Problem 4, 20 points The Stata Center's delicate balance depends on two buckets of water hidden in a secret room. The big bucket has a volume of 25 gallons, and the little bucket has a volume of 10 gallons. If at any time a bucket contains exactly 13 gallons, the Stata Center will collapse. There is an interactive display where tourists can remotely fill and empty the buckets according to certain rules. We represent the buckets asa state machine. The state of the machine is a pair (b,/), where b is the volume of water in big bucket, and I is the volume of water in tle bucket. (a) We informally describe some of the legal operations tourists can perform be- low. Represent each of the following operations as a transition of the state machine The first is done for you as an example. 1. Fill the big bucket. 2. Empty the little bucket. 3. Pour the big bucket into the little bucket. You should have two cases defined in terms of the state (b,l): if all the water from the big bucket fits in the little bucket, then pour all the water. If it doesn't, pour until the little jar is full, leaving some water remaining in the big jar. (b) Use the Invariant Principle to show that, starting with empty buckets, the Stata Center will never collapse. That is, the state (13. x) in unreachable. (In verifying your claim that the invariant is preserved, you may restrict to the representative transitions of part (a).) Problem 4, 20 points The Stata Center's delicate balance depends on two buckets of water hidden in a secret room. The big bucket has a volume of 25 gallons, and the little bucket has a volume of 10 gallons. If at any time a bucket contains exactly 13 gallons, the Stata Center will collapse. There is an interactive display where tourists can remotely fill and empty the buckets according to certain rules. We represent the buckets asa state machine. The state of the machine is a pair (b,/), where b is the volume of water in big bucket, and I is the volume of water in tle bucket. (a) We informally describe some of the legal operations tourists can perform be- low. Represent each of the following operations as a transition of the state machine The first is done for you as an example. 1. Fill the big bucket. 2. Empty the little bucket. 3. Pour the big bucket into the little bucket. You should have two cases defined in terms of the state (b,l): if all the water from the big bucket fits in the little bucket, then pour all the water. If it doesn't, pour until the little jar is full, leaving some water remaining in the big jar. (b) Use the Invariant Principle to show that, starting with empty buckets, the Stata Center will never collapse. That is, the state (13. x) in unreachable. (In verifying your claim that the invariant is preserved, you may restrict to the representative transitions of part (a).)
