Testing finite state machines: State Identification and Verification

(1)

Testing finite state machines:

State Identification and Verification

Automata Learning - reading group December 2013

OREZ

(2)

Mealy Machines

s

¹

s

³

s

²

a/0 b/0

b/1 b/1 a/1

a/0

(3)

Mealy Machines

s

¹

s

³

s

²

a/0 b/0

b/1

a/0

( I , O , S ,δ,λ)

b/1

(4)

Mealy Machines

s

¹

s

³

s

²

a/0 b/0

b/1

a/0

Input

a/0 b/0

b/1

( I , O , S ,δ,λ) I a/0

b/1

(5)

Mealy Machines

s

¹

s

³

s

²

a/ 0 b/ 0

b/ 1

a/ 0

Output

( I , O , S ,δ,λ) O

b/ 1

(6)

Mealy Machines

s

¹

s

³

s

²

a/0 b/0

b/1

a/0

States

( I , O , S ,δ,λ) S

b/1

(7)

Mealy Machines

s

¹

s

³

s

²

a/0 b/0

b/1

a/0

δ: S x I ➝ S

( I , O , S ,δ,λ) δ

b/1

(8)

Mealy Machines

s

¹

s

³

s

²

a/0 b/0

b/1

( I , O , S ,δ,λ) a/0

δ: S x I ➝ S δ

b/1

(9)

Mealy Machines

s

¹

s

³

s

²

a/0 b/0

b/1

( I , O , S ,δ,λ) a/0

δ: S x I ➝ S δ

b/1

(10)

Mealy Machines

s

¹

s

³

s

²

a/0 b/0

b/1

( I , O , S ,δ,λ) a/0

δ: S x I ➝ S δ( ^s

²

,`a’) = ^s

¹

δ

b/1

(11)

Mealy Machines

s

¹

s

³

s

²

a/0 b/0

b/1

( I , O , S ,δ,λ) a/0

δ: S x I ➝ S δ( ^s

²

,`a’) = ^s

¹

δ( ^s

²

,“aaab”) = ^s

³

δ

b/1

(12)

Mealy Machines

s

¹

s

³

s

²

a/0 b/0

b/1 b/1

a/0

s

³

( I , O , S ,δ,λ) δ δ: S x I ➝ S

δ({ ^s

¹

^,s

³

},`b’) = { ^s

³

, ^s

²

} δ( ^s

²

,“aaab”) = ^s

³

δ( ^s

²

,`a’) = ^s

¹

(13)

Mealy Machines

s

¹

s

³

s

²

a/0 b/0

b/1

( I , O , S ,δ,λ) a/0

λ: S x I ➝ O λ

b/1

(14)

Mealy Machines

s

¹

s

³

s

²

a/ 0 b/0

b/1

( I , O , S ,δ,λ) a/0

λ: S x I ➝ O λ( ^s

²

,`a’) = ^`0’

λ

b/1

(15)

Mealy Machines

s

¹

s

³

s

²

a/ 0 b/ 0

b/1

a/ 0

λ: S x I ➝ O ( I , O , S ,δ,λ) λ

b/1

λ(S,`a’) = ^`0’

λ( ^s

²

,“aaab”) = “0000”

(16)

Mealy Machines

s

¹

s

³

s

²

a/0 b/ 0

b/ 1

a/0

λ: S x I ➝ O ( I , O , S ,δ,λ) λ

b/1

λ( ^s

²

,`a’) = ^`0’

λ( ^s

²

,“aaab”) = “0000”

λ({ ^s

¹

^,s

³

},`b’) = { ^`0’ , ^`1’ }

(17)

s

¹

s

³

s

²

a/0 b/0

b/1 b/1 a/1

a/0

(18)

Lack of some informations Goal: Deduce them.

Mean:

Provide input Sequence Observe the output

Testing

a/1 s

¹

s

²

a/0 b/0

b/1 b/1

a/0

(19)

Input Sequence / Observe output Preset Test:

Fixed ahead of time.

Adaptive Test:

Input symbol depends on the

previous observed output symbol.

Tests

a/1 s

¹

s

²

a/0 b/0

b/1 b/1

a/0

(20)

Testing: Problems

a/1 s

¹

s

²

a/0 b/0

b/1 b/1

a/0

State Identification

State Verification

(21)

Testing: State Identification

s

¹

s

²

a/0

b/1 b/1

a/0

State diagram given.

Initial state not identified.

Initial State Identification: Not always possible through testing Distinguishing Sequence

(22)

Testing: State Identification

s

¹

s

³

s

²

a/0 b/0

b/1 b/1 a/1

a/0

(23)

Testing: Problems

a/1 s

¹

s

²

a/0 b/0

b/1 b/1

a/0

(24)

Testing:Questions

a/1 s

¹

s

²

a/0 b/0

b/1 b/1

a/0

Machine is in its initial state.

Goal: Check if it’s in that state.

Unique Input Output (UIO) Sequence

State Identification State Verification

(25)

Testing: Questions

a/1 s

¹

s

²

a/0 b/0

b/1

Existence of the input sequence?

a/0

How long is the sequence?

Complexity to:

Check existence of the sequence, Construct a sequence,

Construct a short one.

State Identification State Verification

b/1

(26)

Testing: Problems

a/1 s

¹

s

²

a/0 b/0

b/1 b/1

a/0

State Identification State Verification

Fault detection (black box), protocol testing

Link:

Distinguishing Sequence

(27)

s

¹

s

³

s

²

a/0 b/0

b/1 b/1 a/1

a/0

State Identification

(28)

Preset Distinguishing sequence: Input x such that the output produced by the

machine is different for each state.

The FSM needs to be reduced, to remove all the equivalent states.

The machine might not have such a sequence

State Identification

s

¹

s

²

a/0

b/1

a/0

Preset Distinguishing sequence

(29)

s

¹

s

³

s

²

a/0 b/0

b/1 b/1 a/1

a/0

State Identification

Reduced

(30)

s

¹

s

³

s

²

s

¹

s

³

s

²

a/0 b/ 0

b/ 1 b/ 1 a/1

a/0

State Identification

Reduced

- ‘b’ distinguish s

¹

from s

²

and s

³

(31)

s

¹

s

³

s

²

s

¹

s

³

s

²

a/ 0 b/0

b/1

a/ 1 b/1

a/0

State Identification

Reduced

- ‘b’ distinguish s

¹

from s

²

and s

¹³

- ‘a’ distinguish s

³

from s

²

(32)

s

¹

s

³

s

²

a/0 b/0

b/1

b/1 a/1

a/0

State Identification

Reduced

But No (Unique) Sequence that distinguish (All states) at once

(33)

s

¹

s

³

s

²

a/0

a/1 a/0

State Identification

Reduced

But No (Unique) Sequence that distinguish (All states) at once - “b... ” distinguish s

¹

from s

²

and s

³

, but

“merges” s

²

and s

³

b/ 0 b/ 1

b/ 1

(34)

s

¹

s

³

s

²

a/ 0 b/0

b/1

a/ 1 b/1

a/ 0

State Identification

Reduced

But No (Unique) Sequence that distinguish (All states) at once - “b...” distinguish s

¹

from s

²

and s

³

, but

“merges” s

²

and s

³

- “a... ” distinguish s

³

from s

¹

and s

²

, but

“merges” s

¹

and s

²

(35)

Preset Distinguishing sequence: Input x such that the output produced by the

machine is different for each state.

THEOREM: It is PSPACE-complete to test whether a given FSM has a preset

distinguishing sequence.

Length of distinguishing sequence can be exponential

Did not get hold of the algorithm.

State Identification

s

¹

s

²

a/0

b/1

a/0

(36)

State Identification

s

¹

s

²

a/0

b/1

a/0

Adaptive Distinguishing Sequence

(37)

State Identification

s

¹

s

²

a/0

b/1

a/0

Adaptive Distinguishing Sequence: Not a

sequence but a decision tree: Input based on the last output.

Rooted tree, n leaves (states), Nodes: Input, Edges: Output.

Length of the sequence is the depth of the tree

Adaptive Distinguishing Sequence

(38)

State Identification

Adaptive Distinguishing Sequence Preset Distinguishing sequence

s

¹

s

⁴

s

³

s

²

s

⁵

s

⁶

b/0

a/0 a/1 b/1

a/0 b/0

b/0

a/1 b/0

a/0 b/0

(39)

a/1

b/0

a/1 b/0

a/1

b/0

State Identification

s

¹

s

⁴

s

³

s

²

s

⁵

s

⁶

b/0

a/0 b/1

a/0 b/0

b/0

a/0

(40)

b/0 a/0

a/0 b/0

b/0

a/0

a/1

b/0

a/1 b/0

a/1

b/0

State Identification

s

¹

s

⁴

s

³

s

²

s

⁵

s

⁶

b/1

(41)

b/0 a/0

a/0 b/0

b/0

a/0

a/1

b/0

a/1 b/0

a/1

b/0

State Identification

s

¹

s

⁴

s

³

s

²

s

⁵

s

⁶

b/1

(42)

State Identification

a

b/1

Initial Current

{s

¹

,s

²

,s

³

,s

⁴

,s

⁵

,s

⁶

}

(43)

b/0 a/0

a/0 b/0

b/0

a/0

a/1

b/0

a/1 b/0

a/1

b/0

State Identification

s

¹

s

⁴

s

³

s

²

s

⁵

s

⁶

b/1

(44)

State Identification

a

b/1

0 Initial Current

{s

¹

,s

³

,s

⁵

}

{s

¹

,s

²

,s

³

,s

⁴

,s

⁵

,s

⁶

}

(45)

b/0 a/0

a/0 b/0

b/0

a/0

a/1

b/0

a/1 b/0

a/1

b/0

State Identification

s

¹

s

⁴

s

³

s

²

s

⁵

s

⁶

b/1

(46)

b/0 a/0

a/0 b/0

b/0

a/0

a/1

b/0

a/1 b/0

a/1

b/0

State Identification

s

¹

s

⁴

s

³

s

²

s

⁵

s

⁶

b/1

(47)

State Identification

a

b/1

0 Initial Current

{s

¹

,s

³

,s

⁵

} {s

²

,s

⁴

,s

⁶

}

{s

¹

,s

²

,s

³

,s

⁴

,s

⁵

,s

⁶

}

(48)

a/1

b/0

a/1 b/0

a/1

b/0

State Identification

s

¹

s

⁴

s

³

s

²

s

⁵

s

⁶

b/0

a/0 b/1

a/0 b/0

b/0

a/0

(49)

State Identification

a

b/1

1 0

{s

²

,s

⁴

,s

⁶

}

Initial Current

{s

¹

,s

³

,s

⁵

} {s

²

,s

⁴

,s

⁶

}

{s

¹

,s

²

,s

³

,s

⁴

,s

⁵

,s

⁶

}

(50)

State Identification

a

b/1

1 0

{s

²

,s

⁴

,s

⁶

}

Initial Current

{s

¹

,s

³

,s

⁵

} {s

²

,s

⁴

,s

⁶

}

{s

¹

,s

²

,s

³

,s

⁴

,s

⁵

,s

⁶

}

(51)

a/1

b/0

a/1 b/0

a/1

b/0

State Identification

s

¹

s

⁴

s

³

s

²

s

⁵

s

⁶

b/0

a/0 b/1

a/0 b/0

b/0

a/0

(52)

a/1

b/0

a/1 b/0

a/1

b/0

State Identification

s

¹

s

⁴

s

³

s

²

s

⁵

s

⁶

b/0

a/0 b/1

a/0 b/0

b/0

a/0

(53)

State Identification

a

b/1

1 0

{s

²

,s

⁴

,s

⁶

}

Initial Current

{s

¹

,s

³

,s

⁵

} {s

²

,s

⁴

,s

⁶

}

{s

¹

,s

²

,s

³

,s

⁴

,s

⁵

,s

⁶

}

(54)

State Identification

a

b/1

1 0

{s

²

,s

⁴

,s

⁶

} {s

³

,s

⁵

,s

¹

} Initial Current

{s

¹

,s

²

,s

³

,s

⁴

,s

⁵

,s

⁶

}

{s

¹

,s

³

,s

⁵

} {s

²

,s

⁴

,s

⁶

}

(55)

State Identification

a

b/1

1 0

{s

²

,s

⁴

,s

⁶

} {s

³

,s

⁵

,s

¹

} Initial Current

{s

¹

,s

²

,s

³

,s

⁴

,s

⁵

,s

⁶

}

(56)

a/1

b/0

a/1 b/0

a/1

b/0

State Identification

s

¹

s

⁴

s

³

s

²

s

⁵

s

⁶

b/0

a/0 b/1

a/0 b/0

b/0

a/0

(57)

a/1

b/0

a/1 b/0

a/1

b/0

State Identification

s

¹

s

⁴

s

³

s

²

s

⁵

s

⁶

b/0

a/0 b/1

a/0 b/0

b/0

a/0

(58)

b 0

State Identification

a

b/1

1 0

{s

²

,s

⁴

,s

⁶

} {s

³

,s

⁵

,s

¹

} Initial Current

{s

²

,s

⁴

,s

⁶

}

{s

¹

,s

²

,s

³

,s

⁴

,s

⁵

,s

⁶

}

(59)

a/1

b/0

a/1 b/0

a/1

b/0

State Identification

s

¹

s

⁴

s

³

s

²

s

⁵

s

⁶

b/0

a/0 b/1

a/0 b/0

b/0

a/0

(60)

State Identification

s

¹

s

⁴

s

³

s

²

s

⁵

s

⁶

b/0

a/0 a/1 b/1

a/0 b/0

b/0

a/1 b/0

a/0 b/0

(61)

b 0

State Identification

a

b/1

1 0

{s

²

,s

⁴

,s

⁶

} {s

³

,s

⁵

,s

¹

} {s

⁴

,s

⁶

,s

¹

} Initial Current

{s

²

,s

⁴

,s

⁶

}

{s

¹

,s

²

,s

³

,s

⁴

,s

⁵

,s

⁶

}

(62)

State Identification

s

¹

s

⁴

s

³

s

²

s

⁵

s

⁶

b/0

a/0 a/1 b/1

a/0 b/0

b/0

a/1 b/0

a/0 b/0

(63)

State Identification

s

¹

s

⁴

s

³

s

²

s

⁵

s

⁶

b/0

a/0 a/1 b/1

a/0 b/0

b/0

a/1 b/0

a/0 b/0

(64)

b

0

State Identification

a

b/1

b 1

0 0

{s

²

,s

⁴

,s

⁶

} {s

³

,s

⁵

,s

¹

} {s

²

,s

⁴

,s

⁶

} {s

⁴

,s

⁶

,s

¹

} {s

²

,s

⁴

,s

⁶

}

Initial Current

{s

¹

,s

²

,s

³

,s

⁴

,s

⁵

,s

⁶

}

(65)

State Identification

s

¹

s

⁴

s

³

s

²

s

⁵

s

⁶

b/0

a/0 a/1 b/1

a/0 b/0

b/0

a/1 b/0

a/0 b/0

(66)

State Identification

s

¹

s

⁴

s

³

s

²

s

⁵

s

⁶

b/0

a/0 a/1 b/1

a/0 b/0

b/0

a/1 b/0

a/0 b/0

(67)

b

0

State Identification

a

b/1

b 1

0 0

{s

²

,s

⁴

,s

⁶

} {s

¹

,s

³

,s

⁵

} {s

²

,s

⁴

,s

⁶

} {s

⁴

,s

⁶

,s

¹

} {s

²

,s

⁴

,s

⁶

} {s

⁵

,s

¹

}

Initial Current {s

¹

,s

²

,s

³

,s

⁴

,s

⁵

,s

⁶

}

Invalid Input

(68)

State Identification

s

¹

s

⁴

s

³

s

²

s

⁵

s

⁶

b/0

a/0 a/1 b/1

a/0 b/0

b/0

a/1 b/0

a/0 b/0

(69)

State Identification

s

¹

s

⁴

s

³

s

²

s

⁵

s

⁶

b/0

a/0 a/1 b/1

a/0 b/0

b/0

a/1 b/0

a/0 b/0

(70)

State Identification

s

¹

s

⁴

s

³

s

²

s

⁵

s

⁶

b/0

a/0 a/1 b/1

a/0 b/0

b/0

a/1 b/0

a/0 b/0

(71)

a 0 1

State Identification

a

b/1

b 1

0 0 {s

¹

,s

²

,s

³

,s

⁴

,s

⁵

,s

⁶

}

{s

²

,s

⁴

,s

⁶

} {s

³

,s

⁵

,s

¹

} {s

²

,s

⁴

,s

⁶

} {s

⁴

,s

⁶

,s

¹

} Initial Current

{s

²

}

{s

⁵

,s

¹

}

(72)

a 0 1

State Identification

a

b/1

b 1

0 0

{s

²

,s

⁴

,s

⁶

} {s

³

,s

⁵

,s

¹

} {s

²

,s

⁴

,s

⁶

} {s

⁴

,s

⁶

,s

¹

} Initial Current

{s

⁶

}

{s

¹

,s

²

,s

³

,s

⁴

,s

⁵

,s

⁶

}

{s

²

}

{s

⁵

,s

¹

}

(73)

a 0 1

State Identification

a

b/1

b 1

0 0

{s

²

,s

⁴

,s

⁶

} {s

³

,s

⁵

,s

¹

} {s

²

,s

⁴

,s

⁶

} {s

⁴

,s

⁶

,s

¹

} Initial Current

{s

¹

,s

²

,s

³

,s

⁴

,s

⁵

,s

⁶

}

{s

²

} {s

⁵

,s

¹

}

{s

⁶

}

s

⁶

(74)

a 0 1

State Identification

a

b/1

b 1

0 0

{s

²

,s

⁴

,s

⁶

} {s

³

,s

⁵

,s

¹

} {s

²

,s

⁴

,s

⁶

} {s

⁴

,s

⁶

,s

¹

} Initial Current

{s

¹

,s

²

,s

³

,s

⁴

,s

⁵

,s

⁶

}

{s

²

} {s

⁵

,s

¹

}

{s

⁶

}

{s

²

,s

⁴

}

s

⁶

(75)

State Identification

a

b/1

b 0

{s

²

,s

⁴

,s

⁶

} {s

³

,s

⁵

,s

¹

} {s

²

,s

⁴

,s

⁶

} {s

⁴

,s

⁶

,s

¹

} Initial Current

{s

¹

,s

²

,s

³

,s

⁴

,s

⁵

,s

⁶

}

{s

²

} {s

⁵

,s

¹

}

0 b

a 0

1 0

1 {s

⁶

}

{s

²

,s

⁴

}

s

⁶

(76)

State Identification

a

b/1

b 0

{s

²

,s

⁴

,s

⁶

} {s

³

,s

⁵

,s

¹

} {s

²

,s

⁴

,s

⁶

} {s

⁴

,s

⁶

,s

¹

} Initial Current

{s

¹

,s

²

,s

³

,s

⁴

,s

⁵

,s

⁶

}

{s

²

} {s

⁵

,s

¹

}

a

b 0

. . .

b

a 0

1 0

1 ...

s

⁶

{s

⁶

}

{s

²

,s

⁴

}

(77)

State Identification

b/1

δ( ^s

⁶

,“aba”) = ^s

²

λ( ^s

⁶

,“aba”) = “100”

a

b 0

Initial Current {s

¹

,s

²

,s

³

,s

⁴

,s

⁵

,s

⁶

}

{s

²

} {s

⁵

,s

¹

} a

b 0

. . .

b

a 0

1 0

1 ...

s

⁶

{s

⁶

}

{s

²

,s

⁴

}

(78)

State Identification

b/1

a

b 0

a

b 0

. . .

b

a 0

1 0

1 ...

s

⁶

(79)

State Identification

Preset Sequence is an Adaptive one Adaptive Sequence might be shorter

FSM might have an Adaptive sequence, but no Preset one.

Length is quadratic, if the seq. exists.

Preset Distinguishing sequence Adaptive Distinguishing Sequence

(80)

State Identification

Existence Algorithm: (Reason about the Initial Set) Initialize State Partition Π with one unique block Iteration:

Pick: Block B, Valid Input a.

Refine the partition: Two states are assigned the same block in the new partition iff

they produce the same output on a, and

move to the same block (of the old partition).

(81)

State Identification

Existence Algorithm:

Initialize State Partition Π with one unique block Iteration:

Pick: Block B, Valid Input a.

Refine the partition...

Adaptive Seq. exists iff final partition is discrete

(82)

State Identification

s

¹

s

⁴

s

³

s

²

s

⁵

s

⁶

b/0

a/0 a/1

a/0 b/0

b/0

a/1 b/0

a/0 b/0

(83)

State Identification

s

¹

s

⁴

s

³

s

²

s

⁵

s

⁶

b/0

a/0 a/1

a/0 b/0

b/0

a/1 b/0

a/0 b/0

Π = {{

s

¹^,

s

²^,

s

³^,

s

⁴^,

s

⁵^,

s

⁶^}}

(84)

State Identification

s

¹

s

⁴

s

³

s

²

s

⁵

s

⁶

b/0

a/0 a/1

a/0 b/0

b/0

a/1 b/0

a/0 b/0

Π = {{

s

¹^,

s

³^,

s

⁵^},{

s

²^,

s

⁴^,

s

⁶^}}

Π = {{

s

¹^,

s

²^,

s

³^,

s

⁴^,

s

⁵^,

s

⁶^}}

(85)

State Identification

s

¹

s

⁴

s

³

s

²

s

⁵

s

⁶

b/0

a/0 a/1

a/0 b/0

b/0

a/1 b/0

a/0 b/0

Π = {{

s

¹^,

s

³^,

s

⁵^},{

s

²^,

s

⁴^,

s

⁶^}}

(86)

State Identification

s

¹

s

⁴

s

³

s

²

s

⁵

s

⁶

b/0

a/0 a/1

a/0 b/0

b/0

a/1 b/0

a/0 b/0

Π = {{

s

¹^,

s

³^,

s

⁵^},{

s

²^,

s

⁴^,

s

⁶^}}

(87)

State Identification

s

¹

s

⁴

s

³

s

²

s

⁵

s

⁶

b/0

a/0 a/1

a/0 b/0

b/0

a/1 b/0

a/0 b/0

Π = {{

s

¹^,

s

³^,

s

⁵^},{

s

²^,

s

⁴^,

s

⁶^}}

Π = {{

s

¹^},{

s

³^,

s

⁵^},{

s

²^,

s

⁴^,

s

⁶^}}

(88)

State Identification

s

¹

s

⁴

s

³

s

²

s

⁵

s

⁶

b/0

a/0 a/1

a/0 b/0

b/0

a/1 b/0

a/0 b/0

Π = {{

s

¹^},{

s

³^,

s

⁵^},{

s

²^,

s

⁴^,

s

⁶^}}

(89)

State Identification

s

¹

s

⁴

s

³

s

²

s

⁵

s

⁶

b/0

a/0 a/1

a/0 b/0

b/0

a/1 b/0

a/0 b/0

Π = {{

s

¹^},{

s

³^,

s

⁵^},{

s

²^,

s

⁴^,

s

⁶^}}

(90)

State Identification

s

¹

s

⁴

s

³

s

²

s

⁵

s

⁶

b/0

a/0 a/1

a/0 b/0

b/0

a/1 b/0

a/0 b/0

Π = {{

s

¹^},{

s

³^,

s

⁵^},{

s

²^,

s

⁴^,

s

⁶^}}

Π = {{

s

¹^},{

s

³^,

s

⁵^},{

s

²^,

s

⁴^},{

s

⁶^}}

(91)

State Identification

s

¹

s

⁴

s

³

s

²

s

⁵

s

⁶

b/0

a/0 a/1

a/0 b/0

b/0

a/1 b/0

a/0 b/0

Π = {{

s

¹^},{

s

³^,

s

⁵^},{

s

²^,

s

⁴^},{

s

⁶^}}

(92)

State Identification

s

¹

s

⁴

s

³

s

²

s

⁵

s

⁶

b/0

a/0 a/1

a/0 b/0

b/0

a/1 b/0

a/0 b/0

Π = {{

s

¹^},{

s

³^,

s

⁵^},{

s

²^,

s

⁴^},{

s

⁶^}}

(93)

State Identification

s

¹

s

⁴

s

³

s

²

s

⁵

s

⁶

b/0

a/0 a/1

a/0 b/0

b/0

a/1 b/0

a/0 b/0

Π = {{

s

¹^},{

s

³^,

s

⁵^},{

s

²^,

s

⁴^},{

s

⁶^}}

Π = {{

s

¹^},{

s

³^},{

s

⁵^},{

s

²^,

s

⁴^},{

s

⁶^}}

(94)

State Identification

s

¹

s

⁴

s

³

s

²

s

⁵

s

⁶

b/0

a/0 a/1

a/0 b/0

b/0

a/1 b/0

a/0 b/0

Π = {{

s

¹^},{

s

³^},{

s

⁵^},{

s

²^,

s

⁴^},{

s

⁶^}}

(95)

State Identification

s

¹

s

⁴

s

³

s

²

s

⁵

s

⁶

b/0

a/0 a/1

a/0 b/0

b/0

a/1 b/0

a/0 b/0

Π = {{

s

¹^},{

s

³^},{

s

⁵^},{

s

²^,

s

⁴^},{

s

⁶^}}

(96)

State Identification

s

¹

s

⁴

s

³

s

²

s

⁵

s

⁶

b/0

a/0 a/1

a/0 b/0

b/0

a/1 b/0

a/0 b/0

Π = {{

s

¹^},{

s

³^},{

s

⁵^},{

s

²^},{

s

⁴^},{

s

⁶^}}

Π = {{

s

¹^},{

s

³^},{

s

⁵^},{

s

²^,

s

⁴^},{

s

⁶^}}

(97)

State Identification

s

¹

s

⁴

s

³

s

²

s

⁵

s

⁶

b/0

a/0 a/1

a/0 b/0

b/0

a/1 b/0

a/0 b/0

Π = {{

s

¹^},{

s

³^},{

s

⁵^},{

s

²^},{

s

⁴^},{

s

⁶^}}

(98)

State Identification

Algorithm 3.2 and

3.3

(99)

State Verification

Machine is in its initial state.

Goal: Check if it’s in that state.

Unique Input Output (UIO) Sequence

(100)

State Verification

Unique Input Output (UIO)

Sequence of a state s⁰ is an input x⁰ such that:

λ( ^s

⁰

,x

⁰

) ≠ λ( ^s ,x

⁰

), ∀s

(101)

State Verification

Unique Input Output (UIO)

Sequence of a state s⁰ is an input x⁰ such that:

λ( ^s

⁰

,x

⁰

) ≠ λ( ^s ,x

⁰

), ∀s

Preset/Adaptive Distinguishing Sequence

Existence of UIO

(102)

State Verification

All these problems are PSPACE- Complete:

Does a given state s of M has a UIO

Does all states of M have a UIO Is there some states of M with UIO

(103)

State Verification

Length: There are machine

whose states have UIO seq. but of exponential length

(104)

Conclusion

PSPACE-Completeness for Preset Seq. and UIO

Polynomial time algorithm for checking existence and

construction of the Adaptive Seq.

Adaptive Seq. might be more useful

(105)

Just to remember ..

s

¹

s

³

s

²

a/0 b/0

b/1 b/1 a/1

a/0

(106)

Assignment:

s

¹

s

³

s

²

b/1 b/0

b/1 a/1 a/0

a/0

(107)

Assignment:

s

¹

s

³

s

²