COMP30112: Concurrency Topics 2.4: More FSP Theory Howard Barringer - - PowerPoint PPT Presentation

Topic 2.4: FSP Theory

COMP30112: Concurrency

Topics 2.4: More FSP Theory

Howard Barringer

Room KB2.20: email: Howard.Barringer@manchester.ac.uk

March 2008

Topic 2.4: FSP Theory

Outline

Topic 2.4: FSP Theory Process Equivalence Examples Strong Bisimilarity Minimising LTSs Weak Bisimilarity Conclusion

Topic 2.4: FSP Theory

Outline

Topic 2.4: FSP Theory Process Equivalence Examples Strong Bisimilarity Minimising LTSs Weak Bisimilarity Conclusion

Topic 2.4: FSP Theory

FSP Process Equivalence

Question: When Are FSP Processes Equivalent? i.e. when do they exhibit the ‘same’ behaviour?

Topic 2.4: FSP Theory

FSP Process Equivalence

Question: When Are FSP Processes Equivalent? i.e. when do they exhibit the ‘same’ behaviour?

• Answer 1: When they have isomorphic LTSs?

Topic 2.4: FSP Theory

FSP Process Equivalence

Question: When Are FSP Processes Equivalent? i.e. when do they exhibit the ‘same’ behaviour?

• Answer 1: When they have isomorphic LTSs?

Two LTSs are isomorphic if they are the same except that the names of the states may be different.

Topic 2.4: FSP Theory

FSP Process Equivalence

Question: When Are FSP Processes Equivalent? i.e. when do they exhibit the ‘same’ behaviour?

• Answer 1: When they have isomorphic LTSs?

Two LTSs are isomorphic if they are the same except that the names of the states may be different. (P =1 Q) if lts(P) is isomorphic to lts(Q) (lts(P) is the LTS of the process expression P)

Topic 2.4: FSP Theory

• Answer 2: When they have the same trace sets?

Topic 2.4: FSP Theory

• Answer 2: When they have the same trace sets?

The trace set of an FSP process is the set of traces of its LTS; i.e. the set of finite sequences a1 · · · an of actions that can label a transition path of the LTS, starting from the start state.

Topic 2.4: FSP Theory

• Answer 2: When they have the same trace sets?

The trace set of an FSP process is the set of traces of its LTS; i.e. the set of finite sequences a1 · · · an of actions that can label a transition path of the LTS, starting from the start state.

(P =2 Q) if traces(lts(P)) = traces(lts(Q)) i.e. the LTSs are trace-equivalent (traces(T) is the set of traces of the LTS T)

Topic 2.4: FSP Theory

• Answer 2: When they have the same trace sets?

The trace set of an FSP process is the set of traces of its LTS; i.e. the set of finite sequences a1 · · · an of actions that can label a transition path of the LTS, starting from the start state.

(P =2 Q) if traces(lts(P)) = traces(lts(Q)) i.e. the LTSs are trace-equivalent (traces(T) is the set of traces of the LTS T)

• Answer 3: When nothing they can interact with can tell the

difference?

Topic 2.4: FSP Theory

• Answer 2: When they have the same trace sets?

The trace set of an FSP process is the set of traces of its LTS; i.e. the set of finite sequences a1 · · · an of actions that can label a transition path of the LTS, starting from the start state.

(P =2 Q) if traces(lts(P)) = traces(lts(Q)) i.e. the LTSs are trace-equivalent (traces(T) is the set of traces of the LTS T)

• Answer 3: When nothing they can interact with can tell the

difference? (P =equiv Q) if ∀R · PR behaves the same as QR

Topic 2.4: FSP Theory

• Answer 2: When they have the same trace sets?

The trace set of an FSP process is the set of traces of its LTS; i.e. the set of finite sequences a1 · · · an of actions that can label a transition path of the LTS, starting from the start state.

(P =2 Q) if traces(lts(P)) = traces(lts(Q)) i.e. the LTSs are trace-equivalent (traces(T) is the set of traces of the LTS T)

• Answer 3: When nothing they can interact with can tell the

difference? (P =equiv Q) if ∀R · PR behaves the same as QR (P =1 Q) = ⇒ (P =equiv Q) = ⇒ (P =2 Q)

• r =1 ⊆ =equiv ⊆ =2

Topic 2.4: FSP Theory

Outline

Topic 2.4: FSP Theory Process Equivalence Examples Strong Bisimilarity Minimising LTSs Weak Bisimilarity Conclusion

Topic 2.4: FSP Theory

Examples

P1 = (a -> Q1 |a -> R1), Q1 = (b -> Q1), R1 = (b -> R1). P2 = (a -> Q2), Q2 = (b -> Q2).

a a b b Q1 R1 P1 a b P2 Q2

Obviously, =1 does not make enough things equal.

Topic 2.4: FSP Theory

Another Example

P = (a -> b -> P). Q = (c -> b -> Q). ||S1 = (P||Q). S2 = ( a -> c -> B | c -> a -> B), B = (b -> S2). S3 = ( a -> c -> B1 | c -> a -> B2), B1 = (b -> S3), B2 = (b -> S3).

a a c c b S1

S2 B a a c c b

B1 B2 S3 b a c a c b

⋆ Are they equivalent? ⋆

Topic 2.4: FSP Theory

And another ...

⋆ Should the following be equivalent? ⋆ P1 = ( a -> Q1), Q1 = ( b -> STOP | c -> STOP). P2 = ( a -> Q2 | a -> R2), Q2 = (b -> STOP), R2 = (c -> STOP). ⋆ (P1 =2 P2)? ⋆ ⋆ (P1 =equiv P2)? ⋆

Topic 2.4: FSP Theory

And another ...

⋆ Should the following be equivalent? ⋆ P1 = ( a -> Q1), Q1 = ( b -> STOP | c -> STOP). P2 = ( a -> Q2 | a -> R2), Q2 = (b -> STOP), R2 = (c -> STOP). ⋆ (P1 =2 P2)? ⋆ Yes, all traces the same ⋆ (P1 =equiv P2)? ⋆

Topic 2.4: FSP Theory

And another ...

⋆ Should the following be equivalent? ⋆ P1 = ( a -> Q1), Q1 = ( b -> STOP | c -> STOP). P2 = ( a -> Q2 | a -> R2), Q2 = (b -> STOP), R2 = (c -> STOP). ⋆ (P1 =2 P2)? ⋆ Yes, all traces the same ⋆ (P1 =equiv P2)? ⋆ No, consider parallel composition with:

Topic 2.4: FSP Theory

And another ...

⋆ Should the following be equivalent? ⋆ P1 = ( a -> Q1), Q1 = ( b -> STOP | c -> STOP). P2 = ( a -> Q2 | a -> R2), Q2 = (b -> STOP), R2 = (c -> STOP). ⋆ (P1 =2 P2)? ⋆ Yes, all traces the same ⋆ (P1 =equiv P2)? ⋆ No, consider parallel composition with: S1 = ( a -> b -> STOP ) + {c}.

Topic 2.4: FSP Theory

And another ...

⋆ Should the following be equivalent? ⋆ P1 = ( a -> Q1), Q1 = ( b -> STOP | c -> STOP). P2 = ( a -> Q2 | a -> R2), Q2 = (b -> STOP), R2 = (c -> STOP). ⋆ (P1 =2 P2)? ⋆ Yes, all traces the same ⋆ (P1 =equiv P2)? ⋆ No, consider parallel composition with: S1 = ( a -> b -> STOP ) + {c}. We need a notion of equivalence called Bisimilarity

Topic 2.4: FSP Theory

Outline

Topic 2.4: FSP Theory Process Equivalence Examples Strong Bisimilarity Minimising LTSs Weak Bisimilarity Conclusion

Topic 2.4: FSP Theory

Strong Bisimilarity

• Processes P and Q are equivalent, i.e. (P =equiv Q) if their LTSs

are strongly bisimilar: (lts(P) ∼ lts(Q))

Topic 2.4: FSP Theory

Strong Bisimilarity

• Processes P and Q are equivalent, i.e. (P =equiv Q) if their LTSs

are strongly bisimilar: (lts(P) ∼ lts(Q))

• LTSs are [strongly] bisimilar if they have

Topic 2.4: FSP Theory

Strong Bisimilarity

• Processes P and Q are equivalent, i.e. (P =equiv Q) if their LTSs

are strongly bisimilar: (lts(P) ∼ lts(Q))

• LTSs are [strongly] bisimilar if they have
• the same alphabet

Topic 2.4: FSP Theory

Strong Bisimilarity

• Processes P and Q are equivalent, i.e. (P =equiv Q) if their LTSs

are strongly bisimilar: (lts(P) ∼ lts(Q))

• LTSs are [strongly] bisimilar if they have
• the same alphabet
• compatible colourings

Topic 2.4: FSP Theory

Strong Bisimilarity

• Processes P and Q are equivalent, i.e. (P =equiv Q) if their LTSs

are strongly bisimilar: (lts(P) ∼ lts(Q))

• LTSs are [strongly] bisimilar if they have
• the same alphabet
• compatible colourings
• A colouring of an LTS is an assignment of a colour to each node.

Topic 2.4: FSP Theory

Strong Bisimilarity

• Processes P and Q are equivalent, i.e. (P =equiv Q) if their LTSs

are strongly bisimilar: (lts(P) ∼ lts(Q))

• LTSs are [strongly] bisimilar if they have
• the same alphabet
• compatible colourings
• A colouring of an LTS is an assignment of a colour to each node.
• The colourings of one or more coloured LTSs are compatible if

Topic 2.4: FSP Theory

Strong Bisimilarity

• Processes P and Q are equivalent, i.e. (P =equiv Q) if their LTSs

are strongly bisimilar: (lts(P) ∼ lts(Q))

• LTSs are [strongly] bisimilar if they have
• the same alphabet
• compatible colourings
• A colouring of an LTS is an assignment of a colour to each node.
• The colourings of one or more coloured LTSs are compatible if
• 1. The starting states of distinct LTSs have the same colour.

Topic 2.4: FSP Theory

Strong Bisimilarity

• Processes P and Q are equivalent, i.e. (P =equiv Q) if their LTSs

are strongly bisimilar: (lts(P) ∼ lts(Q))

• LTSs are [strongly] bisimilar if they have
• the same alphabet
• compatible colourings
• A colouring of an LTS is an assignment of a colour to each node.
• The colourings of one or more coloured LTSs are compatible if
• 1. The starting states of distinct LTSs have the same colour.
• 2. Whenever states P, Q (of the same or distinct LTS) have the

same colour then for any transition P

a

→ P′ there is a transition Q

a

→ Q′ where P′, Q′ have the same colour.

Topic 2.4: FSP Theory

Example: Non-isomorphic Bisimilar LTSs

P = (a -> Q), Q = (a -> P). P = (a -> P). P = (a -> P | a -> Q), Q = (a -> P). P = (a -> Q), Q = (a -> Q |a -> R), R = (a -> P).

⋆ Draw LTSs; calculate colouring? ⋆

Topic 2.4: FSP Theory

Example: Non-isomorphic Bisimilar LTSs

P = (a -> Q), Q = (a -> P). P = (a -> P). P = (a -> P | a -> Q), Q = (a -> P). P = (a -> Q), Q = (a -> Q |a -> R), R = (a -> P).

P Q a a P a P Q a a a R Q P a a a a

Topic 2.4: FSP Theory

Example: Non-isomorphic Bisimilar LTSs

P = (a -> Q), Q = (a -> P). P = (a -> P). P = (a -> P | a -> Q), Q = (a -> P). P = (a -> Q), Q = (a -> Q |a -> R), R = (a -> P).

Q P a a a P a a a P Q a a a a Q R P

Topic 2.4: FSP Theory

Example: Non-isomorphic Bisimilar LTSs

P = (a -> Q), Q = (a -> P). P = (a -> P). P = (a -> P | a -> Q), Q = (a -> P). P = (a -> Q), Q = (a -> Q |a -> R), R = (a -> P).

Q P a a a P a a a P Q a a a a Q R P

All the same colour, so they are bisimilar

Topic 2.4: FSP Theory

And some more ...

P1 = (a -> Q1 |a -> R1), Q1 = (b -> Q1), R1 = (b -> R1). P2 = (a -> Q2), Q2 = (b -> Q2). P3 = (a -> Q3 | a -> R3), Q3 = (b -> b -> b -> Q3), R3 = (b -> R3). P4 = (a -> b -> Q4 | a -> Q4), Q4 = (b -> Q4).

⋆ Draw LTSs; calculate colouring? ⋆

Topic 2.4: FSP Theory

And some more ...

P1 = (a -> Q1 |a -> R1), Q1 = (b -> Q1), R1 = (b -> R1). P2 = (a -> Q2), Q2 = (b -> Q2). P3 = (a -> Q3 | a -> R3), Q3 = (b -> b -> b -> Q3), R3 = (b -> R3). P4 = (a -> b -> Q4 | a -> Q4), Q4 = (b -> Q4).

a a b b Q1 R1 P1 a b P2 Q2 b b b a a b R3 Q3 P3 b b a a P3 Q3

Topic 2.4: FSP Theory

And some more ...

P1 = (a -> Q1 |a -> R1), Q1 = (b -> Q1), R1 = (b -> R1). P2 = (a -> Q2), Q2 = (b -> Q2). P3 = (a -> Q3 | a -> R3), Q3 = (b -> b -> b -> Q3), R3 = (b -> R3). P4 = (a -> b -> Q4 | a -> Q4), Q4 = (b -> Q4).

a a b b Q1 R1 P1 a b P2 Q2 b b b b a a R3 Q3 P3 b b a a P3 Q3

Topic 2.4: FSP Theory

And some more ...

P1 = (a -> Q1 |a -> R1), Q1 = (b -> Q1), R1 = (b -> R1). P2 = (a -> Q2), Q2 = (b -> Q2). P3 = (a -> Q3 | a -> R3), Q3 = (b -> b -> b -> Q3), R3 = (b -> R3). P4 = (a -> b -> Q4 | a -> Q4), Q4 = (b -> Q4).

a a b b Q1 R1 P1 a b P2 Q2 b b b b a a R3 Q3 P3 b b a a P3 Q3

Compatible colourings, so processes are bisimilar

Topic 2.4: FSP Theory

Example: Non-Bisimilar but Trace-Equivalent LTSs

P = ( a -> Q ), Q = ( b -> STOP | c -> d -> P). P = ( a -> Q | a -> R), Q = ( b -> STOP), R = ( c -> d -> P). ⋆ Draw LTSs; calculate colouring? ⋆

Topic 2.4: FSP Theory

Example: Non-Bisimilar but Trace-Equivalent LTSs

P = ( a -> Q ), Q = ( b -> STOP | c -> d -> P). P = ( a -> Q | a -> R), Q = ( b -> STOP), R = ( c -> d -> P).

d a b c Q

STOP

P a a b d c Q R

STOP

P

Topic 2.4: FSP Theory

Example: Non-Bisimilar but Trace-Equivalent LTSs

P = ( a -> Q ), Q = ( b -> STOP | c -> d -> P). P = ( a -> Q | a -> R), Q = ( b -> STOP), R = ( c -> d -> P).

d c b a

STOP

Q P a a b d c Q R

STOP

P

Topic 2.4: FSP Theory

Example: Non-Bisimilar but Trace-Equivalent LTSs

P = ( a -> Q ), Q = ( b -> STOP | c -> d -> P). P = ( a -> Q | a -> R), Q = ( b -> STOP), R = ( c -> d -> P).

d c b a

STOP

Q P a a b d c R

STOP

P Q

Topic 2.4: FSP Theory

Example: Non-Bisimilar but Trace-Equivalent LTSs

P = ( a -> Q ), Q = ( b -> STOP | c -> d -> P). P = ( a -> Q | a -> R), Q = ( b -> STOP), R = ( c -> d -> P).

d c b a Q

STOP

P a a b d c R

STOP

P Q

Topic 2.4: FSP Theory

Example: Non-Bisimilar but Trace-Equivalent LTSs

P = ( a -> Q ), Q = ( b -> STOP | c -> d -> P). P = ( a -> Q | a -> R), Q = ( b -> STOP), R = ( c -> d -> P).

d c b a Q

STOP

P a a b d c R

STOP

P Q

States Q and R on the right cannot have the same colour, and cannot be the same as Q on the left. Not equal!

Topic 2.4: FSP Theory

Computing a colouring

• 1. Start with LTS with all states the same colour.

Topic 2.4: FSP Theory

Computing a colouring

• 1. Start with LTS with all states the same colour.
• 2. Pick a colour c and an action α such that some of the states

with colour c can make the action α to some particular colour

• f state and some states cannot.

Topic 2.4: FSP Theory

Computing a colouring

• 1. Start with LTS with all states the same colour.
• 2. Pick a colour c and an action α such that some of the states

with colour c can make the action α to some particular colour

• f state and some states cannot.
• 3. Give the states with colour c that can make the action α a

new colour.

Topic 2.4: FSP Theory

Computing a colouring

• 1. Start with LTS with all states the same colour.
• 2. Pick a colour c and an action α such that some of the states

with colour c can make the action α to some particular colour

• f state and some states cannot.
• 3. Give the states with colour c that can make the action α a

new colour.

• 4. Repeat from step two until no further colour/action pairs

meet the criteria listed there.

Topic 2.4: FSP Theory

Colouring Example

P = ( a -> Q | b-> T ), Q = ( a -> U | b-> R ), R = ( a -> Q | b-> V ), S = ( a -> U | b-> T ), T = ( a -> S | b-> V ), U = c -> STOP, V = d -> STOP. ⋆ Draw LTS? ⋆

Topic 2.4: FSP Theory

Colouring Example

P = ( a -> Q | b-> T ), Q = ( a -> U | b-> R ), R = ( a -> Q | b-> V ), S = ( a -> U | b-> T ), T = ( a -> S | b-> V ), U = c -> STOP, V = d -> STOP.

d b a c b a a a b a b b V U T Q STOP S R P

Topic 2.4: FSP Theory

Colouring Example

P = ( a -> Q | b-> T ), Q = ( a -> U | b-> R ), R = ( a -> Q | b-> V ), S = ( a -> U | b-> T ), T = ( a -> S | b-> V ), U = c -> STOP, V = d -> STOP. ⋆ Calculate colouring? ⋆

d b a c b a a a b a b b V U T Q STOP S R P

Topic 2.4: FSP Theory

Colouring Example

P = ( a -> Q | b-> T ), Q = ( a -> U | b-> R ), R = ( a -> Q | b-> V ), S = ( a -> U | b-> T ), T = ( a -> S | b-> V ), U = c -> STOP, V = d -> STOP. ⋆ Calculate colouring? ⋆

d b a a c b a b a b b a V U T Q STOP S R P

split on a

Topic 2.4: FSP Theory

Colouring Example

P = ( a -> Q | b-> T ), Q = ( a -> U | b-> R ), R = ( a -> Q | b-> V ), S = ( a -> U | b-> T ), T = ( a -> S | b-> V ), U = c -> STOP, V = d -> STOP. ⋆ Calculate colouring? ⋆

d b a a c b a b a b b a V U T Q STOP S R P

split on a again

Topic 2.4: FSP Theory

Colouring Example

P = ( a -> Q | b-> T ), Q = ( a -> U | b-> R ), R = ( a -> Q | b-> V ), S = ( a -> U | b-> T ), T = ( a -> S | b-> V ), U = c -> STOP, V = d -> STOP. ⋆ Calculate colouring? ⋆

d b a a c b a b a b b a V U T Q STOP S R P

split on b

Topic 2.4: FSP Theory

Colouring Example

P = ( a -> Q | b-> T ), Q = ( a -> U | b-> R ), R = ( a -> Q | b-> V ), S = ( a -> U | b-> T ), T = ( a -> S | b-> V ), U = c -> STOP, V = d -> STOP. ⋆ Calculate colouring? ⋆

d b a a c b a b a b b a V U Q STOP S R P T

split on c

Topic 2.4: FSP Theory

Colouring Example

P = ( a -> Q | b-> T ), Q = ( a -> U | b-> R ), R = ( a -> Q | b-> V ), S = ( a -> U | b-> T ), T = ( a -> S | b-> V ), U = c -> STOP, V = d -> STOP. ⋆ Calculate colouring? ⋆

d b a a c b a b a b b a V T Q STOP S R P U

split on d

Topic 2.4: FSP Theory

Colouring Example

P = ( a -> Q | b-> T ), Q = ( a -> U | b-> R ), R = ( a -> Q | b-> V ), S = ( a -> U | b-> T ), T = ( a -> S | b-> V ), U = c -> STOP, V = d -> STOP. ⋆ Calculate colouring? ⋆

d b a a c b a b a b b a V T Q STOP S R P U

LTS is partitioned

Topic 2.4: FSP Theory

Review ...

• Fact: Isomorphic LTSs are bisimilar

Topic 2.4: FSP Theory

Review ...

• Fact: Isomorphic LTSs are bisimilar
• Fact: Bisimilar LTSs have the same trace sets.

Topic 2.4: FSP Theory

Review ...

• Fact: Isomorphic LTSs are bisimilar
• Fact: Bisimilar LTSs have the same trace sets.
• Fact: The relation ∼ is an equivalence relation, i.e.

reflexive, symmetric, transitive

Topic 2.4: FSP Theory

Review ...

• Fact: Isomorphic LTSs are bisimilar
• Fact: Bisimilar LTSs have the same trace sets.
• Fact: The relation ∼ is an equivalence relation, i.e.

reflexive, symmetric, transitive

• A colouring of an LTS is trivial if distinct states get distinct

colours.

Topic 2.4: FSP Theory

Review ...

• Fact: Isomorphic LTSs are bisimilar
• Fact: Bisimilar LTSs have the same trace sets.
• Fact: The relation ∼ is an equivalence relation, i.e.

reflexive, symmetric, transitive

• A colouring of an LTS is trivial if distinct states get distinct

colours.

• Simplification:

Topic 2.4: FSP Theory

Review ...

• Fact: Isomorphic LTSs are bisimilar
• Fact: Bisimilar LTSs have the same trace sets.
• Fact: The relation ∼ is an equivalence relation, i.e.

reflexive, symmetric, transitive

• A colouring of an LTS is trivial if distinct states get distinct

colours.

• Simplification:
• An LTS may have a non-trivial self-compatible colouring; i.e.

there are distinct states that get the same colouring.

Topic 2.4: FSP Theory

Review ...

• Fact: Isomorphic LTSs are bisimilar
• Fact: Bisimilar LTSs have the same trace sets.
• Fact: The relation ∼ is an equivalence relation, i.e.

reflexive, symmetric, transitive

• A colouring of an LTS is trivial if distinct states get distinct

colours.

• Simplification:
• An LTS may have a non-trivial self-compatible colouring; i.e.

there are distinct states that get the same colouring.

• Simplify LTS by identifying states that have the same colour.

(no problem with transitions due to self-compatibility property)

Topic 2.4: FSP Theory

Review ...

• Fact: Isomorphic LTSs are bisimilar
• Fact: Bisimilar LTSs have the same trace sets.
• Fact: The relation ∼ is an equivalence relation, i.e.

reflexive, symmetric, transitive

• A colouring of an LTS is trivial if distinct states get distinct

colours.

• Simplification:
• An LTS may have a non-trivial self-compatible colouring; i.e.

there are distinct states that get the same colouring.

• Simplify LTS by identifying states that have the same colour.

(no problem with transitions due to self-compatibility property)

• The new LTS will be bisimilar to the original one (Why?)

Topic 2.4: FSP Theory

Outline

Topic 2.4: FSP Theory Process Equivalence Examples Strong Bisimilarity Minimising LTSs Weak Bisimilarity Conclusion

Topic 2.4: FSP Theory

Minimisation of LTSs

An LTS is minimised if every self-compatible colouring is trivial. Fact: Every LTS can be minimised; i.e. for every LTS there is a minimised LTS that is bisimilar with it.

• LTS T not minimised? Then it has a non-trivial colouring.

Topic 2.4: FSP Theory

Minimisation of LTSs

An LTS is minimised if every self-compatible colouring is trivial. Fact: Every LTS can be minimised; i.e. for every LTS there is a minimised LTS that is bisimilar with it.

• LTS T not minimised? Then it has a non-trivial colouring.
• Identify states with the same colour to form T1. Then

T ∼ T1.

Topic 2.4: FSP Theory

Minimisation of LTSs

An LTS is minimised if every self-compatible colouring is trivial. Fact: Every LTS can be minimised; i.e. for every LTS there is a minimised LTS that is bisimilar with it.

• LTS T not minimised? Then it has a non-trivial colouring.
• Identify states with the same colour to form T1. Then

T ∼ T1.

• If T1 is minimised then finish.

Topic 2.4: FSP Theory

Minimisation of LTSs

An LTS is minimised if every self-compatible colouring is trivial. Fact: Every LTS can be minimised; i.e. for every LTS there is a minimised LTS that is bisimilar with it.

• LTS T not minimised? Then it has a non-trivial colouring.
• Identify states with the same colour to form T1. Then

T ∼ T1.

• If T1 is minimised then finish.
• If not we can repeat the procedure to get T2 with

T ∼ T1 ∼ T2.

Topic 2.4: FSP Theory

Minimisation of LTSs

An LTS is minimised if every self-compatible colouring is trivial. Fact: Every LTS can be minimised; i.e. for every LTS there is a minimised LTS that is bisimilar with it.

• LTS T not minimised? Then it has a non-trivial colouring.
• Identify states with the same colour to form T1. Then

T ∼ T1.

• If T1 is minimised then finish.
• If not we can repeat the procedure to get T2 with

T ∼ T1 ∼ T2.

• Termination: Ti has strictly fewer states than the previous

LTS Ti−1

Topic 2.4: FSP Theory

Outline

Topic 2.4: FSP Theory Process Equivalence Examples Strong Bisimilarity Minimising LTSs Weak Bisimilarity Conclusion

Topic 2.4: FSP Theory

Weak Bisimilarity

• ASCII tau is Greek letter τ!
• τ actions are not ‘observable’ so ignore them and simplify?
• Yes. . . sometimes, BUT they can lead to a significant change
• f state.
• Write transition path:

P

τ

→ · · · τ → Q as: P

τ

⇒ Q (zero or more τ transitions going from state P to state Q)

• If there are no τ transitions then P is the same state as Q.
• If a is a non-silent action (i.e. a is not τ)

Write transition path: P

τ

⇒ P′

a

→ Q′

τ

⇒ Q as: P

a

⇒ Q (where P′, Q′ are intermediate states)

Topic 2.4: FSP Theory

• LTSs are weakly bisimilar if they have weakly compatible

colourings.

• The colourings of one or more coloured LTSs are weakly

compatible if

• 1. The starting states of distinct LTSs have the same colour.
• 2. Whenever states P, Q (of the same or distinct LTS) have the

same colour then for any transition path P

a

⇒ P′ there is a transition path Q

a

⇒ Q′ where P′, Q′ have the same colour. (a may be the ‘empty’ action, ǫ).

• Strongly bisimilar LTSs are also weakly bisimilar
• Weak bisimulation is NOT preserved by Choice

Topic 2.4: FSP Theory

Outline

Topic 2.4: FSP Theory Process Equivalence Examples Strong Bisimilarity Minimising LTSs Weak Bisimilarity Conclusion

Topic 2.4: FSP Theory

A little background & history

• Strong bisimulation, the silent action τ, and weak bisimulation

were first introduced by Robin Milner in his process algebra Calculus of Communicating Systems, CCS, (see his book Communication and Concurrency, Prentice Hall (1989)).

Topic 2.4: FSP Theory

A little background & history

• Strong bisimulation, the silent action τ, and weak bisimulation

were first introduced by Robin Milner in his process algebra Calculus of Communicating Systems, CCS, (see his book Communication and Concurrency, Prentice Hall (1989)).

• Many other process algebras have been introduced, used and

studied, with a considerable variety of equivalences.

Topic 2.4: FSP Theory

A little background & history

• Strong bisimulation, the silent action τ, and weak bisimulation

were first introduced by Robin Milner in his process algebra Calculus of Communicating Systems, CCS, (see his book Communication and Concurrency, Prentice Hall (1989)).

• Many other process algebras have been introduced, used and

studied, with a considerable variety of equivalences.

• One of the main competing approaches is due to Tony Hoare,

his process algebra is called Communicating Sequential Processes, CSP, (see his book with this title, Prentice Hall, (1985)).

Topic 2.4: FSP Theory

A little background & history

• Strong bisimulation, the silent action τ, and weak bisimulation

were first introduced by Robin Milner in his process algebra Calculus of Communicating Systems, CCS, (see his book Communication and Concurrency, Prentice Hall (1989)).

• Many other process algebras have been introduced, used and

studied, with a considerable variety of equivalences.

• One of the main competing approaches is due to Tony Hoare,

his process algebra is called Communicating Sequential Processes, CSP, (see his book with this title, Prentice Hall, (1985)).

• FSP uses equivalences of CCS, but the parallel composition
• perator and syntax is more like CSP.

Topic 2.4: FSP Theory

A little background & history

• Strong bisimulation, the silent action τ, and weak bisimulation

were first introduced by Robin Milner in his process algebra Calculus of Communicating Systems, CCS, (see his book Communication and Concurrency, Prentice Hall (1989)).

• Many other process algebras have been introduced, used and

studied, with a considerable variety of equivalences.

• One of the main competing approaches is due to Tony Hoare,

his process algebra is called Communicating Sequential Processes, CSP, (see his book with this title, Prentice Hall, (1985)).

• FSP uses equivalences of CCS, but the parallel composition
• perator and syntax is more like CSP.
• FSP is finite-state; CCS and CSP may be infinite-state