Fixed-point computations

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Live variable analysis was applied to a running example, obtaining this system of equations as a result: In the case of the available expression analysis we obtained another system: The equations of such systems have always the same shape: X = Expr, where Expr is a set expression and X is a variable denoting a set. Assume that the syntax of set expressions is given by the following grammar: In this grammar, the x1,..., xn are elements that could appear in the solutions to the equations. For example, in the first case shown above (live variables), these elements are program variables; in the second case, these elements are arithmetic expressions. 1. In your favourite language, define the necessary data structures to represent set expressions, equations, and systems of equations. 2. Define a function or method that performs a single iteration of the fixed point computation. That is, given an assignment from set variables to actual sets, it returns another assignment with the set variables updated. For example, in Java: 3. Define a function or method that computes the solution of a given system of equations from an initial assignment to the set variables in the system. Use the method implemented in the previous exercise. For example, in Java: 4. Define a function or method that solves each of the systems of equations shown at the beginning of this assignment and outputs the corresponding solutions. Note: The Java and Haskell declarations shown above are just for explanatory purposes. Feel free to use any other programming language to implement the given methods.

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