Lecture 17 - State Machines, Combinatorics, Game Strategies

Question 1

What does a finite state machine have? (4)

Answer 1

• a set of states
• a set of transitions between the states
• an initial state
• zero or more final (“accepting”) states

Question 2

What are side effects of a state machine?

Answer 2

The actions taken on entering/leaving states

Question 3

Can a state machine be between states?

Answer 3

No. State transitions are considered “atomic”

Question 4

1. What are the externally visible parts of a state machine? (3)
2. What’s not externally visible?

Answer 4

1.

• – the events the state machine understands
• – the guard conditions the machine checks
• – the actions that the machine performss

2. Everything else, including the machine’s current state

Question 6

What’s a finite state transducer?

Answer 6

If the action on entering/leaving a state is to output a symbol, we call the FSM a finite state transducer. – it converts (transduces) the sequence of input symbols into a sequence of output symbols

Question 7

What are the types of FSTs? (2)

Answer 7

• Mealy machines
• Moore machines

Question 8

What’s a mealy machine?

Answer 8

(output symbol is associated with transition)

Question 9

What’s a Moore machine?

Answer 9

(output symbol is associated with state)

Question 10

Compare Moore and Mealy machines (3)

Answer 10

• Moore machines in general have many more states than the equivalent Mealy machine –
  
  • because output depends only on current state, not state+input
  • so we need to add states whenever the Mealy machine has different outputs for different transitions out of a state

Question 11

What’s an example of a time when state machines are beneficial?

Answer 11

• Retrieving content based on a user’s mouse click. During retrieval, they click something again.
• The logic might get complicated in an ordinary flow chart, but a state machine wouldn’t accept this 2nd click because it’s in the fetching state, so the 2nd click wouldn’t cause problems.

Question 12

What are the two common approaches for implementing state machines?

Answer 12

• switch statement
• transition table

Question 13

Why does the factorial function keep appearing in things we’ve been doing this semester?

Answer 13

It’s the number of permutations of N items

Question 14

How many permutations are there of k elements from a set of n elements total

Answer 14

n! / (n-k)!

Question 15

Define combination

Answer 15

A combination is a subset of k elements from a set of n elements

Question 16

What is n choose k?

Answer 16

a subset of k elements from a set of n elements

call the size of the set of all such subsets “n choose k”

Question 17

What is this?

Answer 17

n choose k

Question 18

How does the set of k-permutations compare to choose(n,k)?

Answer 18

Question 19

What’s the equation for n choose k?

Answer 19

Question 20

What is the power set?

Answer 20

the set of all subsets of a set, including both the set itself and the empty set

Question 21

What’s the size of the power set?

Answer 21

2^|s| sets

Question 22

What’s the time complexity for generating power sets?

Answer 22

O(2ⁿ)

Question 23

What’s the strategy in take N marble game game strategy?…

• For 4
• In general

Answer 23

Take just enough to leave a multiple of 4

• If players can take 1…N marbles, then whoever can leave a multiple of N+1 marbles for their opponent can force a win
  
  • whichever number K in 1…N your opponent chooses as their move, you choose L = (N+1)-K as your next move

Question 24

Based on the starting circumstances of the take N marbles game, which player can always win?

Answer 24

• If the game starts with a multiple of N+1 marbles, player B can always win
• If the game starts with something other than a multiple of N+1, player A can always win