advertisement

1 Strictly Composite Numbers - Alloy A number c is said to be strictly composite if 1 < r < c and gcd(c, r) = 1 implies r is prime. Create an Alloy model that capture the concepts of prime, composite and strictly composite. Use this model to demonstrate that 30 is stringly composite, while 45 is not. Use meta–programming to test all numbers less than a command line parameter and tabulate the results. 2 Strictly Composite Numbers - Isabelle Use Isabelle to implement prime, composite, and strictly composite. Prove that there are only finitely many strictly composite numbers. Find the largest strictly composite number and prove that it has that property. 3 Hamiltonian Path Create an Alloy model of a graph that captures the concepts of node, edge, path, and directness. Devise a series of Alloy assertions to test that your model is internally consistent. For example, if you allow multipe edges connecting the same (from,to) pair, how does the behavior of the model difer from the case where at most one edge can exist between any (from,to) pair? Formulate the concepts of is connected and has a Hamiltonian path. Encode the Königsburg bridge problem (shown below) using Alloy initializers and demonstrate that it is connected, but does not have any Hamiltonian path. Create the smallest possible modification to the Königsburg bridge problem such that the modifier version is connected and has a Hamiltonian path. Figure 1: The Königsburg bridge problem 1 4 Dijkstra Ordering ment in Isabelle. Extract dijkstra.als from the Alloy jar file. Reimple- 5 Process Priorities - Alloy Implement an Alloy model that has a numerical priority value for each process (for example, starting from dijkstra.als). Create a test for priority inversion, and create two examples: one that displays priority inversion, and one that does not. Choose any of the standard tactics for preventing priority inversion, and demonstrate that the bad behavior of your first example has been corrected. 6 Process Priorities - Isabelle Implement project 5 using Isabelle. 7 Probabilistic Programming in Alloy Implement an Alloy model of a directed graph that captures the concepts of node, edge, path and predecessors. Consider the graph shown below. Figure 2: A State Machine in Alloy Observe that the ABD nodes occurs in 0.6 of all paths terminating at D, 2 while the ACD nodes occurs in 0.4 of all such paths. Write Alloy initializers to represent this graph. Given a set of probabilities pi with N X = 1.0 i=1 create a meta–program that accepts the pi as command line parameters and creates a set of Alloy initializes with that probability distribution. Make a simplifying assumption about the pi so that the resulting Alloy initializer list is reasonably compact. 3