Finite state systems
TDDC05: Embedded Systems
• Inputs: modelled with finite set of
symbols
• Reaction: based on finite sequence of
earlier input symbols
• Time: sequence of reactions
• ⇒ Finite state machines (FSM)
Simulation and Verification
Lecture 2: Finite state automata
as reactive systems
Simin Nadjm-Tehrani
Real-time Systems Laboratory
Department of Computer and Information Science
Embedded systems simulation and verification
Linköping university
28 pages
Spring term 2006
Embedded systems simulation and verification
Linköping university
Example: Cash dispenser
code
amount
rejected
return
Strings
•
•
•
•
approved
card
ok
rejected
rejected
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Over alphabet Σ of symbols
Finite sequence of symbols over Σ
Empty string denoted ε
Definition:
– ε is a string
– if w is a string, and a ∈ Σ then aw is a
string
rejected
money
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Embedded systems simulation and verification
Linköping university
Finite deterministic automata
•
•
•
•
•
Finite alphabet
Finite number of states
Defined initial state
A (number of) final state(s)
Deterministic transitions:
– for each state and each input there is
exactly one next state
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Formal definition
A finite deterministic automaton is a
5-tuple ⟨Q, Σ, δ, q0, F⟩ where
• Q is a finite set of states, q0 ∈ Q
• Σ is a finite (input) alphabet
• δ is a function: Q × Σ → Q that gives
next state for each input from each
state
• F ⊆ Q denotes final state
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The language of Automata
Discretised landing gear
• Extend δ to strings over the alphabet
– δ(q, ε) = q
– δ(q, aw) = δ(δ(q, a), w)
open
close
retract
extend
• A string w is accepted by the automaton M iff
δ(q0, w) ∈ F
• M:s language L(M) = {w| δ(q0, w) ∈ F}
Controller
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Landing gear
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FSM model
r,c
o,r
opened+retracted
e
o,e
c
e
–A language Ls
–A set of reachable states Rs
closed+extended
r
opened+extended
o,c,r,e
c
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Verification
• M satisfies the specification S if
–L(M) ⊆ Ls
–R(M) ⊆ Rs Only those states
allowed by the specification S are
reachable in M
Embedded systems simulation and verification
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Specification
• Automaton M:s reachability set R(M) =
{q| ∃ w. δ(q0, w) = q}
• A specification S:
closed+retracted
o
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• What we will (will not) the system to do
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Non-deterministic FSM
A finite non-deterministic automaton is
a 5-tuple ⟨Q, Σ, δ, q0, F⟩ where
• Q is a finite set of states, q0 ∈ Q
• Σ is a finite (input) alphabet
• δ is a function: Q × Σ → 2Q that gives
possible next state for each input and
each state
• F ⊆ Q describes the final states
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Exercise
Example: Telephone
• Describe the cash dispenser with a nondeterministic finite automaton!
2
connect
4
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ring
dial
0
hang-up
=
=
=
=
=
dial
reciever on
reciever off
idle
busy
connected
ui
qi
qi+1= δ (ui , qi)
Memory
Clock
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Gear:
open
A Moore machine is a tuple
⟨Q, Σ, ∆, δ, λ, q0⟩ where
• Q is a finite set of states, q0 ∈ Q
• Σ is a finite input alphabet
• ∆ is a finite output alphabet
• δ is a function: Q × Σ → Q that gives
the next state for each input
• λ is a function: Q → ∆ that gives the
output in each state
Embedded systems simulation and verification
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open
close
closed
close
Embedded systems simulation and verification
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retract
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Mealy machine
A Mealy machine is a tuple
⟨Q, Σ, ∆, δ, q0⟩ where
• Q, Σ, ∆, q0 as in Moore-machine
retract
retracted
opened
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Moore machine
Landing gear: Moore-machine
Door:
3
Embedded systems simulation and verification
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Dynamic systems with output
yi = λ (ui , qi)
1
hang-up
hang-up
0
1
2
3
4
hang-up
extend
extended
extend
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• δ is a relation: δ ⊆ Q × 2Σ × 2∆ × Q that
for a given state and possible inputs
gives possible output and next state
• For deterministic machines the function
λ is defined based on δ
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Exercise
• Non-determinism:
– Note: δ ⊆ Q × 2Σ × 2∆ × Q is a
notation that describes a relation
between elements in a domain. Same
set of elements can be represented by
a function δ: Q × 2Σ × 2∆ → 2Q
• Relative expressiveness of DFA and
NDFA
• Expressiveness of Moore and Mealy
machines
• Draw the landing gear as a Mealy
machine!
Embedded systems simulation and verification
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FAQs
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Composition
• Assume that the subsystems are
synchronous, i.e. When one takes one
step the others take (at least) one step
• Then one can define the synchronous
composition
• Special consideration if the subsystems
communicate with each other
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Parallel composition
F1
q1
F1
F2
F2
q2
q2
q1
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||-operator for Mealy
Let M1 = ⟨Q1, Σ1, ∆1, δ1, q01⟩ and
• Q = Q1 × Q2
• q0 = ⟨q01,q02⟩
• Σ = Σ1 ∪ Σ2 - ( ∆1 ∪ ∆2)
• ∆ = ∆1 ∪ ∆2
• δ ⊆ Q × 2Σ × 2∆ × Q such that
⟨ ⟨q1,q2⟩, i, o, ⟨q1´,q2´⟩ ⟩ ∈ δ iff
⟨ q1, (i ∪ o) ∩ Σ1, o ∩ ∆1, q1´ ⟩ ∈ δ1 and
⟨ q2, (i ∪ o) ∩ Σ2, o ∩ ∆2, q2´ ⟩ ∈ δ2
M2 = ⟨Q2, Σ2, ∆2, δ2, q02⟩ be two Mealymachines where
∆1 ∩ ∆2 = ∅
Then the synchronous composition
M = M1 || M2 is a Mealy-machine
⟨Q, Σ, ∆, δ, q0⟩ such that:
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Exercise
Serial composition
• Construct the landing gear door/gear
composition as a Mealy machine!
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F1
F2
q1
q2
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Formal definition
Let M1 = ⟨Q1, Σ, ∆1, δ1, λ1, q01⟩ and
M2 = ⟨Q2, ∆1, ∆2, δ2, λ2, q02⟩ be Moore-machines.
• The serial composition M of M1 and M2
= ⟨Q, Σ, ∆, δ, λ, q0⟩ is defined as:
• Q = Q1 × Q2
• ∆ = ∆2
Embedded systems simulation and verification
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• q0 = ⟨q01,q02⟩
• δ(⟨q1,q2⟩, i) =
⟨ δ1(q1,i), δ2(q2, λ1(δ1(q1,i))) ⟩
• λ(⟨q1,q2⟩) = λ2 (q2)
Exercise: Define serial composition for Mealy
and parallel composition for Moore machines!
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