Homework 1 (May 31) Material covered: Episodes 1 and 2 1. When do we say that a function f is computable? (definition) 2. When do way say that a set S is decidable? 3. When do we say that a set S is recursively enumerable (recognizable)? 4. When do we say that a set A is mapping reducible to a set B? 5. When do we say that a set A is Turing reducible to a set B? 6. What is the Kolmogorov complexity of a given number? 7. Let  be some alphabet, S a subset of *, and S’ the complement of S, i.e., S’={x | x* and xS}. Give a brief justification for the following statement: If S is decidable, then S’ is also decidable. [Hint: consider an algorithm (TM) that decides S, and think about how to turn it into an algorithm that decides S’]. 8. In each case below, find (precisely define) a particular function f which is a mapping reduction of A to B. A={w | w is an even-length bit string}, B={w | w is an odd-length bit string} A={0,1}* (i.e. A contains all bit strings),  B={00,11} 9. Show that the acceptance problem is mapping reducible to the halting problem (Slide 2.19) 10. Show that the Kolmogorov complexity problem is Turing reducible to the halting problem (Slide 2.24)