Model Checking Lecture 2 Tom Henzinger

Model Checking

Lecture 2

Tom Henzinger

Model-Checking Problem

I |= S

System model System property

a

a,b b

q1

q3q2

System model: State-transition graph

States Q = { q1, q2, q3 }

Atomic observations A = { a, b }

Transition relation Q Q

Observation function [ ] : Q 2A

Run: sequence of states q1, q2

Observation: set of atomic observations

Trace: sequence of observations {a}, {a,b}

-safety (finite runs) vs. liveness (infinite runs)

-linear time (traces) vs. branching time (runs)

-logic (declarative) vs. automata (executable)

System property: 2x2x2 choices

STL (Safe Temporal Logic)

-safety (only finite runs)

-branching (runs, not traces)

-logic

Defining a logic

1. Syntax:

What are the formulas?

2. Semantics:

What are the models?

Does model M satisfy formula ?

M |=

Propositional logics:

1. boolean variables (a,b) & boolean operators (,)

2. model = truth-value assignment for variables

Propositional modal (e.g. temporal) logics:

1. ... & modal operators (,)

2. model = set of (e.g. temporally) related prop. models

Propositional logics:

1. boolean variables (a,b) & boolean operators (,)

2. model = truth-value assignment for variables

Propositional modal (e.g. temporal) logics:

1. ... & modal operators (,)

2. model = set of (e.g. temporally) related prop. models observations

state-transition graph (“Kripke structure”)

STL Syntax

::= a | | | | U

boolean variable (atomic

observation)

boolean operators

modal operators

STL Model

( K, q )

state-transition graph

state of K

STL Semantics

(K,q) |= a iff a [q]

(K,q) |= iff (K,q) |= and (K,q) |=

(K,q) |= iff not (K,q) |=

(K,q) |= iff exists q’ s.t. q q’ and (K,q’) |=

(K,q) |= U iff exist q0, ..., qn s.t. 1. q = q0 q1 ...

qn 2. for all 0 i < n, (K,qi) |= 3. (K,qn) |=

EX exists next

= AX forall next

U EU exists until

= true U EF exists eventually

= AG forall always

W = ( () U ( ))

AW forall waiting-for (forall weak-

until)

Defined modalities

Important safety properties

Invariance a

Sequencing a W b W c W d

= a W (b W (c W d))

Important safety properties: mutex protocol

Invariance (in_cs1 in_cs2)

Sequencing ( req_cs1

in_cs2 W in_cs2 W in_cs2 W in_cs1 )

Branching properties

Deadlock freedom true

Possibility (a b)

(req_cs1 in_cs1)

CTL (Computation Tree Logic)

-safety & liveness

-branching time

-logic

[Clarke & Emerson; Queille & Sifakis 1981]

CTL Syntax

::= a | | | | U |

CTL Model

( K, q )

fair state-transition graph state of K

CTL Semantics

(K,q) |= iff exist q0, q1, ... s.t.

1. q = q0 q1 ... is an infinite fair run

2. for all i 0, (K,qi) |=

EG exists always

= AF forall eventually

W = ( U ) ( )

U = ( W ) ()

Defined modalities

Important liveness property

Response (a b)

(req_cs1 in_cs1)

If only universial properties are of interest,

why not omit the path quantifiers?

LTL (Linear Temporal Logic)

-safety & liveness

-linear time

-logic

[Pnueli 1977; Lichtenstein & Pnueli 1982]

LTL Syntax

::= a | | | | U

LTL Model

infinite trace t = t0 t1 t2 ...

(K,q) |= iff for all t L(K,q), t |=

(K,q) |= iff exists t L(K,q), t |=

Language of deadlock-free state-transition graph K at state q :

L(K,q) ... set of infinite traces of K starting at q

X next

U U until

= true U F eventually

= G always

W = ( U ) W waiting-for (weak-until)

Defined modalities

Important properties

Invariance a

(in_cs1 in_cs2)

Sequencing a W b W c W d

( req_cs1

in_cs2 W in_cs2 W in_cs2 W in_cs1 )

Response (a b)

(req_cs1 in_cs1)

Composed modalities

a infinitely often a

a almost always a

Where did fairness go ?

Unlike in CTL, fairness can be expressed in LTL !

So there is no need for fairness in the model.

Weak (Buchi) fairness :

(enabled taken )

(enabled taken)

Strong (Streett) fairness :

( enabled ) ( taken )

Starvation freedom, corrected

(in_cs2 out_cs2)

(req_cs1 in_cs1)

CTL cannot express fairness

a a

b b

ba aq0

q1 q2

LTL cannot express branching

Possibility (a b)

So, LTL and CTL are incomparable.

(There are branching logics that can express fairness, e.g. CTL* = CTL + LTL, but they lose the computational attractiveness of CTL.)

Finite Automata

-safety (no infinite runs)

-linear or branching time

-automata (not logic)

Specification Automata

Syntax, given a set A of atomic observations:

S finite set of states

S0 S set of initial states

S S transition relation

: S PL(A) where the formulas of PL are

::= a | |

for a A

Language L(M) of specification automaton

M = (S, S0, , ) :

finite trace t0, ..., tn L(M)

iff

there exists a finite run s0 s1 ... sn of M

such that

for all 0 i n, ti |= (si)

(K,q) |=L M iff L(K,q) L(M)

Linear semantics of specification automata:

language containment

state-transition graph

state of K

specification automaton

finite traces

Invariance specification automaton

in_cs1

in_cs2

Starvation freedom specification automaton

out_cs1

req_cs1

in_cs2req_cs1

in_cs2

in_cs1 req_cs1

in_cs2

Automata are more expressive than logic, because temporal logic cannot count :

This cannot be expressed in LTL.

(How about a (a a) ?)

a true

Checking language containment between finite automata is PSPACE-complete !

L(K,q) L(M)

iff

L(K,q) complement( L(M) ) =

involves determinization (subset construction)

In practice:

1. require deterministic specification automata

2. use monitor automata

3. use branching semantics

Monitor Automata

Syntax:

same as specification automata, except also set E S of error states

Semantics:

define L(M) s.t. runs must end in error states

(K,q) |=C M iff L(K,q) L(M) =

Invariance monitor automaton

in_cs1

in_cs2

in_cs1

in_cs2

Starvation freedom monitor automaton

out_cs1

req_cs1

in_cs2req_cs1

in_cs2

in_cs1 req_cs1

in_cs2

req_cs1

in_cs2

Specification automaton Monitor automaton

M complement(M)

-describe correct traces -describe error traces

-check language containment-check emptiness (linear): (exponential) reachability of error states

“All safety verification is reachability checking.”

Main problem with deterministic specifications and monitor automata:

not suitable for stepwise refinement / abstraction

S1 |= S2 |= S3

“refines”

In practice:

1. require deterministic specification automata

2. use monitor automata

3. use branching semantics

(K,q) |=B M

iff

there exists a simulation relation R Q S s.t. (q,s) R for some initial state s of M

Branching semantics of specification automata:

simulation

states of K

states of M

R Q S is a simulation relation

iff

(q,s) R implies

1. [q] |= (s)

2. for all q’ s.t. q q’ , exists s’ s.t. s s’

and (q’,s’) R.

[Milner 1974]

(K,q) |=L M M language contains (K,q) :exponential check

(K,q) |=B M M simulates (K,q) :quadratic check

X

involves only traces (hence linear !)

involves states (hence branching !)

In practice, simulation is usually the “right” notion.

(If there is language containment, but not simulation, this is usually accidental, not by design.)

Finite Omega-Automata

-safety & liveness (infinite runs !)

-linear or branching time

-automata (not logic)

-specification vs. monitor automata

-linear (language containment) vs. branching (simulation) semantics

We discuss only the linear specification case.

Specification Omega-Automata

Syntax as for finite automata, in addition one of the following acceptance conditions:

Buchi: BA S

coBuchi: CA S

Streett: SA 2S 2S

Rabin: RA 2S 2S

Language L(M) of specification omega-automaton

M = (S, S0, , , A ) :

infinite trace t0, t1, ... L(M)

iff

there exists an infinite run s0 s1 ... of M

such that

1. s0 s1 ... satisfies A

2. for all i 0, ti |= (si)

Let Inf(s) = { p | p = si for infinitely many i }.

The infinite run s satisfies the acceptance condition A

iff

Buchi: Inf(s) BA

coBuchi: Inf(s) CA

Streett: for all (l,r) SA, if Inf(s) l then Inf(s) r

Rabin: for some (l,r) RA, Inf(s) l = and Inf(s) r

Buchi: BA

coBuchi: CA

Streett: (l r)

Rabin: (l r)

(K,q) |=L M iff L(K,q) L(M)

Linear semantics of specification omega-automata:

omega-language containment

infinite traces

Response specification automaton :

(a b) assuming (a b) = false

a b

ba

s1

s2

s3

s0

Buchi condition { s0, s3 }

a a

s0s1

Buchi condition { s0 }

No coBuchi condition

a

Streett condition { ({s0,s1}, {s0}) }

Rabin condition { (,{s0}) }

a a

s0s1

No Buchi condition

coBuchi condition { s0 }

a

Streett condition { ({s1}, ) }

Rabin condition { ({s1}, {s0,s1}) }

a a

s0s1

Buchi condition { s2 }

a

a

s2

-Buchi and coBuchi automata cannot be determinized

-Streett and Rabin automata can be determinized

nondeterministic Buchi =

deterministic Streett = deterministic Rabin =

nondeterministic Streett = nondeterministic Rabin =

omega-regular [Buchi 1960]

Omega-automaton determinization is even harder (conceptually, at least) than finite-automaton determinization [Safra 1989].

So monitor automata and simulation are particularly important.

Omega-automata are strictly more expressive than LTL

Omega-automata: omega-regular languages

LTL: counter-free omega-regular languages

Omega-automata: omega-regular languages = second-order theory of monadic

predicates & successor = omega-regular expressions

LTL: counter-free omega-regular languages = first-order theory of monadic predicates & successor = star-free omega-regular expressions

Structure of the omega-regular languages

Streett = Rabin

Buchi coBuchi

FinitecoFinite

Structure of the counter-free omega-regular languages

positive boolean combinations of and

The location of a linear-time property in the Borel hierarchy indicates how hard (theoretically as well as conceptually) the corresponding model-checking problem is.

positive boolean combinations of and

safety

weak fair

strong fair

response

Model Checking Lecture 2 Tom Henzinger

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