example #1  What language is defined by the RE r = a*  (a + b) ? 12

13 Computer Science Dept. Spring 2016: February 11 CS 154: Formal Languages and Computability © R. Mak Precedence Rules  Consider the RE a  b + c  If it’s (a  b) + c then L(a  b + c ) = {ab, c}.  If it’s a  (b + c) then L(a  b + c ) = {ab, ac}.  To resolve this ambiguity, we use the precedence rules: star-closure is the highest concatenation is the next highest union is the lowest  Therefore, a  b + c is (a  b) + c. 13

14 Computer Science Dept. Spring 2016: February 11 CS 154: Formal Languages and Computability © R. Mak Regular Expression Language Example #2  Let Σ={0, 1}. Find regular expression r such that  RE r must have 00 in it somewhere.  What comes before or after the 00 is arbitrary.  Therefore, r = (0+1)*00(0+1)* 14 has at least one pair of consecutive zeros }

15 Computer Science Dept. Spring 2016: February 11 CS 154: Formal Languages and Computability © R. Mak Regular Expression Language Example #3  Let Σ={0, 1}. Find regular expression r such that  Whenever there’s a 0, it must be followed immediately by a 1.  There may be any number of leading and trailing 1’s.  There can be a 0 at the very end.  Therefore, r = (1*011*)*(0 + λ) + 1*(0 + λ) 15 has no pair of consecutive zeros }

16 Computer Science Dept. Spring 2016: February 11 CS 154: Formal Languages and Computability © R. Mak Regular Expression Language Example #3, cont’d  Alternate view: The RE r can be a repetition of 1’s and 01’s, with a possible 0 at the end.  Therefore, r = (1 + 01)*(0 + λ)  There is more than one RE for a given language.  Two REs are equivalent if they denote the same language. 16 r = (1*011*)*(0 + λ) + 1*(0 + λ)

17 Computer Science Dept. Spring 2016: February 11 CS 154: Formal Languages and Computability © R. Mak Regular Expressions and Regular Languages  Regular expressions and regular languages are the same concept.  For every regular expression r, there is a regular language L = L(r).  The textbook proves this by constructing, for any regular expression r, an NFA that accepts L(r). Recall that any language accepted by an NFA or a DFA is regular. 17 Theorem 3.1

18 Computer Science Dept. Spring 2016: February 11 CS 154: Formal Languages and Computability © R. Mak Construct an NFA from a Regular Expression  NFA accepts ϕ  NFA accepts {λ}  NFA accepts {a}  NFA accepts L(r) 18 Formal Languages and Automata, 5 th ed. Peter Linz Jones & Bartlett, 2012

19 Computer Science Dept. Spring 2016: February 11 CS 154: Formal Languages and Computability © R. Mak Construct an NFA from an RE, cont’d  NFA accepts L(r 1 + r 2 )  NFA accepts L(r 1 r 2 ) 19 Formal Languages and Automata, 5 th ed. Peter Linz Jones & Bartlett, 2012

20 Computer Science Dept. Spring 2016: February 11 CS 154: Formal Languages and Computability © R. Mak Construct an NFA from an RE, cont’d  NFA accepts L(r*) 20 Formal Languages and Automata, 5 th ed. Peter Linz Jones & Bartlett, 2012

21 Computer Science Dept. Spring 2016: February 11 CS 154: Formal Languages and Computability © R. Mak Example: Construct an NFA from an RE  Construct an NFA that accepts L(r), where RE 21 r = (a + bb)*(ba* + λ) M 1 accepts L(a + bb) M 2 accepts L(ba* + λ) Automaton accepts L((a + bb)*(ba* + λ)) Formal Languages and Automata, 5 th ed. Peter Linz Jones & Bartlett, 2012

22 Computer Science Dept. Spring 2016: February 11 CS 154: Formal Languages and Computability © R. Mak Generalized Transition Graph  Generalized transition graph (GTG): A transition graph where the edges are labeled with regular expressions. Example: 22 Formal Languages and Automata, 5 th ed. Peter Linz Jones & Bartlett, 2012

23 Computer Science Dept. Spring 2016: February 11 CS 154: Formal Languages and Computability © R. Mak Generalized Transition Graph, cont’d  The canonical form of a two-state GTG:  The RE covers all possible paths and is the graph’s RE. 23 Formal Languages and Automata, 5 th ed. Peter Linz Jones & Bartlett, 2012

24 Computer Science Dept. Spring 2016: February 11 CS 154: Formal Languages and Computability © R. Mak NFA to RE Conversion  Convert the NFA to a GTG.  If the GTG has more than two states, remove the extra states one at a time. See the procedure in the textbook: 24 Then apply Formal Languages and Automata, 5 th ed. Peter Linz Jones & Bartlett, 2012

25 Computer Science Dept. Spring 2016: February 11 CS 154: Formal Languages and Computability © R. Mak NFA to RE Conversion, cont’d  From an NFA, we can construct a GTG. Recall that any language accepted by an NFA or a DFA is regular.  From a GTG, we can derive a regular expression.  Therefore, for every regular language L, there is a regular expression r such that L=L(r). 25 Theorem 3.2

26 Computer Science Dept. Spring 2016: February 11 CS 154: Formal Languages and Computability © R. Mak Regular Grammars  A right-linear grammar G = (V,T,S,P) has all its production rules of the form where A and B are in V and x is in T*.  A left-linear grammar has all its production rules of the form  A regular grammar is either right- or left-linear. 26

27 Computer Science Dept. Spring 2016: February 11 CS 154: Formal Languages and Computability © R. Mak Regular Grammars, cont’d  If G is a right-linear grammar, then L(G) is a regular language. The proof in the textbook involves constructing automata based on the production rules.  If L is a regular language, then there exists a right-linear grammar such that L = L(G).  The language L is regular if and only if there exists a left-linear grammar G such that L = L(G). 27 Theorem 3.4 Theorem 3.3 Theorem 3.5

28 Computer Science Dept. Spring 2016: February 11 CS 154: Formal Languages and Computability © R. Mak Regular Grammars, cont’d  The language L is regular if and only if there exists a regular grammar G such that L = L(G). 28 Theorem 3.6

29 Computer Science Dept. Spring 2016: February 11 CS 154: Formal Languages and Computability © R. Mak Ways to Describe Regular Languages 29 Formal Languages and Automata, 5 th ed. Peter Linz Jones & Bartlett, 2012

30 Computer Science Dept. Spring 2016: February 11 CS 154: Formal Languages and Computability © R. Mak grep and Java’s regex Package  How does grep work?  What happens when you compile a Java regular expression with Pattern.compile() ?  Note that regular expressions in Java and grep use | instead of + for the union operation. Example: (a|b)*abb 30

31 Computer Science Dept. Spring 2016: February 11 CS 154: Formal Languages and Computability © R. Mak grep and Java’s regex Package, cont’d 1. Parse the RE. 2. Construct an NFA for the RE. 3. Construct a DFA from the NFA. 4. Minimize the DFA. 5. Generate a state transition matrix from the minimized DFA. 6. Use the transition matrix to search for matches. 31 Given a regular expression such as: (a|b)*abb RE  parse  NFA  DFA  minimized DFA  state transition matrix  search

32 Computer Science Dept. Spring 2016: February 11 CS 154: Formal Languages and Computability © R. Mak Midterm Preparation  Understand proof by induction language and language rules grammar: production rules, derivations deterministic and nondeterministic acceptors state transition graphs, state transition matrices applications for string pattern matching NFA to DFA conversions, reducing DFA states regular expressions and regular languages 32

33 Computer Science Dept. Spring 2016: February 11 CS 154: Formal Languages and Computability © R. Mak Midterm Preparation, cont’d  Look over the textbook’s exercises especially the ones with answers in the back Chapters 1, 2, and 3  Know how to use JFLAP automata and grammars Do a manual derivation. 33

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