MAT2ALC Algebra- Linear Codes And Automata

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MAT2ALC Algebra, Linear Codes And Automata

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MAT2ALC Algebra, Linear Codes And Automata

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Course Code: MAT2ALC
University: La Trobe University

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Country: Australia

Question:

Consider the permutations f; g 2 S7 given by the table to the right.

1

2

3

4

5

6

7

1

7

6

4

2

3

5

6

2

5

3

4

7

1

 
Write each of f and g in cycle notation, as a cycle or a product of disjoint cycles.

Answer:
The functions  and  map values of x as illustrated in the table.
For example,  maps 2 and 3 into 7 and 6 respectively.
A cycle is complete when an element is mapped into itself or into another element within the cycle. 
Cycle notations 
Calculate . Write your answer in cycle notation, as a cycle or a product of disjoint cycles. 
The order of a permutation f is the smallest positive integer n such that is the identity.
In other words, the order of a permutation is equal to the lowest common multiple of the lengths of the cycles occurring in its decomposition into disjoint cycles.  
(d) Express each of f and g as products of transpositions, and hence decide whether they are even or odd permutations.    
(e) Without calculating , decide whether it is even or odd. Justify your answer. 
 is an odd permutation because sum of an odd and even digit is always an odd number.
2.Consider the quadrilateral shown at right. All sides are of equal length (so it is a rhombus), but the distance between vertices 1 and 3 is greater than the distance between vertices 2 and 4 (so it is not a square). As with the equilateral triangle and the square, every symmetry of the above quadrilateral may be represented by the permutation of the vertex labels that it induces. Thus, we regard each symmetry of the quadrilateral as an element of S4.    
Cyclical notations

Write out the table for the group . Is it Abelian? 

Id

(13)

(24)

Id

Id

(13)

(24)

(13)

(13)

Id

(1342)

(24)

(24)

(1342)

Id

A permutation is considered as an Abelian if the composite rule is commutative. 
The group is Abelian because for(a) Show that  is associative.
(b) Show that * is commutative.
(c) Find the identity element for *. [Hint: Let e denote the identity. Use the equation x * e = x to determine the value of e. Then show carefully that e is indeed an identity element for *.]
(d) Let  . Find the inverse, , of x with respect to the identity element found in (c). [  
Operation * is associative under
Operation * is commutative if
Therefore the operation is commutative.
Identity element e such that    
To prove whether 6-x is the inverse of x, we operate it with x to find out if it yields back the identity element. 
Therefore, inverse of operation * is 6-x.
Let be the DFA over the alphabet with states , initial state , accepting states , and transition function given by the table at right.

x

a

b

A

B

B

C

x

a

b

A

B

A

C

the set of states
Input alphabets
Accepting states {A, C}

Show how M processes the following words, indicating whether they are accepted or rejected.  

       is accepted since acceptance states are (A,C)   
Hence  accepted because the final state C is one of the acceptance states  is rejected because does not fall under acceptance states.
(c) Let  denote the extended transition function of M. Calculate each of the following. (Final answers only.)      

Give a regular expression (over) for the language 

We need an expression, w, (in terms of a and b) that will map C into itself
That is;
C is mapped into by  
Therefore, the regular expression will be 

Give a regular expression (over {a, b}) for the language of M Using the method of Practice Class 4B, Question 8, construct an NFA recognizing the language), explaining each step of the construction. You do not need to name the states. 

Step 1: State all the states
N is the accepting state
M is the initial state
” is the input alphabet
  is the nondeterministic value
Step 2: state the language 
Step 3: NFA

1

0

a(M0)

N

M

M

a(M1)

M

N

M

a(M2)

M

M

N

 
Let N denote NFA over {0; 1} shown at right.
(a) What is the language of N? Give a regular expression. 
(b) Use the Power Set Construction 16.10 to draw the state diagram of a DFA equivalent to N
Proof Question: Let 
For all  Prove that  is a permutation of
Proof
From
Implying the functional transformation   
Since ‘a’ is a member of universal set (U), then the product  would fall under Z 
References
Bóna, M. (2016). Combinatorics of permutations. Chapman and Hall/CRC.
Hromkovi?, J., & Schnitger, G. (2003, June). Nondeterminism versus determinism for two-way finite automata: generalizations of Sipser’s separation. In International Colloquium on Automata, Languages, and Programming (pp. 439-451). Springer, Berlin, Heidelberg.

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