1. Regular Expression ManipulationFSM Model Sequential Machine Theory Prof. K. J. Hintz Department of Electrical and Computer Engineering Lecture 6 Modifications by Marek Perkowski

2. Null Machine 3 Methods for Proving That a Machine Accepts No Words • By inspection • Any path from the start state to a final state means that at least one word is accepted by the machine • By state diagram manipulation • If a final state is relabeled as a start state, then the machine must accept at least one word

3. Null Machine • By converting the regular expression into a deterministic FA • If possible, FA must accept at least one word • Conversion to FA may not be possible • Machine may have no final states. • There is no path from the initial state to any final state.

4. State Diagram Manipulation A procedure to determine if a machine accepts no strings • Remove all edges (arrows) to the start state. • From the start state, identify all single-step “next states.” • Relabel these “next states” as start states and eliminate the edges used to get there. • go to (b) • If a final state is relabeled as a start state, then the machine must accept at least one word.

5. State Diagram Manipulation

6. State Diagram Manipulation

7. State Diagram Manipulation Does Not Accept Any Word Since There Is No Path From - To +.

8. The Complement Machine • A Complement Machine Accepts All ExpressionsOther Than Those Accepted by the Original Machine • Method • Change all non-final states into final states • Change all final states into non-final states • Leave start state unchanged

9. Language Decidability Methods for Determining If Two Regular Expressions Define the Same Language • Language Enumeration with 1:1 correspondence between the 2 languages. • The regular languages can be accepted by identical FAs. • Generate

10. Language Overlap • If the Overlap Language Is NOT the Null Set, Then There Is Some Word in L1 Which Is Not Accepted by L2 and Vice Versa. • If the Overlap Language Accepts the Null String, Then the Languages Are Not Equal.

11. DeMorgan’s Theorem Applies Equally Well to Sets As Well As Boolean Algebra

12. Regular Expression Equivalence Methodology • Construct the complement machines • Apply DeMorgan’s theorem since it is difficult to form the intersect machine

13. Regular Expression Equivalence Take the Unions of the Complemented and Non-complementedSeveral Times to Determine Whether Loverlap Is the Null Set or Not

14. RE Equivalence Example* Two REs are represented by their equivalent FAs (FA1 does = FA2) *Cohen, Prob. 2, page 233.

15. RE Equivalence Example Form the Complement Machines

16. RE Equivalence Example Make the Product Machine of FA2 and the Complement of FA1.

17. RE Equivalence Example • States of Product Machine, FA1-bar & FA2 • Only One Start State / Multiple Final States

18. RE Equivalence Example

19. Product Machine State Table

20. State Diagram of Product FA1-Bar, FA2

21. Reduced State Diagram Non-Reachable States Removed

22. RE Equivalence Example • Take the Complement Of the Union by changing final states to non-final and vice-versa • No Final States, So Complement FA Accepts No Words

23. RE Equivalence Example Do the Same for the Right Term of Loverlap

24. RE Equivalence Example • Application of Same Procedure to Preceding Machine Also Results in No Recognizable Words. • Since Both Terms of Loverlap are Null, Then REs Are Equivalent Since Their Union Is Null.

25. Moore & Mealy Machines • The Behavior of Sequential Machines Depends on Previous Inputs. • Moore Machine • Output only depends on present state • Mealy Machine • Output depends on both the present state and the present input

26. Moore & Mealy Machines Equivalent Descriptive Methods • Transition (state) table • Transition (state) diagram • Operational descriptions using set theory • (Language recognized by the machine)

27. Comb Ckt Comb Ckt Memory Moore Machine Input Present State Output Output Is Only a Function of Present State

28. input on state/ output A/0 B/1 off off Legend C/0 D/0 etc. Primitive State Diagram, Moore

29. x1x0 10 s1s0/z 00/1 01/0 00 01 Legend 10/1 11/1 etc. Moore Machine State Diagram

30. Comb Ckt Comb Ckt Memory Mealy Machine Input Present State Output Output Is Function of Present State AND Present Input

31. input/output on/0 state A B off/1 off /0 Legend C D etc. Primitive State Diagram, Mealy

32. x1x0 /z 10 /0 s1s0 00 01 00 /1 01 /1 Legend 10 11 etc. Mealy Machine State Diagram

33. Transition Table

34. FSM Design Approaches • “One-Hot” • One flip-flop is used to represent each state • Costly in terms of discrete hardware, but trivial to design • Efficient in FPGAs because FF part of each CLB • Binary Coded State • n flip-flops used to store 2nstates • Most efficient • Need to account for unused states

35. FSM and Clocks • Synchronous FSMs may change state only when a unique input, the clock, occurs • Asynchronous FSMs may change state when input changes • Next state depends on present input and present state for both Moore and Mealy

36. Synchronous versus Asynchronous Machines in Design • Synchronous FSMs • Easier to design, turn the crank • Slower operation • Asynchronous • Harder to design because of potential for races, iterative solutions • Faster operation

37. Mealy “0101” Detector M = ( S, I, O, d, b ) S: { A, B, C, D } I: { ‘0’, ‘1’ } O: { 0, 1 } = { not detected, detected} d: next slide b: next slide

38. Mealy Transition/Output Table Next State/Output

39. A B C D “0101“ State Diagram ‘1’/0 ‘0’/0 ‘0’/0 ‘1’/0 ‘0’/0 ‘1’/0 ‘1’/1 ‘0’/0

40. Moore “0101” Detector M = ( S, I, O, d, l ) S: { A, B, C, D, E } I: { ‘0’, ‘1’ } O: { 0, 1 } = { not detected, detected} d: next slide l: next slide

41. Moore Transition/Output Table Next State

42. D/0 E/1 Moore “0101“ State Diagram ‘0’ ‘1’ ‘0’ detected ‘0’ A/0 B/0 ‘0’ ‘1’ ‘1’ ‘1’ ‘0’ C/0 ‘0’ ‘1’ “01” det “0101” det “010” det

43. Asynchronous FSM Fundamental Mode Assumption • Only one inputcan change at a time • Analysis too complicated if multiple inputs are allowed to change simultaneously • Circuit must be allowed to settle to its final value before an input is allowed to change • Behavior is unpredictable (nondeterministic) if circuit not allowed to settle

44. Asynch. Design Difficulties Delay in Feedback Path • Not reproducible from implementation to implementation • Variable • may be temperature or electrical parameter dependent within the same device • Analog • not known exactly

45. Stable State • PS = present state • NS = next state • PS = NS = Stability • Machine may pass through none or more intermediate states on the way to a stable state • Desired behavior since only time delay separates PS from NS • Oscillation • Machine never stabilizes in a single state

46. Races • A Race Occurs in a Transition From One State to the Next When More Than One Next State Variables Changes in Response to a Change in an Input • Slight Environment Differences Can Cause Different State Transitions to Occur • Supply voltage • Temperature, etc.

47. PS 01 11 00 if Y1 changes first if Y2 changes first 10 desired NS Races

48. Types of Races • Non-Critical • Machine stabilizes in desired state, but may transition through other states on the way • Critical • Machine does not stabilize in the desired state

49. 01 11 00 10 00 critical race Races PS if Y2 changes first if Y1 changes first desired NS non- critical race

50. Asynchronous FSM Benefits • Fastest FSM • Economical • No need for clock generator • Output Changes When Signals Change, Not When Clock Occurs • Data Can Be Passed Between Two Circuits Which Are Not Synchronized • In some technologies, like quantum, clock is just not possible to exist