LTL – model checking

Jonas KongslundPeter MechlenborgChristian Plesner

Kristian Støvring Sørensen

OverviewSystem

Model

Büchi automaton (Asys)

Negation of property

PLTL-formula ()

Normal-form formula

Graph

Generalised Büchi automaton

Büchi automaton (A )

Product automaton (Asys A )

State space

Checking emptiness

Yes! No!

Model checker

Büchi Automata

• Def.: Labelled Büchi Automaton

over sequences infinite ofset

function labelling state ,2:

statesaccept ofset ,

functionn transitio,2:

statesstart ofset ,

states ofset finite ,

symbols ofset ,

),,,,,( :LBA

0

0

Sl

SF

S

SSØ

ØS

Ø

lFSS

S

Büchi Automata 2

• Def.: Run of a LBA

ALBA by the accepted is )(

.0 allfor

)(such that run acceptingan exists thereiff

A,LBA an by accepted is A word

sequence. in theoften infinitely occurs

state accepting oneleast at iff accepting is run A

.0for and where,:

10

10010

wwAL

i

sla

aaw

issSsss

ii

ii

Büchi Automata 3

• Example: Σ={a,b,c,d,e}

{a,d} {b}

{c}

(a|d)(bc+)ω

Büchi Automata 4

• For each PLTL formula φ one can construct an LBA Aφ s.t. Lω(Aφ) is the sequences of sets of atomic propositions that satisfy φ.

• Let Σ=2AP where AP is the set of atomic propositions.

Büchi Automata 5

• Def.: Generalised LBA

sequence. in theoften infinitely occurs Feach

from state accepting oneleast at iff accepting is run A

.,

sets, state acceptance ofset a hasit t except thaLBA an As

i

1

SF),F,(FF ik

Getting Normal

• Eliminate F and G operators

• Make negations adjacent to atomic propositions

• Example:

atruepfalse

atruepfalse

atrueptrue

atruep

ap

ap

alarmproblem

F

FF

FG

FG:

LTL → Normal Form → GLBA → LBA → LBA × LBA → Empty?

• Past operators do not add any expressive power to LTL

• Why are they useful?

• Past operators are not easy expressed with future operators

Getting Normal 2

problemalarmproblem G

LTL → Normal Form → GLBA → LBA → LBA × LBA → Empty?

• Past operators does not add any expressive power to LTL

• Why are they useful?

• Past operators are not easy to translate to normal form

• Possible exponential blowup

Getting Normal 3

problemalarmproblemproblemalarm GFG

problem. abeen has there

unless soundnot must alarm the:propertySafety

LTL → Normal Form → GLBA → LBA → LBA × LBA → Empty?

Normal Form → GLBALTL → Normal Form → GLBA → LBA → LBA × LBA → Empty?

ψ)) U X( (ψ ) (ψψ U

ψ)) UX( ( ψψ U

Overall idea: A node in the graph represents a state, an edge represent a step forward in time. Each node contains formulas that must be true at this time; view these formulas as proof obligations:

• Atomic propositions: check for contradictions

• Conjunctions: check both clauses

• Disjunctions: split into two nodes and allow a nondeterministic choice

• Next: Push proof obligation to the successors

• Until and its evil twin: unfold recursively on demand

Accept states 1LTL → Normal Form → GLBA → LBA → LBA × LBA → Empty?

Definition of strict p U q:

)p|)s(R.ikq|)s(i.(R q Up | s ki Sooner or later, q must happen!

{{q}, {p, q}} Ø

{{p}, {p, q}}

(Remember, every run is accepted, since the set of accept sets is empty)

Accept states 2LTL → Normal Form → GLBA → LBA → LBA × LBA → Empty?

Definition of strict p U q:

)p|)s(R.ikq|)s(i.(R q Up | s ki Sooner or later, q must happen!

{{q}, {p, q}} Ø

{{p}, {p, q}}

Problem: The automaton accepts pω!

Accept states 3LTL → Normal Form → GLBA → LBA → LBA × LBA → Empty?

Definition of strict p U q:

)p|)s(R.ikq|)s(i.(R q Up | s ki Sooner or later, q must happen!

{{q}, {p, q}} Ø

{{p}, {p, q}}

Solution: Insert accept states to break the cycle (not needed for U).

Un-generalizing GLBAs 1LTL → Normal Form → GLBA → LBA → LBA × LBA → Empty?

The generated automaton may have more than one set of accept states (one for each ‘until’ in the original formula):

Un-generalizing GLBAs 2LTL → Normal Form → GLBA → LBA → LBA × LBA → Empty?

Duplicate automaton. When leaving an accept state, jump to next automaton. Keep only one set of accept states.

Un-generalizing GLBAs 3LTL → Normal Form → GLBA → LBA → LBA × LBA → Empty?

Duplicate automaton. When leaving an accept state, jump to next automaton. Keep only one set of accept states.

Un-generalizing GLBAs 4LTL → Normal Form → GLBA → LBA → LBA × LBA → Empty?

Duplicate automaton. When leaving an accept state, jump to next automaton. Keep only one set of accept states.

Un-generalizing GLBAs 5LTL → Normal Form → GLBA → LBA → LBA × LBA → Empty?

Duplicate automaton. When leaving an accept state, jump to next automaton. Keep only one set of accept states.

Combining the two LBAs 1LTL → Normal Form → GLBA → LBA → LBA × LBA → Empty?

Wanted: an automaton accepting the intersection of the two languages:

x

Combining the two LBAs 2LTL → Normal Form → GLBA → LBA → LBA × LBA → Empty?

By the ordinary DFA product construction:

Problem: Requires accept states to be visited at the same time.

Combining the two LBAs 3LTL → Normal Form → GLBA → LBA → LBA × LBA → Empty?

Solution: Use a GLBA with two accept sets, then reduce to an LBA.

The emptiness problemLTL → Normal Form → GLBA → LBA → LBA × LBA → Empty?

How do we do it?

Find an appropriate cycle in the LBA – if no such cycle exists, the language is empty.

Why does this work?

Theorem 17.

Seriously, why?

In order for the language to be non-empty, there must be an infinite run of the automaton that visits an accept state infinitely often. This means that there has to be a reachable cycle containing an accept state.

OverviewSystem

Model

Büchi automaton (Asys)

Negation of property

PLTL-formula ()

Normal-form formula

Graph

Generalised Büchi automaton

Büchi automaton (A )

Product automaton (Asys A )

State space

Checking emptiness

Yes! No!

Model checker

The state space• Example

int i;proctype P1(){

do::true -> atomic( if::(i<2) -> i=i+1

fi)od }

proctype P2(){do::true -> atomic( if::(i!=2) -> i=2

::else -> i=0fi)

od }init{i=0; run(P1); run(P2);}

The state space 2

• A state– all global vars.– local vars. and program counter in all

processes

• State space: all possible simulations from the initial state

• State space must be finite

The state space 3

i=0

i=1 i=2

P1 and P2 enabledP1

and

P2

enab

led

P2 enabled

• Convert states to proposition tables– Get all propositions from the LTL expression– In each state

• Change the lable to the set of all satisfied propositions

State space → LBA

• Propositions:p:= (i <= 0)

q:= (i == 1)

r:= (i >= 2)

State space → LBA 2

i=0

i=1 i=2

p

q r

State space → LBA 3

• Make all paths infinite

• Make all states accepting – Product is now normal DFA product

The rest

• Is in chapter 5

References

• G. J. Holzmann: An improved protocol reachability analysis technique.

• O. Lichtenstein, A. Pnueli: The glory of the past.• R. Gerth et al.: Simple on-the-fly automatic verification of

linear temporal logic.• K. Etessami, G. J. Holzmann: Optimizing Büchi

automata.• A. M. Mikkelsen: On-the-fly model checking in

Design/CPN.• G. J. Holzmann: The model checker SPIN.

Exercises

• Exercises 8, 9, 10 (s3 should be s2), 12

• Derive the semantics of U from the semantics of U, and give an intuitive explanation.