CS E6831 04f HWK #4 Due 10/6/04 PROBLEMS: NOTE: As a matter of good form, and to facilitate checking of results, when specifying an unordered set, order the members lexicographically. For example, the set of states {2, 5, 1, 3} should be written as {1, 2, 3, 5}. Similarly, the blocks of a partition should be ordered by the first member of each. So we would write a partition as {13, 257, 4, 8} NOT, for example, as {4, 257, 8, 13} or {31, 275, 8, 4} 1. If p is a partition ) for machine S, q=m(p) is defined as the smallest partition on S, such that [p,q] is a pp, and r=M(p) is defined as the largest partition such that [r,p] is a pp (partition pair). For the machine below, find: (a) m(12, 3, 45, 6) (b) m(135, 26, 4) (c) m(1, 2, 3, 4, 5, 6) (d) m(123456) (e) M(1456, 23) (f) M(12, 3, 45, 6) (g) M(1, 2, 3, 4, 5, 6) (h) M(123456) X 0 1 ----- 1 6 4 2 6 3 3 5 2 4 1 5 5 4 2 6 3 1 ----- 2. Suppose that [(123, 45, 678), (13, 2, 45, 6, 78)] is a pp (partition pair) for machine M. Specify 4 closed partitions for M. 3. (a) If p1 is a closed partition of machine M, , and if p2>p1, does it necessarily follow that p2 is also a closed partition of machine M? Justify your answer. (b) If p1 is an input consistent partition of machine M, and if p2>p1, does it necessarily follow that p2 is also an input consistent partition of machine M? Justify your answer. 4. Consider the machine below (assuming the input state 11 never occurs). AB 00 01 10 ---------------- 1 2,0 3,0 1,0 2 4,1 3,1 1,1 3 4,0 3,0 1,1 4 2,1 3,1 1,0 --------------- Sketch the lattice of closed partitions. 5. (a) For the machine of problem 4, utilize the work you did to find a good state assignment. (b) draw a block diagram showing a decomposition of this machine, based on the state assignment. Also specify flow tables for the component machines. 6. Consider the problem 4 machine again, from the point of view of partition pairs. Find a 2-block partition p such that [I, p] is a partition pair. What, in general, does [I, p] being a partition pair say about the variables that Y's associated with p are dependent on? For this particular case, write a logic expression for the Y associated with p, assuming that a suitable closed partition among those you found above is used to complete the state assignment. 7. Let pab be the partition that equates only states a and b. For the machine below, assuming m(q) is defined as in problem 1 above, find m(p12), m(p25), and m(p34). Use this information to generate 2-block partitions e and f such [e, f] is a pp. Associate y1 with e and y2 with f, and then write a logic expression for Y2. A B C --------- 1 1 5 2 2 3 5 3 3 3 1 3 4 2 1 3 5 1 4 2 ----------