1. Models of Sequential Systems
Sungho Kang
Yonsei University

2. Outline Introduction to Finite State Machines
Synthesis of Finite State Machines
FSMs : Definition, Notation, and Examples
FSM Minimization of Completely Specified Machines
Graph Algorithms for FSM Traversal
Models of Sequential Systems
FSTs: Strings, Runs, Reachability and Products
FSM Equivalence Checking
Reachability Analysis
Symbolic FSM State Traversal

3. A Verbal FSM Specification
Introduction
The circuit receives a stream of bits, one per clock cycle, on input and at every clock cycle it indicates, on the three outputs Z2Z1Z0 the difference between the number of ones and the number of zeroes in the last three bits received.
The difference is positive if the number of ones exceeded the number of zeroes and it is negative otherwise. The difference is represented as a 2's complement number, with Z2 the sign bit and Z0 the least significant bit.
At reset, the circuit is assumed to have received an arbitrarily long string of zeroes, so that the output is -3.

4. Sign Bit 2’s Complement Assume -2n  i  2n. Then the 2’s complement
Introduction
Assume -2n  i  2n. Then the 2’s complement
representation of an integer i is the binary code for the integer j = 2n - |i|, augmented by an extra bit denoting the sign of i.
Examples:
Sign Bit

5. A Sequential Counter Circuit and Its STG
Introduction
Latches 0,1,2, use n=2.
Circuit
Truth Table
STG (State
Transition
Graph)

6. Systematic FSM Design Paradigm
Introduction

7. Synthesis of Practical FSMs
Synthesis of FSMs
We have learned basic methods for minimizing, encoding, checking equivalence, and synthesizing circuits for realizing completely specified FSMs
Now we must learn to deal with the more practical case of incomplete specification
Our goal is thus to find a least cost circuit that satisfies a partial specification

8. FSM Synthesis--State Encoding
Synthesis of FSM
K-map of
State set
4 = log29
s=(s1,s2,s3,s4)
Code Length
Encoding vector
Characteristic
function of set
Criteria: avoid 1s, promote adjacency

9. Encoding the Transition Functions
FSMs
Given: FSM specified as Cube Table
Find code length and encoding
Replace symbolic states with binary codes
For each row with a 1 in the i column, add the input cube to the formula for i (s).

10. Encoding: Simplification: 11 (or 10) literals x s s d d l 1 2 1 2 1 1
FSM Synthesis--State Encoding
FSMs
Encoding: x s s d d l 1 2 1 2 1 1 1 1 1 1 1 1 1 1 1 1
Simplification:
11 (or 10) literals

11. Exploiting Unreachable State Don’t Cares
FSMs
When there are unreachable states, the encodings of these states are don’t cares
These don’t cares can be used to minimize the logic of the next state and output functions
1’s in the output columns of the cube table are primary reason for “fat” logic--don’t cares minimize this effect

12. Encoding(2 1s) Encoding(4 1s) Conclusion: Avoid codes with 1s
FSM Synthesis--State Encoding
FSMs
Encoding(2 1s)
Encoding(4 1s) d = s s x + s s x + s s x , d = s + s s x ,
Conclusion:
Avoid codes
with 1s 1 1 2 1 2 1 2 2 1 1 2 l = s s x DC = s s ( 11
literals ) 1 2 1 2 d = s s x + s x + s x , d = s + x , l = s s x 1 1 2 1 2 2 1 1 2

13. State Sequences Output Sequences FSM State Sequences
M = < I, S, , S0, O, >
I : Input Alphabet
O : Output Alphabet
S : State Set
S0 : Initial State Set
 : Next State Transition Function
 : Output Function
State Sequences
Output Sequences

14. Formal FSM Specification
STG
Cube Table
Flow Table
M = < I, S, , S0, O, >
I={0,1}, S={S0, S1, S2}, S0={S1}, O={0,1}

15. State Minimization
FSM Minimization
Two FSMs are equivalent if their initial states are equivalent.
Two states are equivalent if there is no sequence of inputs, which, applied to both machines, produces an output difference
The task of replacing two or more equivalent states with a single representative state is called State Minimization

16. sequence of inputs to an FSM M = < I, S, , S0, O, >
Strings and Runs
FSM Minimization
A string is a finite
sequence of inputs to an FSM
M = < I, S, , S0, O, >
where produces the
, that is,
the sequence of states generated
by when starts in state .
Examples:
Are equivalent?

17. Output Sequences
FSM Minimization
Given an input sequence , the run leads to the corresponding output sequence
(From )
(From )

18. Two states and are distinguished by an input string if and only if the
Distinguishing Sequences
FSM Minimization
Two states and are distinguished by an input string if and only if the
corresponding runs lead to output sequences
and
which differ in the last output symbol.
Distinguishes
and

19. or (no distinguishing sequence exists).
Indistinguishable States
FSM Minimization
Two states and are indistinguishable if and only if any input sequence gives the same output sequence , irrespective of whether the FSM starts in state
or (no distinguishing sequence exists).

20. Merge Indistinguishable States
FSM Minimization
Merge

21. Indistinguishable States
FSM Minimization
Two states are indistinguishable if (but not only if) they have the same next state and the same output for all inputs
States and are
indistinguishable
Hence,
State goes to
state for both inputs

22. which differ only in the last output symbol.
Length-k Distinguishing Sequences
FSM Minimization
Two states and are k-distinguished by a length-k input string if and only if the corresponding runs lead to output strings and
which differ only in the last output symbol.
String
3-Distinguishes
and

23. In an n-state FSM ( ), and are equivalent if and only if they are
k-Equivalent States
FSM Minimization
If and are not k-distinguishable (no k-distinguishing sequence exists), they are k-equivalent ( ).
In an n-state FSM ( ), and are
equivalent if and only if they are
n-equivalent ( ).

24. Equivalent States
FSM Minimization
Because all runs from these states either go back and forth from to , outputting a 0, or converge at
Because all runs from these states either go to equivalent states and , outputting a 1, or converge at

25. For any string , the output strings for runs starting
Equivalent FSMs
FSM Minimization
For any string , the output strings for runs starting
from and will be identical.

26. where and are the -successors of and
Theorem
FSM Minimization
if and only if , and ,
where and are the -successors of
and u
Proof (p266): s 1 v
Take as , where
both and are arbitrary t w 1

27. is an equivalence relation (reflexive, symmetric, and transitive).
Equivalence Classes
FSM Minimization
The binary relation
is an equivalence relation (reflexive, symmetric,
and transitive).
We denote the equivalence classes of by

28. Equivalence Classes
FSM Minimization

29. Refinement (Meet) of Two Partitions
FSM Minimization
The “Meet” of two partitions is just the Cartesian Product (next slide)
Example:

30. Refinement (Meet) of Partitions
FSM Minimization
Let and denote the partition into equivalence
classes induced by inputs and
Note the Cartesian Product element
is dropped (by the absorption)

31. Finding Equivalent States
FSM Minimization
Find equivalence class partition
of 1-equivalent states
Extend to equivalence class
partition of (k+1)-equivalent
states
Return partition and length of
longest distinguishing sequence

32. We will use the following example Flow Table
Equivalence Classes
FSM Minimization
We will use the following example
Flow Table
STG

33. Finding Equivalent States
FSM Minimization
Start with maximum partition: all states
in a single block
Flow Table

34. The equivalence classes of are
FSM Minimization
The equivalence classes of are

35. P Finding Equivalent States = { A , C , E }, t = ( E , E , C ), b = (
FSM Minimization
Extend to equivalence class
partition of (k+1)-equivalent
states P 2 = { A , C , E }, t = ( E , E , C ), 1 b = ( 1 , 1 , 1 ), Pb = ( A , C , E )

36. x (k+1)-Equivalence Classes ( ) k + 1
FSM Minimization
Note Cartesian product of set intersections

37. +
(k+1)-Equivalence Classes ( ) x k 1
FSM Minimization

38. Fanin cone (logic cone) Maximal completely connected subgraph
Graph Algorithms
FSM Traversal
Subgraph
Connected subgraph
Fanin cone (logic cone)
Maximal completely connected subgraph
If G is undirected, Clique
Maximal
if it is not a proper subgraph of another subgraph
Strongly connected subgraph
maximal subgraph for which every pair of included vertices lies on a cycle

39. States in reached in one step from Note Breadth First Search
FSM Traversal
States in
reached in one
step from
Note

40. BFS is modified for use in FSM analysis:
BFS For FSMs
FSM Traversal
BFS is modified for use in FSM analysis:
Trace-Back Method to identify shortest paths
BDD data structures for representing large state sets
Length of the longest shortest path from an initial state to any of the reachable states is called the sequential depth of the machine

41. DFS is most useful in detecting cycles of a directed graph.
Depth-First Search
FSM Traversal
Unlike BFS, DFS prioritizes depth of penetration, rather than exhausting the current locale before proceeding.
DFS proceeds recursively, visiting all transitive successors of a vertex, before completing the DFS of the current active vertex.
DFS is most useful in detecting cycles of a directed graph.

42. Earliest in cycle (later)
Depth First Search
FSM Traversal
When started
When completed
Earliest in cycle (later)

43. Procedure DFS(u) { Active Vertex Scan successors
Depth First Search
FSM Traversal
Active Vertex
Scan successors
Procedure DFS(u) {
kpre=kpre+1; preorder[u]=kpre
foreach ( ) {
if(preorder [a]=0 DFS(a). }
kpost=kpost+1; postorder[u]=kpost
When Started When Completed
Recur if unsearched

44. Strongly Connected Components
FSM Traversal
An SCC (Strongly Connected Component) of a directed graph is a subgraph which is maximal with respect to the following property:
“There is a path from every vertex to every other vertex, and back”.
That is, “every pair of vertices lies on a cycle”.

45. Strongly Connected Components
FSM Traversal
When started
When completed
Earliest (preorder index) in cycle

46. Shortest Paths Procedure SHORTEST_PATH(V,E,L,v0) {
FSM Traversal
Procedure SHORTEST_PATH(V,E,L,v0) {
1 for(v  V) v= ; Reached = 
2 S = {v0}; v0=0
3 while (S  ) {
4 * = minsS {v}
V* = {vV | v = *}
5 v = SELECT1(V*)
6 S=S-{v}
Reached = Reached  {v}
7 for (a  ADJ(V,E,v)) {
if (a  Reached ) {
9 S = S  {a}
10 a = min(a, v + Lv,a ) }
11 return ()

47. Models of Sequential Systems
FST (Finite State Transition Structures)
In a given state, FSTs receive an input symbol, and make a transition to a new state
FAs (Finite Automata)
are FSTs which also take notice when a favorable state (called accepting) is entered
FSMs (Finite State Machine)
are FSTs which emit a specified output symbol when they make a transition from one state to another
Regular Languages
Languages are just sets of sequences of input or output symbols (called strings)
Regular languages are just the kind of sets of strings that can be accepted by FAs or generated by FSMs
DFA (Deterministic Finite Automata)
NFA (Nondeterministic Finite Automata)

48. Finite State Transition Structures
FSTs
An FST is defined as a 4-tuple  = <X,S,,S0>
X is the input alphabet
S is the state set or state alphabet
 : S x X  2S is the next state transition function
S0  S is a set of initial states
Definition
FST  = <X,S,,S0> is a deterministic if the image of all pairs (x,s) is a singleton next state
In this case, the specified  does not map any given state into a set of two or more states
If in addition the initial state set S0 of  is a singleton set,  is said to be strongly deterministic
If the mapping  is specified for all pairs of states and input symbols,  is said to be complete

49. Finite State Transition Structures
FSTs
NFAs and -moves
-moves, which represent the further conceptual ability of an NFA to be in more than one state at a time
The concept of -moves is important in the procedure for deciding whether a specified string is accepted by a given FA and for constructing a DFA which accepts the same language as a given NFA
FSTs as labeled diagraphs
Sometimes an FST  = <X,S,,S0> is specified by giving its STG : G=(S,E,L)
The vertices s  S correspond to the states of 
The edge (s,t)  E are directed from s to t and signify a possible state transition (s,t)  E  t = (s,x)
For each edge s, t E of G L(s,t) ={x  X| t  (s,x)} = {x  X| t = (s,x)}

50. Finite State Transition Structures
FSTs
Strings, Tapes and Runs
A run of an FST is a sequence of states which starts in an initial state S0 occurs in response to some possible input sequence
A sequence of inputs is called a string if it is finite and a tape if it is infinite

51. Traversing the Product STG
Equivalence Checking
Equivalence checking requires testing output pairs at every reachable state
A state t is reachable from state s if there exists a string x which produces a run s, ending in state t
The act of identifying the set of reachable states of an FSM is called FSM Traversal
In Traversal, we systematically search the STG from the initial state. Any method can be used but the most efficient is based on BFS (Breadth First Search)

52. Building the Product Machine
Equivalence Checking
Check this condition for
all reachable product states
Comparator
FST ( )
PRODUCT
Common input

53. Building the Product (STG View)
Equivalence Checking
Note output of product machine is 1 for all inputs at
all reachable states

54. Start by finding states and reachable from initial state of product .
Building the Product (STG View)
Equivalence Checking
Start by finding states and reachable from
initial state of product

55. Continue by finding state reachable from
Building the Product (STG View)
Equivalence Checking
Continue by finding state reachable from
previous new state of product

56. Specific to FSMs Equivalence Checking Product Machine Same as BFS

57. Finding a Shortest Error Trace
Equivalence Checking
Returned: run s, which
starts at initial state and
ends at error state.

58. BFS for Error Trace Suppose all outputs were 1 except on edge from
Equivalence Checking
Suppose all outputs were 1 except on edge from
s2 to s1. Then a shortest error trace would be
s=(s5,s4,s1,s2), x=(0,1,0,0)

59. Isomorphism
Equivalence Checking
Two graphs Ga = (Va, Ea) and Gb = (Vb, Eb) are said to be isomorphic if |Va| = |Vb| if the nodes of Va can be relabeled so that the two graphs are identical
A relabeling function is a function  : Va  Vb such that for each node vb  Vb there is exactly one node va  Va such that vb = (va)
Such a function is said to be 1 to 1 and onto and is called an isomorphism

60. BDD Representation of Reached Sets
Reachability Analysis
FSM Traversal requires storage of various
reachable states sets ( , etc.)
For a circuit with 64 latches, there could be
as many as reachable states
It is clearly infeasible to store an “int” to
identify each individual state
Thus BDDs must be used to represent sets
of states

61. BDD Representation of State Sets
Reachability Analysis
Encoding:
(codes packed around origin)

62. BDD Representation of State Sets
Reachability Analysis
Minterm count = path
count (3 paths for all n)
Missing variables
amplify minterm count

63. BDD Representation of State Sets
Reachability Analysis

64. Symbolic FSM State Traversal
Symbolic Traversal
Image
The set of co-domain points
Preimage
Transition relations
Given a deterministic transition function (s,x), T(s,x,t) = i=ni=1 (ti i(s,x))
T(s,x,t)=1 denotes a set of encoded triples s,x,t of state s, input x and x-successor t of s, each representing a transition in the FST if the given FSM
Existential abstraction
Given an m-variable Boolean function f(x1, ,,, xm), xi f = fxi + fxi’ where fxi(fxi’) is positive(negative) cofactor
I(t) = Img(T,C) = xsC(s) T(s,x,t)