Outline

1
Outline

• Composition, Conformance
• Topologies
• Proof of solution
• Node flexibilities
• Examples
• Node minimization
• Windowing
• C-progressive

2
Composition

• Synchronous
• Asynchronous (parallel)
• Mapping Asynchronous to Synchronous

3

• Composition (synthronous or asynchronous)
  involves two steps
• Make the two machines have the same input
  alphabet (support)
• Perform the product

• Synchronous (changing the support)
• Projection Given a language L over the alphabet
  projection is defined as
• Lifting given a language L over the alphabet X,
  lifting to the alphabet is defined as
  

• Asynchronous (changing the support)
• Restriction Given a language L over the
  alphabet , the restriction to X is
  defined as
  where
• Expansion given a language L over X, lifting to
  the alphabet is defined as

4
Mapping Parallel into Synchronous Suppose F is a
FSM on inputs i,v and outputs u and S is an FSM
on inputs i and outputs o.
u
S
F
X
o
o
i
i
v
The semantics are that when an input comes into a
module, it takes an unspecified amount of time
for the module to produce an output. This will be
modeled with a non-deterministic self-loop
labeled with
5
Transitions of F (as an FSM) are one of the forms
(s i/u s) or (s v/u s). For S, its
transitions are of the type (q i/o q). For each,
we convert into automata by creating new
intermediate states between inputs and outputs.
Thus a transition (s i/u s) becomes two
transitions (s i s) (s u s).
Similarly for the others. For S a transition of
the type (q i/o q) becomes (q i q) (q o q).
i/u
s
s
s
q
i
u
i
o
s
s
q
q
6
We want to lift up the language of F to include o
and arbitrary delays so that iuvu becomes for
example And S to include u and v and arbitrary
delays, so that the two languages can contain
similar strings. io becomes for example In
particular, they both can become the same thing
The common symbols act to synchronization
This is done on the automaton F by the following
i1/u1
v1/u2
becomes
Note the alphabet is
7
Similarly for S
i1/o1
becomes
i1
o1
i1
Note that all states are
deterministic since the alphabet is
.
With these conversions, we can do synchronous
composition and get the equivalent expanded
result of parallel composition
Thus we need to implement only one type of
compositional method synchronous, and simply
have a mapping of each machine into its extended
machine to compose in parallel. Finally, we can
take the synchronous solution and map it back
into an FSM.
8
Conformance Simulation Relation (
) Let
and
be two automata over the same alphabet
simulates
(
) if there exists a simulation relation
such that
implies that
Note that
simulates
but these are not equivalent notions. If may be
easier to find a simulation relation than to
prove language containment.
9
Language Containment ( )
To show that
) we typically show that
(i.e.
This requires complementing
which may be hard if
is non-deterministic (subset
construction). So simulation may be easier to
check.
Use of a simulation relation instead of language
containment can allow avoidance of computing
in the construction of . Note that if S
is deterministic or small, then there is no
motivation to avoid computing so using
language containment is fine.

10
Language Equality ( )
Proposition If two FSMs, F1 and F2, and F2 is
deterministic, then
Hence, if we are solving and S is
a deterministic FSM then is the
MSG.
If S is ND, then what is MSG?
Comment Clearly, we need in order
for there to be a solution of .
This requires that supp(S) supp(F), since
otherwise there is a variable v in F but not in
S. Then would be too large.
11
Topologies
12
(No Transcript)
13
(No Transcript)
14
(No Transcript)
15
Communicating the internal state
O1
O1
F
F
I1
I1
cs
U1
U2
U1
U2
O2
O2
X
X
I2
I2
latch_expose
16
Hiding only the outputs
O
I
F
U
V
X
17
Hiding the outputs only
iuvo1
The only way it could become non-deterministic
after hiding o is if iuv iuv. But then o1
o2 which means that the product machine was ND.
iuvo2
18
Solving a language equation Solve
and
where
In particular, find the largest solution X (most
general solution). Theorem A Let A and C be
languages over alphabets
and
respectively. For the equation,
the most general solution is
Theorem B Let A and C be languages over
alphabets
and
respectively. For the equation,
the most general solution is
.
19
Proof We prove Theorem A. Let
. Then
means that
A
I
U
O
X
Thus
is the largest solution of
The proof of Theorem B is similar.
20
Computing the CF for a node -global step
21
Computing CF - local step
22
CF
23
FSM networks computing complete sequential
flexibility (CSF)
Specification S (i,o) Context C
(i,v,u,o) Unknown X (u,v)
Problem Given S and C , find the Most General
Solution (MGS) of
24
FSM Networks
The most general solution (MGS) of
is
In general, MGS is deterministic automaton but
as an FSM it is non-deterministic (NDFSM)
25
Complete Sequential Flexibility (CSF)

• CSF is maximum sub-behavior in MGS which is
  prefix closed and u-progressive.
• For unknown to be an FSM, it must be progressive
  in its inputs

26
Comparison with combinational case
Sequential
Combinational
unknown
27
Extending CF
FSM
O
I
X
u
v
Combinational sub-block
Spec is IO behavior of FSM. Combinational block
is treated as unknown X with inputs u and outputs
v. We derive the CSF for X. It is different than
the CF where the spec is taken to be the
combinational behavior of the FSM, i.e. with
inputs I,CS and outputs O,NS. Also, if we
extract from X a maximum combinational subpart
(combinational projection), it is also different
that CF
28
Algorithm
Algorithm LanguageEquationSolving Input prefix
-closed deterministic S(i.o) and
C(i,v,u,o) Output most general prefix-closed,
progressive X (FSM) begin 01 X Complete
( S, non-accepting ) 02 X DeterminizeCompleme
nt ( X ) 03 X Support (X, (i,v,u,o)) -
raise 03 X Product ( C, X ) 04 X Support
( X, (u,v) ) - hide 05 X Determinize
Complement ( X, u ) 06 return
PrefixProgressive (X ) end
Convert to FSM
29
Examples

• Games
• Nim
• Tic-tac-toe
• Toe-tac-tic
• Board
• Control
• Wolf, goat, cabbage
• Latch splitting

30
Example Coin Game (NIM)
Context describes the state of the game and legal
moves. Its input is random moves by first player
and its output tells if the game is in a losing,
winning or continuing state.
Specification is a 3-state automaton, playing,
won, and lost.

1. Players alternate turns
2. On each turn, player can take 1-n coins from any
   one pile
3. Player who takes last coin loses

Winning strategy Give your opponent a pile of
coins with even number of 1s in bit columns
(except at end)
6 1 1 0 5 1 0 1 3 0 1 1 ____ 2
2 2
Example 6 5 3
31
NIM
.model game-piles .inputs p1 d1 p2 d2 .outputs
out .mv p1,p2,p,pt,ptt 3 .mv d1,d2,d 7 .mv
cs0,cs1,cs2,ns0,ns1,ns2,nh,h 7 .mv
whoseturn,whoseturn1 2 1 2 .mv out 3 OK notOK
done .latch ns0 cs0 .reset cs0 3 .latch ns1
cs1 .reset cs1 2 .latch ns2 cs2 .reset cs2
1 .latch whoseturn1 whoseturn .reset
whoseturn 1 set this to 2 if Player 2 goes
first. .table whoseturn whoseturn1 .default 2 2
1 .table whoseturn d1 d2 d 1 - - d1 2 - -
d2 .table whoseturn p1 p2 ptt 1 - - p1 2 - -
p2
spec.mva
.model spec .inputs out .outputs Acc .mv out 3 OK
notOK done .mv CS,NS 3 a b c .table CS
-gtAcc .default 1 b 0 .table out CS -gtNS OK a
a notOK a b done a c - b b - c c .latch NS
CS .reset CS a .end
32
Map move into a legal move .table ptt cs0 cs1
cs2 pt 0 (1,2,3,4,5,6) - - 0 1 - (1,2,3,4,5,6) -
1 2 - - (1,2,3,4,5,6) 2 0 0 - - 1 1 - 0 - 2 2 - -
0 0 .table pt cs0 cs1 cs2 p 0 (1,2,3,4,5,6) - -
0 1 - (1,2,3,4,5,6) - 1 2 - - (1,2,3,4,5,6) 2 0 0
- - 1 1 - 0 - 2 2 - - 0 0 selects the height of
the pile chosen by player 1 .table p cs0 cs1 cs2
h 0 - - - cs0 1 - - - cs1 2 - - -
cs2 computes h-d. If hltd then 0 .table h d
nh .default 0 6 1 5 6 2 4 6 3 3 6 4 2 6 5 1 5 1
4 5 2 3 5 3 2 5 4 1 4 3 1 4 2 2 4 1 3 3 2 1 3 1
2 2 1 1
The next state ns is due to the move. .table p
nh cs0 ns0 0 - - nh (1,2) - - cs0 .table p nh
cs1 ns1 1 - - nh (0,2) - - cs1 .table p nh cs2
ns2 2 - - nh (0,1) - - cs2 "out" indicates
who the winner is. .table whoseturn ns0 ns1 ns2
out .default OK 1 0 0 0 done 2 0 0 0 notOK .end
33
Lang.script (NIM)
rl fixed.mv stg_extract fixed.mva echo "Synthesis
..." determinize -ci spec.mva spec_dci.mva support
p2(3),d2(7),p1(3),d1(7),out(3) spec_dci.mva
spec_dci_supp.mva support p2(3),d2(7),p1(3),d1(7),
out(3) fixed.mva fixed_supp.mva product
fixed_supp.mva spec_dci_supp.mva p.mva support
p1(3),d1(7),p2(3),d2(7) p.mva p_supp.mva determini
ze -ci p_supp.mva p_dci.mva prefix p_dci.mva
p_dci_pre.mva progressive -i 2 p_dci_pre.mva
x.mva minimize x.mva x-min.mva prefix x-min.mva
x-min.mva echo "Verification ..." support
p2(3),d2(7),p1(3),d1(7),out(3) x.mva
x_supp.mva product x_supp.mva fixed_supp.mva
prod.mva support p2(3),d2(7),p1(3),d1(7),out(3)
spec.mva spec_supp.mva check prod.mva
spec_supp.mva
34
mvsis 02gt source lang.script The STG with 40
states and 110 transitions is written to file
"fixed.mva". Synthesis ... The automaton is
deterministic determinization is not
performed. Product (40 st, 110 trans) x (3 st, 5
trans) -gt (42 st, 112 trans) The automaton is
deterministic determinization is not
performed. Warning The automaton has been
completed before state minimization. State
minimization (22 states, 45 trans) -gt (13
states, 30 trans) Verification ... Product (21
st, 34 trans) x (40 st, 110 trans) -gt (21 st, 34
trans) Warning Automaton "game-pilesspecgame-pi
les" is completed before checking. The behavior
of "game-pilesspecgame-piles" is contained in
the behavior of "spec". mvsis 03gt psa
x-min.mva "game-pilesspec" incomplete (9 st),
deterministic, non-progressive (9 st), and
non-M oore (9 st). 4 inputs (4 FSM inputs) 12
states (12 accepting) 19 trans Inputs
p1(3),d1(7),p2(3),d2(7) mvsis 03gt
35
Example of CSF computation NDFSM represented as
automaton
Inputs p1(3),d1(7) Outputs p2(3),d2(7)
36
Tic-tac-toe
spec.mva
.model spec .inputs out .outputs Acc .mv out 3
.mv CS,NS 3 a b c .table CS -gtAcc .default 1 b
0 .table out CS -gtNS 0 a a 1 a b 2 a c - b b - c
c .latch NS CS .reset CS a .end
spec.mv
.model spec .inputs m1 c1 m2 c2 .outputs out .mv
m1,m2 9 .mv out 3 .table -gtout 0 2 .end
37
.model game-tic-tac-toe .inputs m1 m2 .outputs
out .mv out 3 .mv m1,m2,m 9 .mv
cs0,cs1,cs2,cs3,cs4,cs5,cs6,cs7,cs8 3 .mv
ns0,ns1,ns2,ns3,ns4,ns5,ns6,ns7,ns8 3 .mv
whoseturn,whoseturn1 2 1 2 .latch whoseturn1
whoseturn .reset whoseturn 1 .latch ns0
cs0 .reset cs0 0 .latch ns1 cs1 .reset cs1
0 .latch ns2 cs2 .reset cs2 0 .latch ns3 cs3
.reset cs3 0 .latch ns4 cs4 .reset cs4 2
set this to 0 if player 2 makes the second
move. .latch ns5 cs5 .reset cs5 0 .latch ns6
cs6 .reset cs6 0 .latch ns7 cs7 .reset cs7
0 .latch ns8 cs8 .reset cs8 0
.table illegal whoseturn whoseturn1 0 2 1 0 1 2 1
- whoseturn .table whoseturn m1 m2 m 1 - - m1 2
- - m2 Player makes a illegal move if the
square indicated by mt in not empty .table m cs0
cs1 cs2 cs3 cs4 cs5 cs6 cs7 cs8 illegal .default
1 0 0 - - - - - - - - 0 1 - 0 - - - - - - - 0 2 -
- 0 - - - - - - 0 3 - - - 0 - - - - - 0 4 - - - -
0 - - - - 0 5 - - - - - 0 - - - 0 6 - - - - - - 0
- - 0 7 - - - - - - - 0 - 0 8 - - - - - - - - 0
0 out records if there is a line of 2's (then
out2) or a line of 1's (then out1) .table
whoseturn ns0 ns1 ns2 ns3 ns4 ns5 ns6 ns7 ns8
out .default 0 2 - - 2 - 2 - 2 - - 2 2 2 2 2 - -
- - - - 2 2 2 - - 2 - - 2 - - 2 2 2 - - - 2 - -
- 2 2 2 - - - 2 2 2 - - - 2 2 - - 2 - - 2 - - 2
2 2 - 2 - - 2 - - 2 - 2 2 - - - - - - 2 2 2
2 1 - - 1 - 1 - 1 - - 1 1 1 1 1 - - - - - - 1 1
1 - - 1 - - 1 - - 1 1 1 - - - 1 - - - 1 1 1 - -
- 1 1 1 - - - 1 1 - - 1 - - 1 - - 1 1 1 - 1 - -
1 - - 1 - 1 1 - - - - - - 1 1 1 1
38
.table illegal done cs3 m whoseturn ns3 .default
0 0 1 - - - cs3 0 0 - 3 1 1 0 0 - 3 2 2 0 0 -
(0,1,2,4,5,6,7,8) - cs3 1 - - - - cs3 .table
illegal done cs4 m whoseturn ns4 .default 0 0 1
- - - cs4 0 0 - 4 1 1 0 0 - 4 2 2 0 0 -
(0,1,2,3,5,6,7,8) - cs4 1 - - - - cs4 .table
illegal done cs5 m whoseturn ns5 .default 0 0 1
- - - cs5 0 0 - 5 1 1 0 0 - 5 2 2 0 0 -
(0,1,2,3,4,6,7,8) - cs5 1 - - - - cs5 .table
illegal done cs6 m whoseturn ns6 .default 0 0 1
- - - cs6 0 0 - 6 1 1 0 0 - 6 2 2 0 0 -
(0,1,2,3,4,5,7,8) - cs6 1 - - - - cs6 .table
illegal done cs7 m whoseturn ns7 .default 0 0 1
- - - cs7 0 0 - 7 1 1 0 0 - 7 2 2 0 0 -
(0,1,2,3,4,5,6,8) - cs7 1 - - - - cs7 .table
illegal done cs8 m whoseturn ns8 .default 0 0 1
- - - cs8 0 0 - 8 1 1 0 0 - 8 2 2 0 0 -
(0,1,2,3,4,5,6,7) - cs8 1 - - - - cs8 .end
Once the game gets into a winning
configuration, do not change the state. .table
cs0 cs1 cs2 cs3 cs4 cs5 cs6 cs7 cs8 done .default
0 - - 2 - 2 - 2 - - 1 2 2 2 - - - - - - 1 2 - -
2 - - 2 - - 1 2 - - - 2 - - - 2 1 - - - 2 2 2 -
- - 1 - - 2 - - 2 - - 2 1 - 2 - - 2 - - 2 -
1 - - - - - - 2 2 2 1 - - 1 - 1 - 1 - - 1 1 1 1
- - - - - - 1 1 - - 1 - - 1 - - 1 1 - - - 1 - -
- 1 1 - - - 1 1 1 - - - 1 - - 1 - - 1 - - 1
1 - 1 - - 1 - - 1 - 1 - - - - - - 1 1 1 1 If
there is a winner (done1) then the state remains
unchanged. Otherwise, if mi and whoseturn1,
then csi1. Similarly, if mi and whoseturn2,
then csi2 .table illegal done cs0 m whoseturn
ns0 .default 0 0 1 - - - cs0 0 0 - 0 1 1 0 0 -
0 2 2 0 0 - (1,2,3,4,5,6,7,8) - cs0 1 - - - -
cs0 .table illegal done cs1 m whoseturn
ns1 .default 0 0 1 - - - cs1 0 0 - 1 1 1 0 0 -
1 2 2 0 0 - (0,2,3,4,5,6,7,8) - cs1 1 - - - -
cs1 .table illegal done cs2 m whoseturn
ns2 .default 0 0 1 - - - cs2 0 0 - 2 1 1 0 0 -
2 2 2 0 0 - (0,1,3,4,5,6,7,8) - cs2 1 - - - -
cs2
39
Lang.script (tic-tac-toe)
rl fixed1.mv latch_expose stg_extract
fixed.mva echo "Synthesis ..." determinize -ci
spec.mva spec_dci.mva support cs0(3),cs1(3),cs2(3)
,cs3(3),cs4(3),cs5(3),cs6(3),cs7(3),cs8(3),
whoseturn(2),m1(9),m2(9),out(3) spec_dci.mva
spec_dci_supp.mva support cs0(3),cs1(3),cs2(3),cs3
(3),cs4(3),cs5(3),cs6(3),cs7(3),cs8(3),
whoseturn(2),m1(9),m2(9),out(3) fixed.mva
fixed_supp.mva product fixed_supp.mva
spec_dci_supp.mva p.mva support
cs0(3),cs1(3),cs2(3),cs3(3),cs4(3),cs5(3),cs6(3),c
s7(3),cs8(3), whoseturn(2),m2(9) p.mva
p_supp.mva determinize -ci p_supp.mva
p_dci.mva prefix p_dci.mva p_dci_pre.mva progressi
ve -i 10 p_dci_pre.mva x.mva minimize x.mva
x-min.mva prefix x.mva x-min.mva
40
Wolf, goat, cabbage
.table stop bank in1 csw nsw 0 left
(empty,goat,cabbage) boat left 0 left wolfe
(left,boat) boat 0 right (empty,goat,cabbage)
boat right 0 right wolfe (right,boat) boat 0 -
(empty,goat,cabbage) (left,right) csw 1 - - -
csw .table stop bank in1 csg nsg 0 left
(empty,wolfe,cabbage) boat left 0 left goat
(left,boat) boat 0 right (empty,wolfe,cabbage)
boat right 0 right goat (right,boat) boat 0 -
(empty,wolfe,cabbage) (left,right) csg 1 - - -
csg .table stop bank in1 csc nsc 0 left
(empty,goat,wolfe) boat left 0 left cabbage
(left,boat) boat 0 right (empty,goat,wolfe) boat
right 0 right cabbage (right,boat) boat 0 -
(empty,goat,wolfe) (left,right) csc 1 - - -
csc .table bank nsw nsg nsc out .default
OK right left left - notOK left right right -
notOK right - left left notOK left - right right
notOK right (right,boat) (right,boat)
(right,boat) done map input (in) into any legal
input .table bank in csw csg csc in1 .default
empty right wolfe (right,boat) - - in left wolfe
(left,boat) - - in right goat - (right,boat) -
in left goat - (left,boat) - in right cabbage -
- (right,boat) in left cabbage - - (left,boat)
in .end
.model wolfe .inputs in .outputs out .mv in,in1 4
empty wolfe goat cabbage .mv csw,csg,csc,nsw,nsg,n
sc 3 left right boat .mv bank,bank1 2 left
right .mv out 3 OK notOK done .latch stop1
stop .reset stop 0 .latch nsw csw .reset csw
left .latch nsg csg .reset csg left .latch nsc
csc .reset csc left .latch bank1 bank .reset
bank left .table out stop stop1 .default 0 done -
1 - 1 1 .table stop bank bank1 .default left 0
left right 1 - bank
out
in(boat)
41
spec.mva for wolf-goat-cabbage
lang.script
.model spec .inputs out .outputs Acc .mv out 3 OK
notOK done .mv CS,NS 3 a b c .table CS
-gtAcc .default 1 b 0 .table out CS -gtNS OK a
a notOK a b done a c - b b - c c .latch NS
CS .reset CS a .end
rl wolfe.mv stg_extract fixed.mva echo
"Synthesis ..." determinize -lci spec.mva
spec_dci.mva support in(4),out(3) spec_dci.mva
spec_dci_supp.mva support in(4),out(3) fixed.mva
fixed_supp.mva product -l fixed_supp.mva
spec_dci_supp.mva p.mva support in(4) p.mva
p_supp.mva determinize -lci p_supp.mva
p_dci.mva prefix p_dci.mva p_dci_pre.mva progressi
ve -i 0 p_dci_pre.mva x.mva minimize x.mva
x-min.mva prefix x-min.mva x-min.mva echo
"Verification ..." support in(4),out(3) x.mva
x_supp.mva product x_supp.mva fixed_supp.mva
prod.mva support in(4),out(3) spec.mva
spec_supp.mva check prod.mva spec_supp.mva
42
Wolf, goat, cabbagex.mva
43
Minimized x-min.mva
44
Other Games

• Toe-tac-tic (solvable)
• Like tic-tac-toe
• Except that any player can play either X or O at
  any time
• A player wins when he completes a line or either
  Xs or Os
• Board game (too many states)
• 4 x 4 board
• Each player has 4 pieces which initially at the
  top and bottom rows of the board.
• Any piece can move forward, left or right
• Player wins when he moves one of his pieces to
  the other side
• 12870 reachable states cant do it right now

45
Application - splitting FSM blif files
u
FSM
i
X
v
o
This is just a syntactic change. Nothing has been
done yet.
46
Latch split
mvsis 05gt _split -h Usage _split -v
ltlatch_listgt splits the current network S
into two parts F and X generates the
script to solve the equation F X S -v
toggles verbose default no ltlatch_listgt
the list of latches to be included in X
no spaces are allowed in the latch list
the numbers of latches are
zero-based for example
0,3,5-7,9 mvsis 05gt
47
Latch_split example
S
.model s27.bench .inputs G0 G1 G2 G3 .outputs
G17 .reset G5 0 .latch G10 G5 .reset G6 0 .latch
G11 G6 .reset G7 0 .latch G13 G7 .table G0 G1 G3
G5 G6 G7 G17 .default 0 1 1 - - - - 1 1 - 0 - -
- 1 1 - - - - 1 1 - - - 1 - - 1 - 1 - - 0 -
1 - - 0 - 0 - 1 - - - - 0 1 1 .table G0 G1 G3
G5 G7 G10 .default 0 1 1 - - - 1 1 - 0 - - 1 1
- - 1 - 1 1 - - - 1 1 .table G0 G1 G3 G5 G6 G7
G11 .default 0 0 - - 0 1 - 1 - 0 1 0 - 0
1 .table G1 G2 G7 G13 .default 0 1 0 - 1 - 0 1
1 .end
s27a (X)
s27f (F)
.model s27.bench .inputs G0 G1 G2 G3 G5
G6 .outputs G17 .latch G13 G7 0 .names G0 G1
G3 G5 G6 G7 G17 11---- 1 1-0--- 1 1----1 1 ---1--
1 -1--0- 1 --0-0- 1 ----01 1 .names G0 G1 G3 G5
G7 G10 11--- 1 1-0-- 1 1--1- 1 1---1 1 .names G0
G1 G3 G5 G6 G7 G11 0--01- 1 -010-0 1 .names G1 G2
G7 G13 10- 1 -01 1 .end
.model s27.bench .inputs G0 G1 G2 G3 G7 .outputs
G17 .latch G10 G5 0 .latch G11 G6 0 .names G0
G1 G3 G5 G6 G7 G17 11---- 1 1-0--- 1 1----1
1 ---1-- 1 -1--0- 1 --0-0- 1 ----01 1 .names G0
G1 G3 G5 G7 G10 11--- 1 1-0-- 1 1--1- 1 1---1
1 .names G0 G1 G3 G5 G6 G7 G11 0--01- 1 -010-0
1 .names G1 G2 G7 G13 10- 1 -01 1 .end
48
Language solving script generated by MVSIS
for sequential network "s27.blif" on Wed Feb 18
213553 2004 Command line was "split
0,1". echo "Solving the language equation ...
" solve s27f.blif s27.blif G0,G1,G2,G3,G7 G5,G6
s27x.aut psa s27x.aut echo "Verifying the
containment of the known implementation ...
" read_blif s27a.blif latch_expose stg_extract
s27a.aut support G0,G1,G2,G3,G7,G5,G6 s27a.aut
s27a.aut check s27a.aut s27x.aut read_blif
s27.blif
49
s27x.aut
50
s27x-dcmin.aut
s27a.blif
G0, G1, G2, G3, G7, G5, G6
inputs
outputs
51
FSM networks - Node Minimization

• Given a NDFSM CSF, find the smallest FSM Y,
  such that Y is well-defined and

Y is called a reduction of CSF
52
State graph of X
It generally looks like
53
C-compatibility - dcmin

• Two states and are c-compatible if
  their care sets do not intersect, i.e. the care
  set of one is completely contained in the dont
  care set of the other.

54
A simple state reduction method-dcmin

• Let be the relation for
  the incomplete CSF X, and compute
• i.e. those states and inputs for which there
  exists a next state and output (the next state
  can be either accepting or not).  
• Order this BDD with the u variables first, and
  let be the unique functions below the
  u variables pointed to.
• Two states and are c-compatible if
  and only if for all i,
  i.e. they have no minterm u in
  common.
• So is a clique of states that can't be
  merged, i.e. are not c-compatible and must have
  different colors.
• Then the c-incompatibility graph is
  which has to be
  colored.
• Suppose is the assignment of states
  s  to colors c. The new automaton relation for X
  is then
•    

55
Simple state reduction
Note that this is a simple coloring problem in
contrast to the compatibilities problem normally
associated with state minimization for
incompletely specified FSMs. In contrast, here a
group of states is c-compatible iff they are
pair-wise c-compatible.
56
Other ideas on reduction of CSF

• This problem is similar to SOP minimization when
  using CF to minimize the node in the
  combinational network.
• Many cost functions are possible. If we try to
  minimize the number of states in CSF, it is the
  problem of minimizing a PNDFSM
• T. Kam et. al., DAC 1994.
• We might want to look for a good implementation
  directly, rather than first minimizing the number
  of states.
• Similarly, for a node in the combinational
  circuit, looking for a small SOP, or the minimum
  number of literals in FF, may be misleading.
• A specialized algorithm has been developed to
  check whether a combinational solution (a
  single-state reduction) exists.
• The problem is reduced to SAT with as many
  variables as there are states transitions in
  the CSF. Solution is practical for, say, 100
  states and 500 transitions.
• A similar algorithm can be developed to check
  whether a 2 or 3 state solution exists
• more variables, the SAT problem is harder

57
Iterative language solving

• The problem of computing the CSF can be
  iterative.
• Given F and S
• Split F into F1 and F2
• Solve F1 X S.
• If we can reduce X to a smaller implementation
  than F2, replace F2
• Solve F2 X S
• If we can reduce X to a smaller implementation
  than F1, replace F1
• Set F F1 F2
• If either F1 or F2 has changed, go to 2

58
FSM Windowing
X1
FSM1
i
X2
X
FSM2
FSM3
X3
X X1 X2 X3
59
CompositionallyProgressive

• X should be compositionally progressive
  (c-progressive) with F
• i.e. for every product state cs of X F, the
  next state ns and output o should be defined for
  all i.
• Roland Jiang has proposed a way to use this to
  additionally trim the solution X during the
  subset construction. But he is not 100 sure it
  is right.
• Nina and Tiziano have another method for trimming
  and have proved that the largest c-progressive
  solution can contain well-defined FSM
  sub-behaviors that are not c-progressive.
• Roland has demonstrated that the above paper is
  wrong.
• Being c-progressive does not necessarily imply no
  combinational loops
• To hear a more detailed discussion, attend MVSIS
  weekly meeting Friday, 11-1pm in DOP center
  library (fishbowl)
• There might be a connection here with
  omega-automata.

60
Future developments

• Objective is to push to the limit, the size of
  application that can be done
• Keep multi-level MV structure, given in MVSIS, as
  long as possible (lecture on this later)
• Use SAT in subset construction
• The bottleneck looks to be extracting good
  sub-behavior of CSF (reduction)
• A sub-graph of the CSF usually not good enough
• Simplified (dcmin) state minimization of CSF
  may be good first step?
• Try for a good sub-behavior more directly without
  constructing CSF
• Try hierarchy and windowing applied to FSM network