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Translating from LTL to automataBook: Chapter 6

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Why translating?

Want to write the specification in some logic.

Want model-checking tools to be able to check the specification automatically.

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Preprocessing

Convert into normal form, where negation only applies to propositional variables.

¬ becomes }¬. ¬} becomes ¬. What about ¬ ( U )? Define operator R (Release) such that

¬ ( U ) = (¬) R (¬), ¬ ( R ) = (¬) U (¬).

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Semantics of p R q

p

q q qqq

¬p¬p¬p¬p

p

q qqq

¬p¬p¬p¬p

Almost like q W p, but not exactly:

p R q:

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Replace ¬T by F, and ¬F by T. Replace ¬ ( Ç ) by (¬) Æ (¬)

and ¬ ( Æ ) by (¬) Ç (¬)

1.Convert to Negation Normal Form

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2. Eliminate !, },

Replace ! by (¬ ) Ç . Replace } by (T U ). Replace by (F R ).

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Example

Translate ( } P ) ! ( } Q ) Eliminate implication ¬( } P ) Ç ( } Q ) Eliminate , } :

¬( F R ( T U P ) ) Ç ( F R ( T U Q ) ) Push negation inwards:

(T U (F R ¬ P ) ) Ç ( F R ( T U Q ) )

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The data structure

Incoming

New Old

NextName

Incoming = Set of nodes with an edge to this node

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The main idea

U = Ç ( Æ O ( U ) ) R = Æ ( Ç O ( R ) ) This separates the formulas to two

parts: One holds in the current state, and The other in the next state.

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How to translate?

Take one formula from “New” and add it to “Old”.

According to the formula, either Split the current node into two, or Evolve the node into a new version.

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Splitting

Incoming

New Old

Next

Incoming

New Old

Next

Incoming

New Old

Next

Copy incoming edges, update other field.

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Evolving

Incoming

New Old

Next

Incoming

New Old

Next

Copy incoming edges, update other field.

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Possible cases:

U , split: 1. Add to New, add U to Next.2. Add to New.Because U = Ç ( Æ O ( U )).

R , split:1. Add to New.2. Add to New, R to Next.Because R = Æ ( Ç O ( R )).

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More cases:

Ç , split:1. Add to New.2. Add to New.

Æ , evolve:1. Add to New.

O , evolve:1. Add to Next.

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Example} p

(T U p)Old: {}New: {T U p}Next: {}

Old: {T U p}New: {T}Next: {T U p}

Old: {T U p}New: {p}Next: {}

Old:{T U p}New:{}Next:{T U p}

Old:{T U p}Next:{T U p}

Old:{T U p, p}New:{}Next:{}

Old:{T U p, p}Next:{}

T pp

Old:{}New:{}Next:{}

Old:{}Next:{}

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How to start?

Incoming

New Old

Next

init

aU(bUc)

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Incoming

init

aU(bUc)

Incoming Incoming

aU(bUc)aU(bUc) bUc

aU(bUc)

a

init init

Old

Next

New

Old

Next

New

Keep track of nodes evolving from original formula (init)

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Incoming

aU(bUc)bUc

init initIncoming Incoming

aU(bUc), bUc

aU(bUc),bUc

c

(bUc)

b

Old

Next

Newinit

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When to stop splitting?

When “New” is empty. Then compare against a list of existing

nodes “Nodes”: If a node with the same “Old”, “Next” exists,

just add the incoming edges of the new versionto the old one.

Otherwise, add the node to “Nodes”. Generate a successor with “New” set to “Next” of father.

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a,aU(bUc)

aU(bUc)

Incoming

init

Incoming

aU(bUc)

Creating a successor node.

Old

Next

New

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How to obtain the automaton?

There is an edge from node X to Y if X is in the incoming list of Y

Initial node is init.

Incoming

New Old

Next

Node X

Node Y

a, b, ¬c

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Old:New: aU(bUc)Next:

Old: aU(bUc)New:aNext: aU(bUc)

Old: aU(bUc),(bUc)New:bNext:(bUc)

Old: aU(bUc)New:(bUc)Next:

Old: aU(bUc),(bUc)New:cNext:

Old: aU(bUc),aNew:Next: aU(bUc)1

Old: aU(bUc),(bUc),bNew:Next:(bUc)2

Old: aU(bUc),(bUc),cNew:Next:3

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Old:New: aU(bUc)Next:

Old: aU(bUc),aNew:Next: aU(bUc)1

Old: aU(bUc),(bUc),bNew:Next:(bUc)2

Old: aU(bUc),(bUc),cNew:Next:3

If we continue here, we re-create nodes 1,2,3. So ‘1’ will be added to the ‘incoming’ list of 1,2,3.

Old:New: bUcNext:

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Old: bUcNew: bNext: bUc

Old: bUcNew: cNext:

Old: bUc,bNew: Next: bUc

Old: bUc,cNew: Next:If we continue here,

we re-create nodes 4,5. So ‘4’ will be added to the ‘incoming’ list of 4,5.

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Old: New:Next:

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The resulting nodes.

Initial states: All nodes with incoming edge from “init”.

a, aU(bUc) b, bUc, aU(bUc) c, bUc, aU(bUc)

b, bUc c, bUc

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Reminder: Buchi automata

A=<, S, , I, F> Alphabet (finite). S: States (finite). : S x x S ) S is the transition relation. I µ S are the Initial states. F µ S is a set of accepting states.

An infinite word is accepted in A if it passes an infinite no. of times in at least one of the F states

A

A

B BS0 S1

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Generalized Buchi automata

A=<, S, , I, F> Alphabet (finite). S: States (finite). : S x x S ) S is the transition relation. I µ S are the Initial states. F µ 2S is a set of sets of accepting states.

An infinite word is accepted in A if it passes an infinite no. of times in at least one state in element of F

A

A

B BS0 S1

F1 = {S0}F2 = {S0,S1}

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Generalized Büchi automaton Multiple accepting sets, one for each

Until sub-formula ( U ) such that The Old field contains

or The Old field doesn’t contain U

Accepting states

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Accepting w.r.t. bUc

All nodes with c in their past (Old), or without (commitment for) bUc in their past.

a, aU(bUc) b, bUc, aU(bUc) c, bUc, aU(bUc)

b, bUc c, bUc

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Acceptance w.r.t. aU(bUc)

All nodes with bUc in their past, or without aU(bUc) in their past.

a, aU(bUc) b, bUc, aU(bUc) c, bUc, aU(bUc)

b, bUc c, bUc

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Acceptance w.r.t. aU(bUc)

bUcaU(bUc)

a b c

b c

Accepting sets:

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De-generalization of GBA

Turn a generalized Büchi automaton into a Büchi automaton

Consider as many copies of the automaton as the number of accepting sets

Replace incoming edges from accepting states with edges to the next copy

Each cycle must go through every copy Each cycle must contain accepting states

from each accepting set

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Example

T

a b

T

a b

T

1

1,2

2

F a F b

1,2 correspond to F1 and F2, the accepting sets

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Example

T

a b

T

a

T

T

a b

T

b

T

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Example

T

a b

T

a

T

T

a b

T

b

T

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Example

T

a b

T

a

T

T

a b

T

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Example

T

a b

T

a

T T

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Example

T

a b

T

a

TT