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. 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
Free and bounded variables
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12. Syntax
Be careful with substitution!
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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
where x0 is a fresh variable
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21. Examples of proofs
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22. Exercises
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23. Exercises
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