1. Formal Methods in software development
a.y.2016/2017
Prof. Anna Labella
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2. OBDD [H-R cap.6] Representing boolean functions
A formula can be represented by a boolean function,
where its variables are boolean variables, as in circuits.
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3. OBDD [H-R cap.6]
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4. Exercises
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5. Binary decision trees
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6. Exercises
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7. Exercises
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8. BDD Removal of duplicated terminals Removal of redundant tests
(they are nomore trees)
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9. BDD
Removal of duplicated non terminals
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10. BDD Eliminate duplicated terminals Eliminate redundant tests
Eliminate duplicated non terminals
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11. BDD: how do we introduce operations? Composing BDD
constants
variables ☐ 1
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12. BDD: sums and products
We can substitute terminals by non terminals and compose functions
e.g.
In the conjunction we substitute 1 by the BDD of the other function
In the disjunction we substitute 0 by the BDD of the other function
In the negation we swap 1 and 0
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13. Exercises
Let f be represented by
describe
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14. BDD Satisfiability: to reach 1 via a coherent path
Validity: it is not possible to reach 0 via a coherent path
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15. OBDD: ordered binary decision diagrams
Equivalence?
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16. OBDD Normal form: order and then reduce
Theorem. We have a unique result
Hence there is a canonical form
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17. OBDD An example: the parity function on 4 variables Repeated variables
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Repeated variables

18. OBDD
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order without repetitions

19. OBDD
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20. OBDD Composition is nomore directly allowed
Absence of redundant variables
Test for semanic equivalence with a compatible ordering
Test for validity (reducibility to B1)
Test for implication (reducibility of f. g to B0)
Test for satisfiability (non reducibility to B0)
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21. Exercises
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22. OBDD: the algorithm reduce
It identifies equal nodes going bottom up
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23. OBDD: the algorithm apply
It exploits operations
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24. OBDD: the algorithm apply
Shannon expansion theorem
ƒ = x ƒ[0/x] + x ƒ[1/x]
In general form
ƒ op g = xi ( ƒ[0/xi] op g[0/xi] + xi ( ƒ[1/xi] op g[1/xi])
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25. OBDD: the algorithm apply
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26. OBDD: the control structure for the algorithm apply
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27. OBDD: the algorithm apply
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28. OBDD: the algorithm restrict
restrictig f to f[0/x] and to f[1/x]
restrict(0, x, Bf) works as follows: For each node n labeled
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29. OBDD: restrict exercise
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30. OBDD: the algorithm exists
 f = f[0/x] + f[1/x]
 f = f[0/x] . f[1/x]
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31. OBDD: the algorithm exists exercise
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32. OBDD: the algorithm exists exercise
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33. 2/23/2019

34. Complexity
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