Can we “paint” streets or roads to provide “easy” directions to a destination?
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Transcript of Can we “paint” streets or roads to provide “easy” directions to a destination?
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Can we “paint” streets or roads to provide “easy” directions to a destination?
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Can we “paint” streets or roads to provide “easy” directions to a destination?
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• Geometry
• Algebra– Linear, Basic, Algebraic Structures
• Theory Of Computation
• Discrete Mathematics
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Discussion Topics:
• Finite State Machines– Input Strings
• Underlying Directed Graphs– Reducible States
• Matrix Applications– Multiplication
• Synchronization sequence
• Road Coloring
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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
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• Used to design software and hardware
• Can describe patterns:– Language patterns– Dance patterns– Musical patterns
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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
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-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
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• 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)
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• 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.
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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?
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• 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
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001
100
010
B
100
010
001
R
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• 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.
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- 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
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1
2
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• [2,q0]=BB
• [2,q1]=BBR
• [2,q2]=BBRB
2
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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.
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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?
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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?
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For Example 2
• What is the synchronizing sequence for [M,q1]?
• What is the synchronizing sequence for [M,q5]?
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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
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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.
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Avraham Trakhtman• Credited for solving the “Road Coloring
Problem”
• Russian Israeli Immigrant
• 63 year old former security
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…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
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Thank You for Listening
Nichole CavallaroAshley Meyers
& Britton Milner