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Class 13: State Machines cs2102: Discrete Mathematics | F17 uvacs2102.github.io David Evans | University of Virginia Return Exam 1 Today No problem set due this week! PS6 will be posted soon and due Friday, Oct 20.

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Plan State Machines Exam 1 Read this week: MCS Chapter 6

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State Machine

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Example = , = → , → ) =

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M =

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State Machine M = (, : × , ) ∈ ) What does it mean if is total?

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State Machine M = (, : × , ) ∈ ) What does it mean if is not a function?

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= = ) = Counter = (, : × , ) ∈ )

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Is your computer a state machine?

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http://www.wolframalpha.com/input/?i=2%5E(2%5E41)

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Modeling as State Machine

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Modeling as State Machine How accurate is this model? How useful is this model?

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State Machine Executions

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State Machine Executions The execution of a state machine, = (, ⊆ ×, ) ∈ ) is a (possibly infinite) sequence of states, () , 3 , … , 5 ) that:

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Two Steps Forward, One Step Back = ℕ =

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Two Steps Forward, One Step Back = =

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Reachable States A state is reachable if it appears in some execution. The execution of a state machine, = (, ⊆ ×, ) ∈ ) is a (possibly infinite) sequence of states, () , 3 , … , 5 ) that: 1. ) = ) (it begins with the start state) 2. ∀ ∈ 0, 1, … , − 1 . > → >?3 ∈ (if and are consecutive states in the sequence, there is an edge → ∈ .

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= , ∈ ℤ, ∈ {, }} = , → + 1, ∈ ℤ} ∪ , → − 2, ∈ ℤ} ) = (0, ) What states are reachable?

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= , ∈ ℤ, ∈ {, }} = , → + 1, ∈ ℤ} ∪ , → − 2, ∈ ℤ} ) = (0, )

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Exam 1

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0 20 40 60 80 <80 [80,85) [85,90) [90,95) [95,100) >=100 Exam 1 Histogram

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Results by Question 0 (Name) 1 (Inference Rules) 2 (Satisfiability) 3 (Logical Equivalence) 4 (Well Ordering) 5 (Relations) 6 (Proofs) 7 (Induction) Average 30.0 9.374 9.887 9.947 8.632 8.726 8.252 8.792 Median 30 10 10 10 8 10 9 10

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Proofs • To prove by induction, need to state a predicate with an induction variable • Need to write proofs to convince: showing the structure of your argument is more important than showing a bunch of algebra ∷= whatever is here must use n (and that is what you are doing induction on)

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Returning Exam 1 Group 1 Group 2 Group 3 Group 4 aa8dp (Akanksha) ea6dc (Elmo) kai4vb (Kelsey) qed4wg (Quinn) aap7sc (Alex) eae4bf (Eliza) kc4au (Kunal) rac4ad (Raina) … … … … dz7as (Daniel) jyy3gx (Josh) pw2hm (Peter) zpg9js (Zach) Front of Classroom

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Returning Exam 1 Group 1 Group 2 Group 3 Group 4 aa8dp (Akanksha) ea6dc (Elmo) kai4vb (Kelsey) qed4wg (Quinn) aap7sc (Alex) eae4bf (Eliza) kc4au (Kunal) rac4ad (Raina) … … … … dz7as (Daniel) jyy3gx (Josh) pw2hm (Peter) zpg9js (Zach) How can we efficiently sort ~80 people by UVA id? Front of Classroom

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Returning Exam 1 Group 1 Group 2 Group 3 Group 4 aa8dp (Akanksha) ea6dc (Elmo) kai4vb (Kelsey) qed4wg (Quinn) aap7sc (Alex) eae4bf (Eliza) kc4au (Kunal) rac4ad (Raina) … … … … dz7as (Daniel) jyy3gx (Josh) pw2hm (Peter) zpg9js (Zach) Front of Classroom

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Charge Read MCS Chapter 6 PS6 will be posted soon, due Oct 20