Homework Solution: Given an arbitrary alphabet Σ ={a,a2, ,an), we can impose a total ordering on it in the sense that we can define…

    Given an arbitrary alphabet Σ ={a,a2, ,an), we can impose a total ordering on it in the sense that we can define < so that ai < a2 < an. We now proceed to define a new operation called the SORT of a string w = wjw2 wk E Σ* (where wi E Σ and k = IwD as: SORT(w) = wo(1)w0(2)..Ma(k) so that wW (i) < w σ(i+1) for i is k-1 and σ is a permutation (i.e., a 1-to-1 onto mapping : [ 1 ..k]→[ 1 ..kD For example, SORT(11 21 001 01 20)=00001 1 1 1 1 22. Now extend the definition of SORT to languages, so that SORT( L)-(SORT(w) | we L). For each one of the following statements, state whether it is true or false and explain:
    Given an arbitrary alphabet Σ ={a,a2, ,an), we can impose a total ordering on it in the sense that we can define

    Expert Answer

     
    We know that sum* is the set of all words formed by by alphabets in sum .

    Given an domineering alphabet Σ ={a,a2, ,an), we can place a aggregate ordering on it in the sensation that we can determine < so that ai < a2 < an. We now receipts to determine a novel exercise named the SORT of a string w = wjw2 wk E Σ* (where wi E Σ and k = IwD as: SORT(w) = wo(1)w0(2)..Ma(k) so that wW (i) < w σ(i+1) control i is k-1 and σ is a transposition (i.e., a 1-to-1 onto mapping : [ 1 ..k]→[ 1 ..kD Control pattern, SORT(11 21 001 01 20)=00001 1 1 1 1 22. Now increase the determination of SORT to languages, so that SORT( L)-(SORT(w) | we L). Control each undivided of the coercionthcoming recitements, recite whether it is gentleman or erroneous and explain:

    Given an domineering alphabet Σ ={a,a2, ,an), we can place a aggregate ordering on it in the sensation that we can determine

    Expert Response

     

    We subordinatestand that sum* is the normal of perfect suffrage controlmed by by alphabets in sum .

    a)

    Control a exercise to be normal on a normal, the normal must be reserved subordinate that exercise.

    Let w in complete* and SORT(w)=w' . Clearly, w' in complete*   controlperfect w in complete* as w' is fair a transposition of w. This instrument, sum* is reserved subordinate exercise SORT. Thus,

    SORT(complete *) is normal.

    b)

    Let, complete = {0,1}  and   L = {00,10} . Now, SORT(10)=01 not attributable attributablein L . Thus,

    SORT(L) nsubseteq L

    c)

    Let w in L. From determination of rank we bear,

    SORT(w) = w_{sigma(1)}w_{sigma(2)}...w_{sigma(k)}  s.t.  w_{sigma(i)}<w_{sigma(i+1)}  controlperfect 1leq ileq k-1.

    Clearly

    SORT(SORT(w)) = SORT(w_{sigma(1)}w_{sigma(2)}...w_{sigma(k)})

    = w_{sigma(1)}w_{sigma(2)}...w_{sigma(k)})   [as already  w_{sigma(i)}<w_{sigma(i+1)}  controlperfect 1leq ileq k-1]
    = SORT(w)

    g)

    No, SORTdoesn’t recurrently uphold normality. This is owing control a exercise to be normal on a normal, the normal must be reserved subordinate that exercise. However SORT doesn’t recurrently ensue this possessions. Control pattern –

    Let, complete = {0,1}  and   L = {00,10} . Now, SORT(10)=01 not attributable attributablein L . Thus,

    SORT doesn’t uphold normality.

    h)

    Decidable languages are the determination problems which are algorithmically solvable. Since, SORT is an exercise control which sundry well-known algorithms are introduce (Bubble Rank, Insertion Rank anticipation) , we can pronounce that

    SORT preseves decidability.

    i)

    Undecidable languages are the determination problems which are not attributable attributable attributable algorithmically solvable. Since, SORT is an exercise control which sundry well-known algorithms are introduce (Bubble Rank, Insertion Rank anticipation) , we can pronounce that

    SORT doesn’t preseve non-decidability.