1. Formal Methods in software development
a.y.2016/2017
Prof. Anna Labella
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2. Semantics syntactical data F functional symbols- constants
P predicate symbols
An interpretation M
a non empty set (domain) A
F  a set of functions fM on A (An)
P  a set of relations PM on A (An)
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3. Semantics
A subset is the interpretation of a 1-ary predicate A n-ary relation is the interpretation of a n-ary predicate
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4. An example: arithmetics
syntactical data
F: f1(-,-), f2(-,-), f0(-), c
P: - = -, P1(-,-)
An interpretation M
Natural numbers (domain)
Functions : - + -, - . -, s(-), 0
Predicates: - = -, - ≤ -
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5. Semantics environments Assigning values l: var  elements of A
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6. Semantics
Summing up, we are given with
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7. An example: arithmetics 2
f1(c,x)=x always true in the interpretation
f2(c,x)=x sometimes true sometimes false in the interpretation depending on the assignment
x(f1(c,x)=x) true in the interpretation
x(f2(c,x)=x) true in the interpretation
x(f2(c,x)=x) false in the interpretation
x(f1(c,x)=f1(c,x)) valid
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8. Semantics
An interpretation M is a model of  M |=l  :
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9. Semantics: satisfiability
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10. Exercises
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11. Semantics: validity
is valid if for every intepretation and for every environment
M |= 
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12. Properties Soundness  |-   M |=  Completeness M |=    |- 
Indecidability
Compactness
Expressivity
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13. Second order logic Existential second order logic
Universal second order logic
Peano’s arithmetics
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14. Specification, verification and logics
[H-R ch.3]
Logic provides:
A framework for modelling systems
A specification language for describing properties to be verified
A verification method to ascertain whether the description of the system satisfies the properties
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15. Possibilities of approaching model verification
Proof-based
Γ |-- φ
Γ is the description while φ is the property to be satisfied
Degree of automation:
Fully automatic
Full behaviour
Sequential
Reactive ….
A priori
Model based
M |= φ
M is a finite model
(only one)
Manual
One property
Concurrent
Terminating
…..
A posteriori
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16. We are possibly dealing with
Γ |-- φ
Γ |= φ
M |= φ
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17. We are possibly dealing with
Γ |-- φ
Γ |= φ
M |= φ
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18. We are possibly dealing with
Γ |-- φ
Γ |= φ
M |= φ
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19. Model Checking Concurrent systems Reactive systems Temporal aspects
Automatic
Based on a builded model
Verifying satisfiability
of properties
A posteriori
Provides a counterexample
Concurrent systems
Reactive systems
Temporal aspects
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20. A formula can change its truth value
A model M is a transition system
We model our system using the description language of the model checker
We code the property to be verified in the same language and the model checker should say whether
M,s |= φ or not for a given state s
In this last case it is often possible to have a counterexample
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21. Models and states A model M is a transition system
A model M is an abstraction: it can describe very different things
We have states and and transitions between them. An assignment statement can make the model move from one state to another one
We can think of a transition system as a set S of states together with a binary relation
   S ✕ S
and a labeling function L: S  P(atoms)
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22. A transition system 1
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23. A transition system 2
unwinding
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