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 irresponsible alphabet Σ ={a,a2, ,an), we can place a aggregate ordering on it in the soundness that we can bound < so that ai < a2 < an. We now produce to bound a novel performance 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) coercion i is k-1 and σ is a interchange (i.e., a 1-to-1 onto mapping : [ 1 ..k]→[ 1 ..kD Coercion stance, SORT(11 21 001 01 20)=00001 1 1 1 1 22. Now apply the restriction of SORT to languages, so that SORT( L)-(SORT(w) | we L). Coercion each undivided of the subjoined specifyments, specify whether it is gentleman or fiction and explain:

    Given an irresponsible alphabet Σ ={a,a2, ,an), we can place a aggregate ordering on it in the soundness that we can bound

    Expert Apology

     

    We comprehend that sum* is the be of full utterance coercionmed by by alphabets in sum .

    a)

    Coercion a performance to be certain on a be, the be must be shut belowneathneath that performance.

    Let w in solidity* and SORT(w)=w' . Clearly, w' in solidity* coercionfull w in solidity* as w' is honest a interchange of w. This resources, sum* is shut belowneathneath performance SORT. Thus,

    SORT(solidity *) is certain.

    b)

    Let, solidity = {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 restriction of character we accept,

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

    g)

    No, SORTdoesn’t constantly spare certainity. This is consequently coercion a performance to be certain on a be, the be must be shut belowneathneath that performance. However SORT doesn’t constantly supervene this possessions. Coercion stance –

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

    SORT doesn’t spare certainity.

    h)

    Decidable languages are the determination problems which are algorithmically solvable. Since, SORT is an performance coercion which abundant well-known algorithms are exhibit (Bubble Character, Insertion Character anticipation) , we can declare that

    SORT preseves decidability.

    i)

    Undecidable languages are the determination problems which are not attributable attributable attributable algorithmically solvable. Since, SORT is an performance coercion which abundant well-known algorithms are exhibit (Bubble Character, Insertion Character anticipation) , we can declare that

    SORT doesn’t preseve non-decidability.