1. Deterministic Finite State Machines Chapter 5

2. Languages and Machines 2

3. Regular Languages Regular Language Regular Expression Finite State Machine L Accepts 3

4. Finite State Machines Vending Machine: An FSM to accept $.50 in change Drink is 25 cents No more than 50 cents can be deposited X Other FSM examples? 4

5. Definition of a DFSM M = (K, , , s, A), where: K is a finite set of states  is an alphabet s  K is the initial state A  K is the set of accepting states, and  is the transition function from (K  )toK state input symbol state Quintuple  Cartesian product 5

6. Accepting by a DFSM Informally, M accepts a string w iff M winds up in some element of A when it has finished reading w. The language accepted by M, denoted L(M), is the set of all strings accepted by M. 6

7. Configurations of DFSMs A configuration of a DFSM M is an element of: K   * It captures the two things that can make a difference to M’s future behavior: its current state the input that is still left to read. The initial configuration of a DFSM M, on input w, is: (s M, w), where s M is the start state of M 7

8. The Yields Relations The yields-in-one-step relation |- M : (q, w) |- M (q', w') iff w = a w' for some symbol a  , and  (q, a) = q' The relation yields, |- M *, is the reflexive, transitive closure of |- M. C 1 |- M * C 2 : M can go from C 1 to C 2 in 0 or more steps. 8

9. Computations Using FSMs A computation by M is a finite sequence of configurations C 0, C 1, …, C n for some n  0 such that: C 0 is an initial configuration, C n is of the form (q,  ), for some state q  K M, C 0 |- M C 1 |- M C 2 |- M … |- M C n. 9

10. Accepting and Rejecting A DFSM M accepts a string w iff: (s, w) |- M * (q,  ), for some q  A. A DFSM M rejects a string w iff: (s, w) |- M * (q,  ), for some q  A M. The language accepted by M, denoted L(M), is the set of all strings accepted by M. Theorem: Every DFSM M, on input s, halts in |s| steps. 10

11. An Example Computation An FSM to accept odd integers: even odd even q 0 q 1 odd On input 235, the configurations are: (q 0, 235) |- M (q 0, 35) |- M Thus (q 0, 235) |- M * (q 1,  ) 11

12. Regular Languages A language is regular iff it is accepted by some FSM. 12

13. A Very Simple Example L = {w  { a, b }* : every a is immediately followed by a b }. 13

14. A Very Simple Example L = {w  { a, b }* : every a is immediately followed by a b }. 14

15. A Very Simple Example L = {w  { a, b }* : every a is immediately followed by a b }. M = (K, , , s, A) = ({q 0, q 1, q 2 }, {a, b}, , q 0, {q 0 }), where  = {((q 0, a), q 1 ), ((q 0, b), q 0 ), ((q 1, a), q 2 ), ((q 1, b), q 0 ), ((q 2, a), q 2 ), ((q 2, b), q 2 )} 15

16. Parity Checking L = {w  { 0, 1 }* : w has odd parity}. 16

17. Parity Checking L = {w  { 0, 1 }* : w has odd parity}. M = (K, , , s, A) = ({q 0, q 1 }, {0, 1}, , q 0, {q 1 }), where  = {((q 0, 0), q 0 ), ((q 0, 1), q 1 ), ((q 1, 0), q 1 ), ((q 1, 1), q 0 )} 17

18. No More Than One b L = {w  { a, b }* : w contains no more than one b}. 18

19. No More Than One b L = {w  { a, b }* : w contains no more than one b}. M = (K, , , s, A) 19

20. Checking Consecutive Characters L = {w  { a, b }* : no two consecutive characters are the same}. 20

21. Checking Consecutive Characters L = {w  { a, b }* : no two consecutive characters are the same}. 21

22. Dead States L = {w  { a, b }* : every a region in w is of even length} 22

23. Dead States L = {w  { a, b }* : every a region in w is of even length} M = (K, , , s, A) 23

24. Dead States L = {w  { a, b }* : every b in w is surrounded by a ’s} 24

25. The Language of Floating Point Numbers is Regular Example strings: +3.0, 3.0, 0.3E1, 0.3E+1, -0.3E+1, -3E8 The language is accepted by the DFSM: 25

26. A Simple Communication Protocol 26

27. Controlling a Soccer-Playing Robot 27

28. A Simple Controller 28

29. Programming FSMs Cluster strings that share a “future”. Let L = {w  { a, b }* : w contains an even number of a ’s and an odd number of b ’s} What states are needed? How many states are there? 29

30. Even a’s Odd b’s 30

31. Vowels in Alphabetical Order L = {w  { a - z }* : all five vowels, a, e, i, o, and u, occur in w in alphabetical order}. 31

32. Vowels in Alphabetical Order L = {w  { a - z }* : all five vowels, a, e, i, o, and u, occur in w in alphabetical order}. O u 32

33. Programming FSMs L = {w  { a, b }* : w does not contain the substring aab }. 33

34. Programming FSMs L = {w  { a, b }* : w does not contain the substring aab }. Start with a machine for  L: How must it be changed? 34

35. Programming FSMs L = {w  { a, b }* : w does not contain the substring aab }. Start with a machine for  L: How must it be changed? 35

36. A Building Security System L = {event sequences such that the alarm should sound} 36

37. FSMs Predate Computers The Prague Orloj, originally built in 1410 The Chinese Abacus, 14 century AD The Jacquard Loom, invented in 1801 37

38. Finite State Representations in Software Engineering A high-level state chart model of a digital watch. 38

39. Making the Model Hierarchical 39

40. The Missing Letter Language Let  = { a, b, c, d }. Let L Missing = {w : there is a symbol a i   not appearing in w}. Try to make a DFSM for L Missing : 40