1. Formal Methods in software development
a.y.2016/2017
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. Examples Every student preparing his homework must be left alone
x (((x)  ψ(x))  χ(x))
For every natural number there is number greater than it
x (N(x)  y (N(y) G(y,x))
x y G(y,x)
There are people who do not like their dog
x (x, d(x))
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4. Syntax Inductive term definition BNF t::= x | c | f(t1,t2,…,tn)
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5. Syntax Inductive definition of wff BNF
::= p(t1,t2,…,tn) | t1=t2 | () | () |
() | () | x | x
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6. Syntax
One can always do without function symbols, using predicates and equality
f(x)=y is the same as
F(x,y)  z (F(x,z)  y=z)
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7. Peano’s arithmetics Predicates: - = -, - ≤ -
Functions : - + -, - . -, s(-),…
Constants: 0
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8. Exercises
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9. Syntax
Free and bounded variables
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10. Syntax
Open and closed formulas
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11. Syntax
Be careful with substitution!
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12. Syntax
Free and bounded variables
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13. Natural deduction 1 A special predicate: equality =(-,-) or - = -
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14. Natural deduction 2
Proof rules for equality
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15. Natural deduction 3
We can prove symmetry
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16. Natural deduction 3
And also transitivity
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17. Exercises
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18. Natural deduction 1 Proof rules for universal quantification
where t is free for x in φ
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where x0 is a fresh variable

19. Natural deduction 2 Proof rules for existential quantification
where t is free for x in φ
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where x0 is a fresh variable

20. Examples of proofs
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where x0 is a fresh variable

21. Examples of proofs
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22. Exercises
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23. 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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24. 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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25. 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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26. Semantics environments Assigning values l: var  elements of A
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27. Semantics
Summing up, we are given with
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28. 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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29. Semantics
An interpretation M is a model of  M |=l  :
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30. Semantics: satisfiability
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31. Exercises
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32. Semantics: validity
is valid if for every intepretation and for every environment
M |= 
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33. Properties Soundness  |-   M |=  Completeness M |=    |- 
Indecidability
Compactness
Expressivity
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34. Second order logic Existential second order logic
Universal second order logic
Peano’s arithmetics
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