Homework 5: Symbolic Evaluation of Boolean Expressions

Due: Wednesday, Oct 14, 2020 at 11:59PM

200 pts.

Overview

Write a Racket function reduce that reduces boolean expressions (represented in Racket notation) to simplified form. For the purposes of this assignment, boolean expressions are Racket expressions constructed from:

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A description of each of these phases follows. The reduce function has type BoolRacketExp -> BoolRacketExp.

Conversion to if form

A boolean expression (BoolExp) can be converted to if form (a boolean expression where the only constructor is make-If) by repeatedly applying the following rewrite rules in any order until no rule is applicable.

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The provided function parse: input -> BoolExp takes a Racket expression and returns the corresponding BoolExp.

Normalization

An ifExp is normalized iff every sub-expression in test position is either a variable (symbol) or a constant (true or false). We call this type NormIfExp .

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Before you start writing normalize, write the template corresponding to the inductive data definition of NormIfExp.

Symbolic Evaluation

The symbolic evaluation process, implemented by the function eval: NormIfExp environment -> NormIfExp, reduces a NormIfExp to simple form. In particular, it reduces all tautologies (expressions that are always true) to true and all contradictions (expressions that are always false) to false.

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We recommend applying the rules in the order shown from the top down until no more reductions are possible (using the constraint on the final rule). Note that the last rule should only be applied once to a given subexpression.

Conversion to Boolean Form

The final phase converts an expression in (not necessarily reduced or normalized) If form to an equivalent expression constructed from variables and { true, false, And, Or, Not, Implies, If. This process eliminates every expression of the form

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where X , Y , and Z are arbitrary If forms. This set of rules is Church-Rosser, so the rules can safely be applied using simple structural recursion.

Points Dsitribution

• convertToIf (10%)
• normalize (20%)
• eval (20%)
• convertToBool (10%)
• reduce (40%)