Class 15: Recursive Data Structures

cs2102: Discrete Mathematics
University of Virginia, Fall 2017

See course site for notes:
https://uvacs2102.github.io

Transcript

1. Class 15: Recursive Data Structures cs2102: Discrete Mathematics | F17 uvacs2102.github.io David Evans University of Virginia
2. Plan for Week Recursive Data Structures: Structural Induction (MCS Chapter 7) o Book data structures: strings (balanced strings), non-negative integers, arithmetic expressions o Class data structures: lists (today), trees (Thurs) You should definitely read and understand what is in the book (and ask questions about anything), but we will (mostly) do different examples in class.
3. Finishing Proof from Thursday…
4. Correctness of Slow Exponentiation ∷= ℕ × ℕ ∷= { , ⟶ , − 1 | ∀ ∈ ℕ, ∀ ∈ ℕ3} 6 ∷= (1, ) What preserved invariant would be useful?
5. Proving Preserved Invariant = , ∷= = ;<= ∷= { , ⟶ , − 1 | ∀ ∈ ℕ, ∀ ∈ ℕ3}
6. Invariant is Preserved ∷= { , ⟶ , − 1 | ∀, ∈ ℕ3} = , ∷= = ;<= Does this prove slow_power(a, b) =
7. Invariant Principle = , ∷= = ;<= Base case: If a preserved invariant is true for the start state, it is true for all reachable states.
8. Invariant Principle = , ∷= = ;<= Prove slow_power(a, b) = Base case: (6 = 1, ) = ;<= since = 1 = 6 and − = 0. Inductive case: is a preserved invariant
9. Partial Correctness = , ∷= ∈ ℕ ∧ = ;<= Prove slow_power(a, b) = If machine reaches a terminating state A = (, 0) the result is correct: = ;. What else do we need to show?
10. Termination = , ∷= ∈ ℕ ∧ = ;<= Prove slow_power(a, b) = Eventually reaches state A = , 0
11. Reachability Reaches state (, ): ∷= { , ⟶ , − 1 | ∀, ∈ ℕ3} 6 ∷= (1, )
12. Reachability Reaches state (, ): ∷= { , ⟶ , − 1 | ∀, ∈ ℕ3} 6 ∷= (1, ) Is slow exponentiation fast enough? In = steps, the terminating state is reached.
14. Fast Exponentiation Read this example in the book 6.3.1 - same idea, but more complex preserved invariant needed.
15. Defining Data Structures
16. Main Idea We can show a property holds for all objects of a data type by: 1. Showing the property holds for all base objects. 2. Showing that all the ways of constructing new objects, preserve the property
17. Recall from Class 7:
18. Simplest Complex Data Type
19. Making Pairs
20. Making Pairs without Lists
21. Making Pairs without Lists
22. Making a Triple
23. Making a Quadruple
24. Multuples A Quintuple is a Pair where the second part is a Quadruple A Sextuple is a Pair where the second part is a Quintuple A Septuple is a pair where the second part is a Sextuple An Octuple is group of octupi A ? is a Pair where the second part is a …? 24
25. Lists 25 A List is a Pair where the second part is a List. But, then we can not create any lists!
26. Lists 26 A List is either: a Pair where the second part is a List or, the empty list null in LISP, None in Python, for mathematical lists
27. Defining List Definition 1. A List is an ordered sequence of objects. Definition 2. A List is (1) either a Pair where the second part is a List or (2) the empty list. Are these equivalent definitions?
28. Charge Thursday: Structural Induction on lists Problem Set 6 due Friday Send any questions you want for Thursday’s class