Regular Expression Manipulation FSM Model

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Regular Expression ManipulationFSM Model

• Sequential Machine Theory
• Prof. K. J. Hintz
• Department of Electrical and Computer Engineering
• Lecture 5

Modifications by Marek Perkowski
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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

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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.

Give examples
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Reachability Analysis
Reachability analysis will show that this machine
will never reach the final state starting from
initial state. Show it.

Discuss reachability for computer hardware and
robotics
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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.

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State Diagram Manipulation

• Does Not Accept Any
• Word Since
• There Is No
• Path From
• - To .

The final state is not reachable
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The Complement Machine

• A Complement Machine Accepts All Expressions
  Other 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

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Language Decidability

• Methods for Determining If Two Regular
  Expressions Define the Same Language
• Language Enumeration with 11 correspondence
  between the 2 languages.
• The regular languages can be accepted by
  identical FAs.
• Generate

This example demonstrates why it is useful to be
able to check if language is empty
This is like xor of languages
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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.

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DeMorgans Theorem

• Applies Equally Well to Sets As Well As Boolean
  Algebra

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Regular Expression Equivalence

• Methodology
• Construct the complement machines
• Apply DeMorgans theorem since it is difficult to
  form the intersect machine

Show some special cases when it is easy to create
the intersect machine
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Regular Expression Equivalence

• Take the Unions of the Complemented and
  Non-complemented Several Times to Determine
  whether Language overlap is the Null Set or not

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RE Equivalence Example

• Two REs are represented by their equivalent FAs
  (FA1 does FA2)

Cohen, Prob. 2, page 233.
state
Try to use first machine minimization method and
isomorphism
Obviously looking at these two machines we see
that they are equivalent, but how to prove it?
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RE Equivalence Example

• Form the Complement Machines

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RE Equivalence Example

• Make the Product Machine of FA2 and the
  Complement of FA1.

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RE Equivalence Example

• States of Product Machine, FA1-bar FA2
• Only One Start State / Multiple Final States

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Product Machine State Table

• States of Product Machine,
• FA1-bar FA2

State Diagram of Product
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Reduced State Diagram

• Non-Reachable States 2,4,6, are Removed

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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

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RE Equivalence Example

• Do the Same for the Right Term of Loverlap

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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.

Finally we proved that these two machines are
equivalent without the need to minimize them.
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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

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Moore Mealy Machines

• Equivalent Descriptive Methods
• Transition (state) table
• Transition (state) diagram
• Operational descriptions using set theory
• (Language recognized by the machine)

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Moore Machine
Input
Present State
Output
Output Is Only a Function of Present State
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Primitive State Diagram, Moore
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Moore Machine State Diagram
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Mealy Machine
Input
Present State
Output
Output Is Function of Present State AND Present
Input
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Primitive State Diagram, Mealy
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Mealy Machine State Diagram
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Transition Table

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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 2n states
• Most efficient
• Need to account for unused states

Review the one-hot coded machines and transition
from non-deterministic to deterministic. Discuss
parallel state machines and similar diagrams like
Petri nets
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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

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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

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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

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Mealy Transition/Output Table
Next State/Output
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0101 State Diagram
1/0
0/0
This machine detects sequence whenever it
appears, mention smart house for disabled and
heart attack devices
0/0
1/0
0/0
1/0
1/1
0/0
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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

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Moore Transition/Output Table
Next State
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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
One state more.
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Sequential Machines Problems

• Three Problems of Sequential Machines
• State minimization problem
• Determine all equivalent states of a sequential
  machine, and,
• Eliminate redundant states
• Machine Decomposition
• Separate large machines into an interconnected
  set of smaller machines
• Easier to design and analyze small machines

Instead to minimize, create minimal machine in
first run.
Instead to decompose, create decomposed machine
in first run.
Good but not always realistic advises
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Sequential Machine Problems

• State assignment problem
• There is no guidance on which binary number to
  assign to which state in a primitive state table
• Complexity of implementation is dependent on
  mapping of states to binary numbers
• Unsolved problem
• Design all machines and compare
• Benefit of decomposition of large machine into
  smaller machines.

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Set Theoretic Description

• Moore Machine is an ordered quintuple

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Set Theoretic Description

• Mealy Machine is an ordered quintuple

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Recursive Definitions of Delta

• State Transition for Moore Mealy
• Single-valued, else not deterministic.
• At least a partial function
• Not necessarily injective or surjective
• Shields nomenclature

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Recursive Definitions of Delta

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Recursive Definitions of Beta

• Causal, No Output for No Input.
• For a Given Input Sequence, There Will Be a
  Deterministic Output Sequence of the Same Length
  As the Input.

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Recursive Definitions of Lambda

• Same Caveats As Beta

sk
sk-1
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Possible Exam Problems

• Find if two state machines (of any type, Mealy,
  Moore or Rabin-Scott) describe the same regular
  language.
• Find if a machine describes an empty language.
• Find a regular language accepted by arbitrary
  type state machine, specified in any way (graph,
  table,etc.).
• Find the intersection, the union, the difference
  of two machines M1 and M2.
• Find a negation of a machine M.

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Possible Exam Problems

• Find Mealy Machine for arbitrary sequence
  detection.
• Find Moore Machine for arbitrary sequence
  detection, finite or infinite sequence over
  arbitrary alphabet.
• Convert a Mealy Machine to an equivalent Moore
  Machine.
• Convert a Moore Machine to an equivalent Mealy
  Machine.