# Optimizing CTL Model checking + Model checking TCTL - PowerPoint PPT Presentation

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Optimizing CTL Model checking + Model checking TCTL. CS 5270 Lecture 9. A(FG p) not AF( AG p). Today…. Summary Optimizations for model checking ROBDDs TCTL- Syntax Semantics Algorithm for MC Optimizations. Summary: Model checking CTL. Optimization. The principal one:

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Optimizing CTL Model checking + Model checking TCTL

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## Optimizing CTL Model checking+Model checking TCTL

CS 5270Lecture 9

Lecture 8

Lecture 8

### Today…

• Summary

• Optimizations for model checking

• ROBDDs

• TCTL-

• Syntax

• Semantics

• Algorithm for MC

• Optimizations

Lecture 8

Lecture 8

### Optimization

• The principal one:

• Reduce to a problem with Boolean variables/Boolean formulæ

• Is this reasonable?

• Well – most modelling is done like this – even when you do have (non-boolean) variables

• + efficiencies from efficient operations on boolean functions

Lecture 8

### States as boolean formulæ

• Encode states using m boolean variables.

• Allows for 2m states.

• For example: m=3:

• S={s1,s2,s3,s4,s5,s6,s7,s8}

• Propositional booleans a,b,c:

• S={000,001,010,011,100,101,110,111}

• S = {abc, abc, abc , … }

Lecture 8

### Transitions as boolean formulæ

• Encode (s,s’) using before and after propositional boolean variables

• a,b,c and a’,b’,c’.

• For example: (s1,s4):

• (s1,s4) = (abc)  (a’b’c’)

Lecture 8

### Sufficient for modelling?

• Encode another mutual exclusion protocol

• Two processes, P1 and P2 share booleans

• Co-operate for mutual exclusion

• Third process T1 monitors and sets a turn variable

• System is parallel composition:

P1 || P2 || T1

Lecture 8

### Co-operative mutex: Process P1

P1 =

if (idle1) {

wait1 = true;

idle1 = false;

} else if (wait1 & idle2) {

active1 = true;

wait1 = false;

} else if (wait1 & wait2 & (!turn)) {

active1 = true;

wait1 = false;

}

if (active1) {

CritSect();

idle1 = true;

active1 = false;

}; ( followed by P1 )

Lecture 8

### Co-operative mutex: Process P2

P2 =

if (idle2) {

wait2 = true;

idle2 = false;

} else if (wait2 & idle1) {

active2 = true;

wait2 = false;

} else if (wait2 & wait1 & turn) {

active2 = true;

wait2 = false;

}

if (active2) {

CritSect();

idle2 = true;

active2 = false;

}; ( followed by P2 )

Lecture 8

### Co-operative mutex: Process T1

if (idle1 & wait2) {

turn = true;

} else if (idle2 & wait1) {

Turn = false;

}; ( followed by T1 )

(P1 || P2 || T1); System;

T1 =

System =

Lecture 8

Lecture 8

### Transitions as predicates

• P1 =

(i1w1’i1’)  (w1i2a1’w1’)

 (w1w2ta1’w1’)  (a1i1’a1’)

• P2 =

(i2w2’i2’)  (w2i1a2’w2’)

• (w2w1ta2’w2’)  (a2i2’a2’)

• T1 =

(i1w2t’)  (i2w1t’)

• Lecture 8

Lecture 8

Lecture 8

Lecture 8

Lecture 8

Lecture 8

Lecture 8

Lecture 8

Lecture 8

### History…

• The ROBDD optimization originally by Bryant (86) – paper on boolean graphs

• The application to model checking by McMillan (Originally in late 80’s – subject of thesis in 1992)

• smv – Symbolic model verifier – originally by McMillan

Lecture 8

### Today…

• Summary

• Optimizations for model checking

• ROBDDs

• TCTL-

• Syntax

• Semantics

• Algorithm for MC

• Optimizations

Lecture 8

### Regional transition system (RTS)

• Given TATTS = (s,s0,Act, ), then the RTS is a quotiented transition system

RTS = (Ř,Ř0, Act,), where

Ř= {(s,[v]t) | (s,v)s [v]tREGv}, and

Ř0= {(s,[v]t) | (s,v)s0 [v]tREGv}, and

• finally, (s,[v]t)  (s’,[v’]t) if and only if there is a transition (s,v) (s’,v’) in TATTS.

a

a

Lecture 8

### Regional transition system (RTS)

• Notation:

Ř – a set of regions

ř – a particular region in the set: (s,[v]t)

r – a particular valuation: (s,v)

Lecture 8

Lecture 8

### Kripke structure/model for TCTL

• Def: A TCTL model over a set of atomic propositions AP is the 4-tuple (Ř,Δ,AP,L)

• Ř – finite set of regions from RTS

• Δ ŘŘ - a total transition relation

• AP – a finite set of atomic propositions

• L: Ř→ 2AP – A labelling function which labels each region with the propositions true in that region

Note that the propositions may include clock constraints…

Lecture 8

### TCTL- syntax

• Given pAP, xX (model clock variables), zZ (property clock variables), (XZ) (clock constraints), then p and  are TCTL- formulæ, and if 1 and 2 are TCTL- formulæ then so are:

• 1

• 1  2

• 1  2

• z in 1

• A( 1U 2 )

• E( 1U 2 )

Lecture 8

### TCTL examples

• Note: temporal operators can be subscripted:

• A( 1U<72 ) means 1 holds until (within 7 time units) 2 becomes true.

• Implemented as: z in A( (1z<7) U2 )

• A( alarm U<7boiler-off): the alarm is on until (within 7 time units) the boiler-off is signaled.

• EF<7( alarm ) = E( true U<7alarm): the alarm will be on within 7 time units.

Lecture 8

### Semantics of TCTL

• Expressed in terms of a model, and the modelling relation² which links a model, a composite stater=(s,v) and a formula clock valuation with a property.

• M,(r,f)²P - means that (TCTL) property P holds in (or is satisfied in) state r in the case of a formula valuation f for a given model M

Lecture 8

### (Inductive) definition of ²

M,(r,f)²P  pL(ř)

M,(r,f)²  v  f ²

M,(r,f)²1 (M,(r,f)²1 )

M,(r,f)²1  2  M,(r,f)²1, and

M,(r,f)²2

M,(r,f)²1  2  M,(r,f)²1, or

M,(r,f)²2

Lecture 8

### (Inductive) definition of ²

• M,(r,f)²z in 1 M,(r,z in f)²1

• The notation z in f asserts that z is reset to 0 whenever it appears in the formula f

• M,(r,f)² A( 1 U2 ) for every path p from r, for some j, M,(j)²2, and i<j, M,(i)²1  2.

Lecture 8

### (Inductive) definition of ²

• M,(r,f)² E( 1 U2 )  for one path p from r, for some j,

M,(j)²2, and

i<j, M,(i)²1  2.

• Note that in both EU and AU, the condition up until 2 is 1  2. and not just 1!!

Lecture 8

Lecture 8

### Model checking TCTL

• Definition of a labelling algorithm in the notes – not much different from CTL

• The only problem is this definition uses a least fixpoint iteration over an infinite set…

• In practice use the region construction…

Lecture 8

### Optimization for TCTL MC

• We have already seen the steps to create a (finite) regional automaton

• Apart from that there is no magic bullet, and real-time model checking has an equivalent region-space explosion

• For this reason, limit the size of systems

• … so far …

Lecture 8

### Uppaal – more formally

• TCTL, but with restrictions that amount to only safety (reachability) formulæ:

• Set of clock constraints Z in formula is {}

• Syntax just AG() and EF() (outer level)

•  ::= a | x op n |  | 12 (op {,,,,})

• a is a location in the model

• Other properties (bounded liveness…) require extended models/automatons:

• compare system model with other test model

Lecture 8