4. for each of the following statements, answer true or false and prove that your answer is correct. (a) (10 points) if a language l ⚆ {0, 1} ∗ is decidable, then every language l 0 ⚆ l is decidable. (b) (10 points) if f : {0, 1} ∗ → z is an upper semi-computable function, then the function g : {0, 1} ∗ → z given by g(x) = 2 − f(x) is lower semi-computable. (c) (10 points) the language {x ∈ {0, 1} ∗ : c(x) > |x|/3} is co-c. e. (d) (10 points) every language l ⚆ {0, 1} ∗ is either c. e. or co-c. e. or both.

Read this excerpt from helen keller’s autobiography, the story of my life. have you ever been at sea in a dense fog, when it seemed as if a tangible white darkness shut you in, and the great ship, tense and anxious, groped her way toward the shore with plummet and sounding-line, and you waited with beating heart for something to happen? i was like that ship before my education began, only i was without compass or sounding-line, and had no way of knowing how near the harbour was. "light! give me light! " was the wordless cry of my soul, and the light of love shone on me in that very hour. . the morning after my teacher came she led me into her room and gave me a doll. the little blind children at the perkins institution had sent it and laura bridgman had dressed it; but i did not know this until afterward. when i had played with it a little while, miss sullivan slowly spelled into my hand the word "d-o-l-l." i was at once interested in this finger play and tried to imitate it. when i finally succeeded in making the letters correctly i was flushed with childish pleasure and pride. running downstairs to my mother i held up my hand and made the letters for doll. i did not know that i was spelling a word or even that words existed; i was simply making my fingers go in monkey-like imitation. in the days that followed i learned to spell in this uncomprehending way a great many words, among them pin, hat, cup and a few verbs like sit, stand and walk. based on this excerpt, which words best describe helen keller?