2. Can we “paint” streets or roads to provide “easy” directions to a destination?

4. Geometry Algebra –Linear, Basic, Algebraic Structures Theory Of Computation Discrete Mathematics

5. Discussion Topics: Finite State Machines –Input Strings Underlying Directed Graphs –Reducible States Matrix Applications –Multiplication Synchronization sequence Road Coloring

6. Is a model of behavior composed of a finite number of states, transitions between those states, and actions The formal definition of an FSM: M=[Q,Σ,δ,s,F] Q- Set of states Σ- Alphabet δ- Transitions s- Start State F- Final State Deterministic: Each pair of state and input symbol there is one and only one transition to the next state

7. Used to design software and hardware Can describe patterns: –Language patterns –Dance patterns –Musical patterns

8. The Input String is comprised of symbols in the alphabet of that language. A valid input string would bring you to the final state in that machine. For example: abbabbbabb would be string accepted by this machine

9. -No declared start state or final state -For this example: There are two elements in this alphabet (R,B) therefore there are two edges leaving each state An Input string : BRRBRRBRBRBB would bring you back to the state from which you started

10. If someone is currently located at state p and follows the instructions w then that person will move to state q. We will use the notation pw to indicate the state the machine, M, will be in if it is currently in state p and then input w is processed. Consequently, pw=q indicates that if M is in state p then the input string w will move M to state q. For example, if we start in state q1 (p) and use the string RRBRB(w) as directions we will end in state q0(q)

11. A pair of states [p,q] is reducible if there is an input string w such that pw = qw. In other words, if we have someone currently at state p and someone else at state q and they both follow the same instructions w then they will meet at a common state.

12. There are no reducible states for this machine. This machine has reducible states for each vertex. For example: q0 and q2 reduce to one vertex (q1) for the input string BR Can you determine the input string needed to reduce q0 and q1 to q2?

13. Transitions for underlying directed graphs can be represented in the form of a matrix If given a graph with unlabeled transitions we can determine them if given the transition matrices The entry b ij, i referring to the i th row and j referring to the j th column; When b ij =1, it signifies a blue transition from qi to qj

15. Let w be the input word BRRB. We can represent this through matrix multiplication: If the product matrix has two 1’s in the same column then the states associated with the two rows in which these 1’s appear are a synchronizing pair.

16. -A sequence of characters from the alphabet that when processed will move to the specified state, regardless of which state the sequence was originated from. -In other words, if you start with a person at each state and they all follow the same instructions (the synchronizing sequence) they will arrive at the specified state in the same number of steps -This implies that every pair of states must be reducible

17. 1 2

18. [2,q0]=BB [2,q1]=BBR [2,q2]=BBRB 2

19. From the previous example we said that one of the synchronizing sequences was BB, for this example: B= 1 0 0 BB= 1 0 0 0 0 1 1 0 0 1 0 0 1 0 0 The column 1’s in BB indicates that regardless of whether someone starts at q0, q1, or q2 the instructions of BB will lead to state q0.

20. For Example 1 Can you find a synchronizing word of length 4 for [M, q3]? Can you find a synchronizing word of length 5 for [M, q2]? Can you write the transitional matrix for this underlying directed graph?

21. If we think of an edge marked B as a blue road and an edge marked R as a red road The number of possible road-colorings for the underlying directed graph would be the number of colors raised to the number of vertices (2 3 ). How many of the 8 possible colorings have a synchronizing word?

22. For Example 2 What is the synchronizing sequence for [M,q1]? What is the synchronizing sequence for [M,q5]?

23. We define road-colorable graphs as being 1.Strongly Connected -If p and q are any two vertices then there is a path from p to q. 2.A periodic digraphs -The largest integer that is divisible the length of each cycle is 1 (relatively prime). 3.Uniform out-degree -All vertices have the same out-degree

24. A periodic This graph has cycles of length 4 and 6. Every cycle has an even length. That is the length of each cycle, is divisible by 2. This graph has cycles of length 3, 4 and 5. The largest integer that divisible by each cycle length is 1. Each length is relative prime. This is an a periodic graph.

26. Avraham Trakhtman Credited for solving the “Road Coloring Problem” Russian Israeli Immigrant 63 year old former security

28. …Links… Article on Avraham Trakhtman http://www.iht.com/bin/printfriendly.php?id =11292773http://www.iht.com/bin/printfriendly.php?id =11292773 The solution/proof http://arxiv.org/pdf/0709.0099v4 Ideas about these concepts http://www.math.siu.edu/budzban/pub/BD- AMS-Notices-05.pdf http://www.math.siu.edu/budzban/pub/BD- AMS-Notices-05.pdf

29. Thank You for Listening Nichole Cavallaro Ashley Meyers & Britton Milner