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The course staff is providing functions parse and unparse in the file parse.rkt that rkt that convert boolean expressions in Racket notation to a simple inductively defined type called BoolExp and vice-versa. The coding of parse and unparse is not difficult, but it is tedious (like most parsing) so the course staff is providing this code rather than asking students to write it. The Racket primitive read: -> RacketExp is a procedure of no arguments that reads a Racket expression from the console. DrRacket pops up an input box to serve as the console when (read) is executed.

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• a boolean constant true and false ;
• a symbol S representing a boolean variable;
• (make-Not X) where X is a BoolExp;
• (make-And X Y) where X and Y are BoolExps BoolExps;
• (make-Or X Y) where X and Y are BoolExps BoolExps;
• (make-Implies X Y) where X and Y are BoolExps BoolExps; or
• (make-If X Y Z) where X , Y , and Z are BoolExps BoolExps.

A bool-RacketExp BoolRacketExp is either:

• a boolean constant true or false;
• a symbol S;
• (list 'not X) where X is a bool-RacketExp BoolRacketExp ;
• (list op X Y) where op is 'and , 'or , or 'implies where X and Y are bool-RacketExps BoolRacketExps;
• (list 'if X Y Z) where X, Y, and Z are bool-RacketExps BoolRacketExps.

The provided functions parse and unparse have the following signatures.

parse: bool-RacketExpBoolRacketExp -> BoolExp
unparse: BoolExp -> bool-RacketExpBoolRacketExp

The course staff is also providing a very simple test file\ for the eval and reduce functions and a file\ containing a sequence of raw input formulas (to be parsed by parse function in parse.ss\). A good solution to this problem will include much more comprehensive test data for all functions, including some much larger test cases for reduce. The normalize function is difficult to test on large data because the printed output for some important normalized trees (represented as DAGs (Directed Acyclic Graphs) in memory) is so large.

Given a parsed input of type BoolExp , the simplification process consists of following four phases:

• Conversion to if If form implemented by the function convert-to-if convertToIf.
• Normalization implemented by the function normalize.
• Symbolic evaluation implemented by the function eval.
• Conversion back to conventional boolean form implemented by the function convert-to-bool convertToBool.

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

Conversion to if form

A boolean expression (BoolExp) can be converted to if form ) 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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In these rules, X and Y denote arbitrary BoolExpsBoolExps}. The conversion process always terminates (since each rewrite strictly reduces the number of logical connectives excluding {{make make-If) and  and yields a unique answer independent of the order in which the rewrites are performed. This property is called the Church-Rosser property, after the logicians (Alonzo Church and Barkley Rosser) who invented the concept.

Since the reduction rules for this phase are Church-Rosser, you can write the function convert-to-if using convertToIf using simple structural recursion. For each of the boolean operators And, Or, Not, Implies, and ifIf, reduce the component expressions first and then applying the matching reduction (except for if If for which there is no top-level reduction).

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(check-expect  (convert-to-ifConverToIf (make-Or (make-And 'x 'y) 'z))    (make-If (make-If 'x 'y false) true 'z))
(check-expect  (convert-to-ifConverToIf (make-Implies 'x (make-Not 'y))   (make-If 'x (make-If 'y false  true) true))

We suggest simply traversing the tree using the structural recursion template for type BoolExp and converting all structures (other than if If) to the corresponding if structures.

Write an inductive data definition and template for boolean formulas in if form, naming this type ifExp. (Note: make-If is  is the only constructor, other than variables and constants, for ifExp.

The provided function parse: input -> boolExpBoolExp takes a Racket expression and returns the corresponding boolExp 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 norm-ifExp .

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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

• convert-to-if ConverToIf (10%)
• normalize (20%)
• eval (20%)
• convert-to-bool (10%)
• reduce (40%)