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Software Verification 2 Automated Verification. Prof. Dr. Holger Schlingloff Institut für Informatik der Humboldt Universität and Fraunhofer Institut für Rechnerarchitektur und Softwaretechnik. Recap: LTS. LTS=( , S, , S 0 )  is a nonempty finite alphabet

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Software Verification 2 Automated Verification

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## Software Verification 2Automated Verification

Prof. Dr. Holger Schlingloff

Institut für Informatik der Humboldt Universität

and

Fraunhofer Institut für Rechnerarchitektur und Softwaretechnik

### Recap: LTS

• LTS=(, S, , S0)

•  is a nonempty finite alphabet

• S is a nonempty finite set of states

•   S    S is the transition relation

• S0  S is the set of initial states

remark: sometimes a pseudo state s0S is used instead of S0S;sometimes there is only a single initial state s0S

• state = (program counter(s), variable valuation)transition = (state, instruction, state)

• S0 can be written as a predicate on variables and pc’s

• init: (pc==  x==0  y<=5  ...)

•  can be written as a predicate on current and next variables

• : ((pc==  x‘==x+1)  (pc== x‘==x+2)  ...)

### Boolean Equivalences

next(state):= case

inp=0: state;

inp=50 & state=s0: s50;

inp=50 & state=s50: s0;

esac;

((inp==0  state‘==state) 

(inp==50  state=s0  state‘== s50) 

(inp==50  state=s50  state‘==s0) )

((inp==0  state‘==state) 

(inp==50  (state=s0  state‘== s50 ) 

(state=s50  state‘== s0 )

)

)

• Parallel transition system / state machine

• T=(T1,...,Tn)

• all state sets must be pairwise disjoint

• Global TS associated with parallel TS: T=(, S, , S0), where

• = i

• S=S1 ...  Sn

• S0=S10 ... Sn0

• ((s1,...,sn), a, (s1’,...,sn’))   iff for all Ti,

• if a  i, then (si, a, si’)  i, and

• if a  i, then si’= si

• Complexity (size of this construction)? Correctness???

### Correctness

• T=(T1,...,Tn), T =T1 ...  Tn

• Intuitively: T accepts/generates exactly those sequences which are accepted/generated by all Ti

• projection of run onto the alphabet of a transition system: =123...|Ti =if (1i) then 1 (23...)|Ti else (23...)|Ti

• Show: T acc  iffi (Ti acc | Ti )

• can also be used as a definition

• ### Parallel State Machines

• Parallel state machine

• T=(T1,...,Tn), i=2E  C  2A

• What is the global state machine associated with a parallel state machine? (“flattening”)

• synchronization by common e[c]/a is not an option

• possible choices: synchronize or compete on common input events (triggers)?

• what if an effect contains sending of a trigger?

(“run-to-completion-semantics”: tedious formalization)

### Introducing Data

• Simple state machines

• E: set of events, C: set of conditions, A: set of actions

• a simple state machine is an LTS where =2E  C  2A

• Extended state machine: Assume a first-order signature (D, F, R) with finite domains D and a set V of program variables on these domains. An ESM is a simple state machine where

• a guard is a quantifier-free first-order formula on (D, F, R) and V

• an action is an assignment V=T

• Attention: the effect of a transition is a set of actions!Parallel execution introduces nondeterminism.

### Introducing Hierarchies

• In a UML state machine, a state may contain other states

• powerful abstraction concept

• semantics can be tedious

### Introducing Visibility Scopes

• A state machine can be part of a class or module

• all variables are visible within the module only

• modules may be nested

• Classes or modules can be parameterized

• instances of classes are objects

### Introducing Fairness

• LTSs cannot specify that something will eventually happen

• only maximal sequences are accepted (terminating or infinite)

• want to express that in infinite runs, certain states must occur infinitely often

• Just LTS=(LTS,J), where J=(J1,...,Jm), JiS(justice requirements)

• for each JiJ each infinite run must contain infinitely many sJi

• Fair LTS=(LTS,F), where F=(F1,...,Fm), Fi=(Pi,Qi), PiS, QiS(compassion requirements)

• for each FiF and each infinite run it holds that if it contains infinitely many sPi, then it also contains infinitely many sQi

• Cf. automata theory: Büchi- and Rabin-acceptance

### Example: Peterson’s Mutual Exclusion

{t=0; x=0; y=0;

{0:while(true){NC1: skip; 1:x=1; 2:t=1;

3:await(t==0  y==0); C1: skip;

4:x=0;}

||

{0:while(true){NC2: skip; 1:y=1; 2:t=0;

3:await(t==1  x==0); C2: skip;

4:y=0;}

}

### Summary: Finite State Modeling Concepts

• We discussed

• (parallel) while-Programs with finite domains

• Labeled transition systems

• Simple state machines

• Parallel transition systems / state machines

• UML state machines

• Object-oriented concepts

• Fairness Constraints (justice, compassion)

• Mutual simulation possible

• but may be tedious; cross-compiler technology