1. Formal Methods in software development
a.y.2017/2018
Prof. Anna Labella
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2. The predicate calculus [H-R ch.2]
The need for a richer language
Terms and formulas
quantifiers
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3. Syntax Inductive term definition BNF t::= x | c | f(t1,t2,…,tn)
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4. Syntax Inductive definition of wff BNF
::= p(t1,t2,…,tn) | t1=t2 | () | () |
() | () | x | x
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5. 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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6. 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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7. 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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8. Semantics environments Assigning values l: var  elements of A
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9. Semantics
Summing up, we are given with
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10. 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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11. Semantics
An interpretation M is a model of  M |=l  :
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12. Semantics: satisfiability
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13. Exercises
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14. Semantics: validity
is valid if for every intepretation and for every environment
M |= 
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15. Properties Soundness  |-   M |=  Completeness M |=    |- 
Indecidability
Compactness
Expressivity
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16. Second order logic Existential second order logic
Universal second order logic
Peano’s arithmetics
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17. 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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18. 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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19. We are possibly dealing with
Γ |-- φ proof theory
Γ |= φ semantic entailment
M |= φ satifiability
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20. We are possibly dealing with
Γ |-- φ
Γ |= φ
M |= φ
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21. We are possibly dealing with
Γ |-- φ
Γ |= φ
M |= φ
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22. 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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23. A formula can change its truth value
We build a model M
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 |= φ or not
Time could change the truth value of a formula
M,s |= φ or not for a given state s
In this last case it is often possible using the model checker to have a counterexample
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24. Models and states
A model M is an abstraction: it can describe very different things and omits lot of particulars
A model M is a transition system
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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25. A transition system 1
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26. A transition system 2
unwinding
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27. Linear and branching time
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28. Exercises
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