Radu Grosu SUNY at Stony Brook Finite Automata as Linear Systems Observability, Reachability and More.
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Radu Grosu SUNY at Stony Brook
Finite Automata as Linear Systems
Observability, Reachability and More
• HSCC Conference: a witness of the fascinating
- convergence between control and automata theory.
• Hybrid Automata: an outcome of this convergence
- modeling formalism for systems exhibiting both discrete and continuous behavior,
- successfully used to model and analyze embedded and biological systems.
Convergence of Theories
Lack of Common Foundation for HA
• Mode dynamics:
- Linear system (LS)
• Mode switching:
- Finite automaton (FA)
• Different techniques:
- LS reduction
- FA minimization
Stimulated
UVs v
UVv
EVv
0 /RV tv
R
x Ax Bu
v
v V
Cx
/ di ts
volt
age(
mv)
time(ms)
• LS & FA taught separately: No common foundation!
• Finite automata can be conveniently regarded as time invariant linear systems over semimodules:
- linear systems techniques generalize to automata
• Examples of such techniques include:
- linear transformations of automata,
- minimization and determinization of automata as observability and reachability reductions
-“Z”-transform of automata to compute associated regular expression through Gaussian elimination.
Main Conjecture of this Talk
Finite Automata as Linear Systems
,states X input alphab - finite set of finite et
- transition relation X X,
- starting final staand sets o tes
,
S,
f
F X
Consider a fin M = (X, , ,ite automaton wit,F) h:S
Finite Automata as Linear Systems
, - finite set of states X, finite input alphabet
- transition relation X X,
- starting and final sets of states S,F X
Consider a finite automaton M = (X, ,
For each inpu
,S,F) with:
,
:
a
(a) X X boolean matrix A(a)
A = A(a) a
- represent as a
- write wh a(x) =
e
r
e
t lette
1 x = a 0
r a
ifotherwise
Finite Automata as Linear Systems
M
t t0
t t x (n+1) = x x =
S
y = x C = F
Now define the linear sy L = [S,A,C]:
(n)A,(n)
stem
(n)C,
Finite Automata as Linear Systems
M
t t0
t t
x (n+1) = x x = S
y = x C = F
Now define the linear system L = [S,A,C]:
(n)A,(
Example: consider following
n
a
) (n)C,
utomaton:
x3 x2x1
a
ab
b0
0 1 0 1A(a) = 0 1 0 , x ( ) = 0
0 0 0 0
0 0 1 0A(b) = 0 0 0 , C( ) = 1
0 0 1 1
L1
Polynomials and their Operations
*strings in (the i - powers:
- coefficien
nput strings)
matrices and vectors over B
s t :
A, C, x(n) and y(n) are polynomials w
ith:
Polynomials and their Operations
* - powers: strings in (the input strings)
- coefficients: matrices and vectors over B
(A(a
Addition and multipli
A, C, x(n) and y(n) are polynomials
done over polynomials
with:
cation:
ˆ
2
A(a)A(a)aa + A(a)A(b)ab + A(b)A(a)ba + A(b)A(
)a + A(b)b) =
A(aa)aa + A(ab)ab + A(ba
)ba + A
b)
(b
bb =
b)bb
Boolean Semimodules
- (B,+,0) commutative idempotent monoid (ois a
is
r),
- (B, ,1) commutative i a
distr
dempotent monoid (and),
- mult
ibutes oiplication v
aer
B is a doubly idempotent, commutative se
miring:
is an
ddition,
- 0 annihilator: 0 a = 0
Boolean Semimodules
- (B,+,0) is a commutative idempotent monoid (or),
- (B, ,1) is a commutative idempotent monoid (and),
- multiplication distributes over a
B is a doubly idempotent, commutative semiring:
n
- r(x+y) = rx + ry, (r+s)x = rx
ddition,
- 0 is an annihilator: 0 a
+ sx, (rs)x =
- 1x = x, 0x =
= 0
0
r(sx)
,
B is a semimodule over scalars in B:
Boolean Semimodules
- (B,+,0) is a commutative idempotent monoid (or),
- (B, ,1) is a commutative idempotent monoid (and),
- multiplication distributes over a
B is a doubly idempotent, commutative semiring:
n
ddition,
- 0 is an annihilator: 0 a = 0
- r(x+y) = rx + ry, (r+s)x = rx + sx, (rs)x = r(sx),
- 1x = x, 0x = 0
B is a semimodule over scalars in B:
additive multiplic No atiNo and ve inte: ver ! ses
Observability
t n
0
-1 t0 0[y(0) y(1) ... y(n-1)] = x C AC ... A C ] = x (1)
Let L = [S,A,C] be an n-state automaton. It's output:
L is observable if x is uniquely determined by
[ O
(1).
Observability
t n-1 t0 0
0
[y(0) y(1) ... y(n-1)] = x C AC ... A C] = x (1)
Let L = [S,A,C] be an n-state automaton. It's output:
[ O
L is observable if x is uniquely determin
Exampl t
ed by (
e: obserh v
1).
le abi i
n
1
2
3
1
b bb b ba aε a a a
x
A C
0 1 1 1 0 0 1 O =1 1 0 1 0 0 01 0 1 0
xx 0 0 1
ty matrix of O L is:
x3 x2x1
a
ab
b
L1
Linear Dependence
0
i0
and therefore has a single initial state, if all rows O in O are distinct
- if L is deterministic x is uniquely determine
d
Initial vector x selects a sum of rows from O. He
nce:
Linear Dependence
0
0 i
- if L is deterministic and therefore has a single initial state, x is uniquely determined if all rows O in
O
are
-
di
if
stinc
t
Initial vector x selects a sum of rows from O. He
nce:
i i
i I i J
i
0
i
i
iI,J [1..n]. I J
b an
O b O
d
a
and has several initial states, if t
L is nondeterministic x is n here are ot uniquely boolean a determined
:
(2)
,
Linear Dependence
0
0 i
- if L is deterministic and therefore has a single initial state, x is uniquely determined if all rows O in O are distinct
- if
Initial vector x selects a sum of rows from O. He
nce:
i i
i I i J
i
0
i
i i
b and
I,J [1..n]. I J a O b O
L is nondeterministic and has several initial states, x is not uniquely determined if there are boolean a :
(2)
Linea
,
r dep
- Def (2) ge linear dependence in vector spacneralizes es
(2) for finite I,J and any endence:
vector set.
Linear Dependence
0
0 i
- if L is deterministic and therefore has a single initial state, x is uniquely determined if all rows O in O are distinct
- if
Initial vector x selects a sum of rows from O. He
nce:
i i
i I i J
i
0
i
i i
b and
I,J [1..n]. I J a O b O
L is nondeterministic and has several initial states, x is not uniquely determined if there are boolean a :
(2)
Linea
,
r de
II,J [1..n]. I J span(O )
- Def (2) generalizes linear dependence in vector
is consequently:- Linear independence
spaces
(2) for finite I,J and any vector spendence:
et.
Jspan(O ) = {0}
Basis in Boolean Semimodule
(a) Y is independent, (b) span(Y) = X
An ordered set of vectors Y is a basis for X if:
Basis in Boolean Semimodule
n
(a) Y is independent, (b) span(Y) = X
An ordered set of vectors Y is a basis for X if:
If X B has a basis Theorem (Basis) Y then Y is un e. iqu
Basis in Boolean Semimodule
n
1
2
n
(a) Y is independent, (b) span(Y) = X
A C
0 1 1 1 0 0 1 O =1 1 0
a a b bε a b a b a bx x 1
An ordered set of vectors Y is a basis for X if:
Theorem (Basis) If X B has a basis Y then Y is unique.
3
1 2 3[x x x
0 0
]
01 0 1 0 0 0
:
x 1
row bas is
x3 x2x1
a
ab
b
L1
Basis in Boolean Semimodule
n
1
2
n
(a) Y is independent, (b) span(Y)
a a b bε a b a b aA C
0 1 1 1 0b
x 0 1 O =1 1 0x
= X
1
An ordered set of vectors Y is a basis for X if:
Theorem (Basis) If X B has a basis Y then Y is uniqu e.
3
1 2 3[x x x ]: [C( ) AC(a) AC(b)]:
0 0 01 0 1 0 0 0 1
x
row basis, column basis .
x3 x2x1
a
ab
b
L1
Observability Reduction by Rows
x1
ab
b
x2
a
x3
a
x4
ax5
a
b
bb
0
0 1 0 0 0 0 0 1 0 0 1 00 1 0 0 0 0 0 0 1 0 0 0
0 0 1 0 0 0 0 0 0 1 0 00
A(a) A(b)
0 1 0 0 0 0 0 0 1 0 10 1 0 0 0 0 0 0 1 0
x ( ) C( )
0 1
L2
Observability Reduction by Rows
x1
ab
b
x2
a
x3
a
x4
ax5
a
b
bb
0
0 1 0 0 0 0 0 1 0 0 1 00 1 0 0 0 0 0 0 1 0 0 0
0 0 1 0 0 0 0 0 0 1 0 00
A(a) A(b)
0 1 0 0 0 0 0 0 1 0 10 1 0 0 0 0 0 0 1 0
x ( ) C( )
0 1
1
2
3
4
5
O
0 0 1 1 1 1 1 10 1 1 1 1 1 1 10 1 1 1 1
a a b ba bε b a b a bb b b b b bxxxx
1 1 11 1 1 1 1 1 1 11 1 1 1 1 1x 1 1
L2
Observability Reduction by Rows
x1
ab
b
x2
a
x3
a
x4
ax5
a
b
bb
0
0 1 0 0 0 0 0 1 0 0 1 00 1 0 0 0 0 0 0 1 0 0 0
0 0 1 0 0 0 0 0 0 1 0 00
A(a) A(b)
0 1 0 0 0 0 0 0 1 0 10 1 0 0 0 0 0 0 1 0
x ( ) C( )
0 1
1
2
3
4
5
O
0 0 1 1 1 1 1 10 1 1 1 1 1 1 10 1 1 1 1
a a b ba bε b a b a bb b b b b bxxxx
1 1 11 1 1 1 1 1 1 11 1 1 1 1 1x 1 1
L2
Observability Reduction by Rows
x1
ab
b
x2
a
x3
a
x4
ax5
a
b
bb
0
0 1 0 0 0 0 0 1 0 0 1 00 1 0 0 0 0 0 0 1 0 0 0
0 0 1 0 0 0 0 0 0 1 0 00
A(a) A(b)
0 1 0 0 0 0 0 0 1 0 10 1 0 0 0 0 0 0 1 0
x ( ) C( )
0 1
1
2
3
4
5
O
0 0 1 1 1 1 1 10 1 1 1 1 1 1 10 1 1 1 1
a a b ba bε b a b a bb b b b b bxxxx
1 1 11 1 1 1 1 1 1 11 1 1 1 1 1x 1 1
t t
T
Define linear tran
1 0 00 1 00 1 00 0 10 0
x = x T:
1
sf
L2
Observability Reduction by Rows
x1
ab
b
x2
a
x3
a
x4
ax5
a
b
bb
0
0 1 0 0 0 0 0 1 0 0 1 00 1 0 0 0 0 0 0 1 0 0 0
0 0 1 0 0 0 0 0 0 1 0 00
A(a) A(b)
0 1 0 0 0 0 0 0 1 0 10 1 0 0 0 0 0 0 1 0
x ( ) C( )
0 1
1
2
3
4
5
O
0 0 1 1 1 1 1 10 1 1 1 1 1 1 10 1 1 1 1
a a b ba bε b a b a bb b b b b bxxxx
1 1 11 1 1 1 1 1 1 11 1 1 1 1 1x 1 1
( 1) ( 1) ( )( ) ( )
t t t
t -1
-1
t
t
t
t
t0 0
T
Define
x x T x AT1 0 00 1 0 x AT x A0 1 0
x ( ) x ( )T0 0 10 0
linear transf
T
x
1 C( ) C )
=
T
x T:
(
n n n
n n
L2
Observability Reduction by Rows
x1
ab
b
x2
a
x3
a
x4
ax5
a
b
bb
0
0 1 0 0 0 0 0 1 0 0 1 00 1 0 0 0 0 0 0 1 0 0 0
0 0 1 0 0 0 0 0 0 1 0 00
A(a) A(b)
0 1 0 0 0 0 0 0 1 0 10 1 0 0 0 0 0 0 1 0
x ( ) C( )
0 1
1
2
3
4
5
O
0 0 1 1 1 1 1 10 1 1 1 1 1 1 10 1 1 1 1
a a b ba bε b a b a bb b b b b bxxxx
1 1 11 1 1 1 1 1 1 11 1 1 1 1 1x 1 1
0
t t
T A (a) A(b) x ( ) C( )
A(x)
1 0 00 1 0 0 1 0 1 00 1 0
0 1 0
Define linear transf x
0 0 1 0 00 1 00 1 0 0 0 1 0 10 0 1
0
= x T
0
1
:
t t
0 0T Tx ( ) = x ( = [A(x)T] C( ) = [C() )]T
L2
Observability Reduction by Columns
x1
ab
b
x2
a
x3
a
x4
ax5
a
b
bb
0
0 1 0 0 0 0 0 1 0 0 1 00 1 0 0 0 0 0 0 1 0 0 0
0 0 1 0 0 0 0 0 0 1 0 00
A(a) A(b)
0 1 0 0 0 0 0 0 1 0 10 1 0 0 0 0 0 0 1 0
x ( ) C( )
0 1
1
2
3
4
5
O
0 0 1 1 1 1 1 10 1 1 1 1 1 1 10 1 1 1 1
a a b ba bε b a b a bb b b b b bxxxx
1 1 11 1 1 1 1 1 1 11 1 1 1 1 1x 1 1
0
t t
T A (a) A(b) x ( ) C( )
A(x)
0 0 10 0 0 0 0 0 0 10 1 1
0 0 0
Define linear transf x
1 0 0 0 00 1 10 1 1 0 1 1 1 01 1 1
1
= x T
1
1
:
t t
0 0T Tx ( ) = x ( = [A(x)T] C( ) = [C() )]T
L2
Mixed Observability Reduction
x1
ab
b
x2
a
x3
a
x4
ax5
a
b
bb
0
0 1 0 0 0 0 0 1 0 0 1 00 1 0 0 0 0 0 0 1 0 0 0
0 0 1 0 0 0 0 0 0 1 0 00
A(a) A(b)
0 1 0 0 0 0 0 0 1 0 10 1 0 0 0 0 0 0 1 0
x ( ) C( )
0 1
1
2
3
4
5
O
0 0 1 1 1 1 1 10 1 1 1 1 1 1 10 1 1 1 1
a a b ba bε b a b a bb b b b b bxxxx
1 1 11 1 1 1 1 1 1 11 1 1 1 1 1x 1 1
0
t t
T A (a) A(b) x ( ) C( )
A(x)
0 0 10 0 0 0 0 0 0 10 1 0
0 1 0
Define linear transf x
1 1 0 0 00 1 00 1 0 0 1 0 1 01 1 0
1
= x T
1
0
:
t t
0 0T Tx ( ) = x ( = [A(x)T] C( ) = [C() )]T
L2
Original and Reduced Automata
x1
ab
b
x2
a
x3
a
x4
ax5
a
b
bb x1
a,b bx2
ax3
a
b
DFA L21 by rows
x3
a,bx2
a,bx1
b
NFA L22 by columns
L2
x3
a,bx2
a,bx1
b
NFA L23 mixed
L2
Original and Reduced Automata
a
21 21,0 21
21 21,0 21
t t21 t
22t
23
'
' ' '
= =
x = x T L 0 0 1 0 0 1L = [A , x , C ] 0 1 1 0 1 0
1 1 1 1 1 0 L
Let in where
= [A , x ,
Then
C ]
T T
x1
ab
b
x2
a
x3
a
x4
ax5
a
b
bb x1
a,b bx2
ax3
b
DFA L21 by rows
L2
x3 x2 x1
NFA L22 by columns
a,b a,b bx3
a,bx2
a,bx1
b
NFA L23 mixed
L2
Row Basis but No Column Basis
x1
a
bb
x2
a
x5
ax3
a
x7
a
b bb
L3
x4
x6
1
2
4
7
5
3
6
x 1 0 1 1 0 0x 0 0 1 1 0 0 0
aa b bε a b ab a b axx
x
0 0 0 1 0 1 00 1 1 1 0 0 0
0 0 1 0 0 0 01 1 0 0 1 0 1
x 1 0
O
x
0
0 0 0 0 0
Row Basis but No Column Basis
x1
a
bb
x2
a
x5
ax3
a
x7
a
b bb
L3
x4
x6
1
2
4
7
5
3
6
x 1 0 1 1 0 0x 0 0 1 1 0 0 0
aa b bε a b ab a b axx
x
0 0 0 1 0 1 00 1 1 1 0 0 0
0 0 1 0 0 0 01 1 0 0 1 0 1
x 1 0
O
x
0
0 0 0 0 0
Row Basis but No Column Basis
x1
a
bb
x2
a
x5
ax3
a
x7
a
b bb
L3
x4
x6
1
2
3
4
5
6
7
aa b bO ε a b ab a b ax 0 0 0 1 0 1 0x 0 1 1 1 0 0 0x 1 0 1 1 0 0 0x 0 0 1 1 0 0 0x 0 0 1 0 0 0 0x 1 1 0 0 1 0 1x 1 0 0 0 0 0 0
1
2
3
4
5
6
7
0 0 00 1 01 0 0
bb b
0 11 01 01 00 0 0
0 0
a bb a
1 01 01 01 00 11 0
0 0 0
aε a a
axxx
O
01 1 11
00 0 0
xxxx 0 0 0
Row Basis but No Column Basis
x1
a
bb
x2
a
x5
ax3
a
x7
a
b bb
L3
x4
x6
1
2
3
4
5
6
7
aa b bO ε a b ab a b ax 0 0 0 1 0 1 0x 0 1 1 1 0 0 0x 1 0 1 1 0 0 0x 0 0 1 1 0 0 0x 0 0 1 0 0 0 0x 1 1 0 0 1 0 1x 1 0 0 0 0 0 0
1
2
3
4
5
6
7
0 0 00 1 01 0 0
bb b
0 11 01 01 00 0 0
0 0
a bb a
1 01 01 01 00 11 0
0 0 0
aε a a
axxx
O
01 1 11
00 0 0
xxxx 0 0 0
• Theorem (Cover): Finding a (possibly mixed) basis T for OL is equivalent to finding a minimal cover for OL.
- either as its set basis cover or as its Karnaugh cover.
• Theorem (Complexity): Determining a cover T for OL
is NP-complete (set basis problem complexity).
• Theorem (Rank): The row (= column) rank of OL is the size of the set cover T (size of Karnaugh cover).
Observabilty Reduction
t t t t n-1 t t0 0 0 [y(0) y(1) ... y(n-1)] = C [x A x ... (A x ] = C (3)
Let L = [S,A,C] be an n-state automaton. It's output:
L is reachable if C is uniquely determined by
.
) R
(3)
Reachability: Dual of Observability
Reachability: Dual of Observability
t t t t n-1 t t0 0 0 [y(0) y(1) ... y(n-1)] = C [x A x ... (A x ] = C (3)
Let L = [S,A,C] be an n-state automaton. It's output:
) R
L is reachable if C is uniquely deter
Example:
mined by
the rea
(
ch
3 .
il
)
ab
1
2
3
n
1
t0
t
t t1 2 3 0 0 0
b bb b b(A )
1 0 0 0 0 0 0R =0 1 0 1 0 0 00 0 1 0 0 0 1
[x x x [
a aε a a ax x
x x
x
x
ity mat of L is:
x
] = ( ) A (a) A
rix
Row (b)] basis col basis.
x3 x2x1
a
ab
b
L1
• DFA Minimization: Is a particular case of observability reduction (single initial state requires distinctness only)
• NFA Determinization: Is a particular case of reachability transformation (take all distinct columns as “basis”)
• Minimal automata: Are related by linear maps (but not by graph isomorphisms!). Better definition of minimality
• Other techniques: Are easily formalized in this setting: Pumping lemma, NFA to RE, Z-transforms, etc.
Observabilty, Reachability and More
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