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• a boolean constant true or false;
• a symbol S;
• (list 'not X) where X is a bool-SchemeExp ;
• (list op X Y) where op is 'and , 'or , or 'implies where X and Y are {{bool-SchemeExp}}sSchemeExps;
• (list 'if X Y Z) where X, Y, and Z are {{bool-SchemeExp}}sSchemeExps.

The provided functions parse and unparse have the following signatures.

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The course staff is also providing a very simple test file for the eval function 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.

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Conversion to if form

A boolean expression (boolExp) can be converted to if form by repeatedlyapplying repeatedly applying the following rewrite rules in any order until no rule is applicable.

(make-Not  X)   	=>	(make-If  X  false true)
(make-And  X  Y)	=>	(make-If  X  Y  false)
(make-Or  X  Y)		=>	(make-If  X  true  Y)
(make-Implies  X  Y)	=>	(make-If  X  Y  true)

In these rules, X and Y denote arbitrary boolExps}. The conversion process always terminates (since each rewrite strictly reduces the number of logical connectives in the expressionexcluding {{make-If) 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 simple structural recursion. For each of the boolean operators And, Or, Not, Implies, and Implies if, reduce the component expressions first and then applying the matching reduction (except for if for which there is no top-level reduction).

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We suggest simply traversing the tree using the structural recursion template for type boolExp and converting all structures (other than if}}s) 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 the only constructor, other than variables and constants, for ifExp.

A boolExp is either:

• a boolean value true and false;
• a symbol s representing a boolean variable;
• (make-Not M) where M is a boolExp;
• (make-And M N) where M and N are boolExps;
• (make-Or M N) where M and N are boolExps;
• (make-Implies M N) where M and N are boolExps; or
• (make-If M N P) where M, N, and P are boolExps.

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

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(make-If  true  X  Y)	   =>	X
(make-If  false  X  Y)	   =>	Y
(make-If  X  true  false)  =>	X
(make-If  X  Y  Y) 	   =>	Y
(make-If  X  Y  Z)	   =>	(make-If  X  Y\[X <\- true\]  Z\[X <\- false\])

Wiki MarkupThe notation {{M\[X <- N\]}} means {{M}} with all occurrences of the symbol {{X}} replaced by the expression {{N}}. It is very costly to actually perform these subtitutions on =norm-if-form= data. To avoid this computational expense, we simply maintain a list of bindings which are pairs consisting of symbols (variable names) and boolean values {{{true}}, {{false}}. The following data definition definition formally defines the {{binding}} {true, false. The following data definition definition formally defines the binding type.

A binding is a pair (make-binding s v) where s is a symbol (a variable) and v is a boolean value (an element of { true, false }.

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