FSA’s Defining FSA’s Computing with FSA’s Defining L(M) Defining language class LFSA

1. FSA’s • Defining FSA’s • Computing with FSA’s • Defining L(M) • Defining language class LFSA • Comparing LFSA to set of solvable languages (REC)

2. Finite State Automata New Computational Model

3. Tape • We assume that you have already seen FSA’s in CSE 260 • If not, review material in reference textbook • Only data structure is a tape • Input appears on tape followed by a B character marking the end of the input • Tape is scanned by a tape head that starts at leftmost cell and always scans to the right

4. Data type/States • The only data type for an FSA is char • The instructions in an FSA are referred to as states • Each instruction can be thought of as a switch statement with several cases based on the char being scanned by the tape head

5. Example program 1 switch(current tape cell) { case a: goto 2 case b: goto 2 case B: return yes } 2 switch (current tape cell) { case a: goto 1 case b: goto 1 case B: return no; }

6. FSA M=(Q,S,q0,A,d) Q = set of states = {1,2} S = character set = {a,b} don’t need B as we see below q0 = initial state = 1 A = set of accepting (final) states = {1} A is the set of states where we return yes on B Q-A is set of states that return no on B d = state transition function 1 switch(current tape cell) { case a: goto 2 case b: goto 2 case B: return yes } 2 switch (current tape cell) { case a: goto 1 case b: goto 1 case B: return no; } New model of computation

7. 1 switch(current tape cell) { case a: goto 2 case b: goto 2 case B: return yes } 2 switch (current tape cell) { case a: goto 1 case b: goto 1 case B: return no; } a b 1 2 2 2 1 1 Textual representations of d * d(1,a) = 2, d(1,b)=2, d(2,a)=1, d(2,b) = 1 {(1,a,2), (1,b,2), (2,a,1), (2,b,1)}

8. 1 switch(current tape cell) { case a: goto 2 case b: goto 2 case B: return yes } 2 switch (current tape cell) { case a: goto 1 case b: goto 1 case B: return no; } a,b 2 1 a,b Transition diagrams Note, this transition diagram represents all 5 components of an FSA, not just the transition function d

9. Exercise • FSA M = (Q, S, q0, A, d) • Q = {1, 2, 3} • S = {a, b} • q0 = 1 • A = {2,3} • d: {d(1,a) = 1, d(1,b) = 2, d(2,a)= 2, d(2,b) = 3, d(3,a) = 3, d(3,b) = 1} • Draw this FSA as a transition diagram

10. a a b 2 1 b b 3 a Transition Diagram

11. Computing with FSA’s

12. a a b 2 1 b b 3 a Computation Example * Input: aabbaa

13. Computation of FSA’s in detail • A computation of an FSA M on an input x is a complete sequence of configurations • We need to define • Initial configuration of the computation • How to determine the next configuration given the current configuration • Halting or final configurations of the computation

14. Given an FSA M and an input string x, what is the initial configuration of the computation of M on x? (q0,x) Examples x = aabbaa (1, aabbaa) x = abab (1, abab) x = l (1, l) a a b 2 1 b b 3 a Initial Configuration FSA M

15. (1, aabbaa) ├M (1, abbaa) config 1 “yields” config 2 in one step using FSA M (1,aabbaa) ├ M (2, baa) config 1 “yields” config 2 in 3 steps using FSA M (1, aabbaa) ├ M (2, baa) config 1 “yields” config 2 in 0 or more steps using FSA M Comment: ├M determined by transition functiond There must always be one and only one next configuration If not, M is not an FSA a a b 2 1 b b 3 a Definition of ├M 3 * FSA M

16. Halting configuration (q, l) Examples (1, l) (3, l) Accepting Configuration State in halting configuration is in A Rejecting Configuration State in halting configuration is not in A a a b 2 1 b b 3 a Halting Configurations * FSA M

17. a a b FSA M b b a FSA M on x • Two possibilities for M running on x • M accepts x • M accepts x iff the computation of M on x ends up in an accepting configuration • (q0, x) ├M (q, l) where q is in A • M rejects x • M rejects x iff the computation of M on x ends up in a rejecting configuration • (q0, x) ├M (q, l) where q is not in A • M does not loop or crash on x • Why? * *

18. a a b FSA M b b a Examples • For the following input strings, does M accept or reject? • l • aa • aabba • aab • babbb

19. Notation from the book d(q, c) = p dk(q, x) = p k is the length of x d*(q, x) = p Examples d(1, a) = 1 d(1, b) = 2 d4(1, abbb) = 1 d*(1, abbb) = 1 d*(2, baaaaa) = 3 a a b 2 1 b b 3 a Definition of d*(q, x) FSA M