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

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

• Geometry

• Algebra– Linear, Basic, Algebraic Structures

• Theory Of Computation

• Discrete Mathematics

Discussion Topics:

• Finite State Machines– Input Strings

• Underlying Directed Graphs– Reducible States

• Matrix Applications– Multiplication

• Synchronization sequence

• Road Coloring

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- Alphabetδ- Transitionss- Start StateF- Final State

Deterministic: Each pair of state and input symbol there is one and only one transition to the next state

• Used to design software and hardware

• Can describe patterns:– Language patterns– Dance patterns– Musical patterns

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

-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

• 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)

• 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.

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?

• 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 bij, i referring to the ith row and j referring to the jth column;

• When bij=1, it signifies a blue transition from qi to qj

001

100

010

B

100

010

001

R

• Let w be the input word BRRB. We can represent this through matrix multiplication:

001

100

010

B

100

010

001

R

100

010

001

R

001

100

010

B

010

001

100

BRRB

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.

- 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

1

2

• [2,q0]=BB

• [2,q1]=BBR

• [2,q2]=BBRB

2

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.

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?

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 (23). How many of the 8 possible colorings have a synchronizing word?

For Example 2

• What is the synchronizing sequence for [M,q1]?

• What is the synchronizing sequence for [M,q5]?

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

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.

Avraham Trakhtman• Credited for solving the “Road Coloring

Problem”

• Russian Israeli Immigrant

• 63 year old former security

…Links…

• Article on Avraham Trakhtman

• http://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

Thank You for Listening

Nichole CavallaroAshley Meyers

& Britton Milner