Recall that the Kolmogorov complexity K(s) of a string s E {0,1}" (with respect to some fixed programming language) is defined as the bit length of a shortest program that, when run on the empty string e as input, outputs s. (a) Prove that, for all n > 1, Is 0,1}" : K(s) 1, there exists a binary string of length n whose Kolmogorov complexity is at least n. (c) A company has released a new compression algorithm COMPRESS that will compress cu- ery file to at most half its original size. Determine, with justification, if an accompanying lossless decompression algorithm can exist for this compression algorithm. (A decompression algorithm is lossless if it always outputs the original file, given the compression algorithm's output when run on the original file.) Recall that the Kolmogorov complexity K(s) of a string s E {0,1}" (with respect to some fixed programming language) is defined as the bit length of a shortest program that, when run on the empty string e as input, outputs s. (a) Prove that, for all n > 1, Is 0,1}" : K(s) 1, there exists a binary string of length n whose Kolmogorov complexity is at least n. (c) A company has released a new compression algorithm COMPRESS that will compress cu- ery file to at most half its original size. Determine, with justification, if an accompanying lossless decompression algorithm can exist for this compression algorithm. (A decompression algorithm is lossless if it always outputs the original file, given the compression algorithm's output when run on the original file.)<><>