Homework 6: Symbolic Evaluation of Boolean Expressions

Due: Friday Monday, Feburary 26March 1, 2010

Extra Credit (100 pts.)

Overview

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

...

The course staff is providing function functions parse and unparse in the file foo parse.ss that convert boolean expressions in Scheme 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 Scheme primitive read: -> SchemeExp is a procedure of no arguments that reads a Scheme expression from the console. DrScheme pops up an input box to serve as the console when (read) is executed.

These parsing functions rely on the following Scheme data definitions:. Given

(define-struct Not (arg))
(define-struct And (left right))
(define-struct Or (left right))
(define-struct Implies (left right))
(define-struct If (test conseq alt))

...

• 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 {{boolExp}}s boolExps;
• (make-Or X Y) where X and Y are {{boolExp}}s boolExps;
• (make-Implies X Y) where {{X and Y are {{boolExp}}s boolExps; or
• (make-If X Y Z) where X , Y , and Z are {{boolExp}}s boolExps.

A bool-SchemeExp is either:

• 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.

parse: bool-SchemeExp -> boolExp
unparse: boolExp -> bool-SchemeExp

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

• Conversion to if form implemented by the function convert-to-if .
• 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 .

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

Conversion to if form

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

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 form implemented by the function convert-to-if.
• 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.

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

Conversion to if form

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

(make-Not  X)   

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

The conversion process always terminates (since each rewrite strictly
reduces the number of logical connectives in the expression) 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.

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 excluding {{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 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).

The following examples illustrate the conversion process:

(check-expect  (convert-to-if (make-Or (make-And 'x 'y) 'z))    (make-If (make-If 'x 'y false) true 'z))
(check-expect  (convert-to-if (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}}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 provided function {{parse: input -> boolExp}} takes a Scheme 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 =norm-ifExp=.

For example, the =ifExp=

the conversion process:

(check-expect  (convert-to-if (make-Or (make-And 'x 'y) 'z))    (make-If (make-If 'x 'y false) true 'z))
(check-expect  (convert-to-if (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) 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.

The provided function parse: input -> boolExp takes a Scheme 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 norm-ifExp .

For example, the ifExp (make-If (make-If X Y Z) U V))

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is

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not

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a

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norm-ifExp

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because

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it

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has

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an

...

If

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construction

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in

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test

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position.

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In

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contrast,

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the

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equivalent

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ifExp

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(make-If X (make-If Y U V) (make-If Z U V))

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is

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normalized

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and

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hence

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is

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an

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norm-ifExp

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.

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The

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normalization

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process,

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implemented

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by

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the

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function

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normalize:

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ifExp

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->

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norm-ifExp

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eliminates

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all

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if

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constructions

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that

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appear

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in

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test positions inside if constructions. We perform this transformation by repeatedly applying the following rewrite rule (to any portion of the expression) until it is inapplicable:

(make-If  (make-If  X  Y  Z)  U  V)  =>	 (make-If  X  (make-If  Y  U  V) (make-If Z  U  V)).

This transformation always terminates and yields a unique answer independent of the order in which rewrites are performed. The proof of this fact is left as an optional exercise.

In the normalize function, it is critically important not to duplicate any work, so the order in which reductions are made really matters. Do NOT apply the normalization rule above unless U and V are already normalized, because the rule duplicates both U and V. If you reduce the consequent and the alternative (U and V in the left hand side of the rule above) before reducing the test, normalize runs in linear time (in the number of nodes in the input); if done in the wrong order it runs in exponential time in the worst case. (And some of our test cases will exhibit this worst case behavior.)

Hint: define a sub-function head-normalize that takes three norm-ifExps X, Y, and Z and constructs a norm-ifExp equivalent to (makeIf X Y Z). This help function processes X because the test position must be a variable or a constant, yet X can be an arbitrary norm-ifExp. In contrast, (head-normalize X Y Z) never even inspects Y and Z because they are already normalized and the normalizing transformations performed in head-normalize never place these expressions in test position.

The following examples illustrate how the normalize and head-normalize functions behave:

(check-expect (head-normalize 'x 'y 'z) (make-If 'x 'y 'z))
(check-expect (head-normalize true 'y 'z) (make-If true 'y 'z))
(check-expect (head-normalize false 'y 'z) (make-If false 'y 'z))
(check-expect (head-normalize (make-If 'x 'y 'z) 'u 'v) (make-If 'x (make-If 'y 'u 'v) (make-If 'z 'u 'v)))
(check-expect (head-normalize (make-If 'x (make-If 'yt 'yc 'ya) (make-If 'zt 'zc 'za)) 'u 'v)
(make-If 'x (make-If 'yt (make-If 'yc 'u 'v) (make-If 'ya 'u 'v)) (make-If 'zt (make-If 'zc 'u 'v) (make-If 'za 'u 'v))))

(check-expect (normalize true) true)
(check-expect (normalize false) false)
(check-expect (normalize 'x) 'x)
(check-expect (normalize (make-If 'x 'y 'z)) (make-If 'x 'y 'z))
(check-expect (normalize (make-If (make-If 'x 'y 'z) 'u 'v)) (make-If 'x (make-If 'y 'u 'v) (make-If 'z 'u 'v)))

Once a large formula has been normalized, do not try to print it unless you know that the formula is small! The printed form can be exponentially larger than the internal representation (because the internal representation can share subtrees).

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

Symbolic Evaluation

The symbolic evaluation process, implemented by the function eval: norm-if-form environment -> norm-if-form, reduces a norm-if-form 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.

Symbolic evaluation applies the following rewrite rules to an expression until none is applicable (with one exception discussed below):

(make-If (make-If X Y Z) U V) => (make-If X (make-If Y U V) (make-If Z U V)).

This transformation always terminates and yields a unique answer
independent of the order in which rewrites are performed.
The proof of this fact is left as an optional exercise.

In the =normalize= function, it is critically important not to duplicate any work, so
the order in which reductions are made really matters.  Do *NOT* apply
the normalization rule above unless =U= and =V= are already
normalized, because the rule duplicates both =U= and =V=.  If you
reduce the _consequent_ and the _alternative_ (=U= and =V= in the left
hand side of the rule above) before reducing the _test_, =normalize=
runs in linear time (in the number of nodes in the input); if done in
the wrong order it runs in exponential time in the worst case.  (And
some of our test cases will exhibit this worst case behavior.)

Hint: define a sub-function head-normalize that takes three =norm-ifExp=s =X=, =Y=, and =Z=
and constructs a =norm-ifExp= equivalent to =(makeIf X Y Z)=.  This help function processes
=X= because the =test= position must be a variable or a constant, yet =X= can be an
arbitrary =norm-ifExp=. In contrast, =(head-normalize X Y Z)= never even inspects =Y=
and =Z= because they are already normalized and the normalizing transformations performed
in =head-normalize= never place these expressions in =test= position.

The following examples illustrate how the =normalize= and =head-normalize= functions behave:

(check-expect (head-normalize 'x 'y 'z) (make-If 'x 'y 'z))
(check-expect (head-normalize true 'y 'z) (make-If true 'y 'z))
(check-expect (head-normalize false 'y 'z) (make-If false 'y 'z))
(check-expect (head-normalize (make-If 'x 'y 'z) 'u 'v) (make-If 'x (make-If 'y 'u 'v) (make-If 'z 'u 'v)))
(check-expect (head-normalize (make-If 'x (make-If 'yt 'yc 'ya) (make-If 'zt 'zc 'za)) 'u 'v)
(make-If 'x (make-If 'yt (make-If 'yc 'u 'v) (make-If 'ya 'u 'v)) (make-If 'zt (make-If 'zc 'u 'v) (make-If 'za 'u 'v))))

(check-expect (normalize true) true)
(check-expect (normalize false) false)
(check-expect (normalize 'x) 'x)
(check-expect (normalize (make-If 'x 'y 'z)) (make-If 'x 'y 'z))
(check-expect (normalize (make-If (make-If 'x 'y 'z) 'u 'v)) (make-If 'x (make-If 'y 'u 'v) (make-If 'z 'u 'v)))

Once a large formula has been normalized, do not try to print it unless you know that the formula is small!
The printed form can be exponentially larger than the internal representation (because the internal representation can share subtrees).

Before you start writing =normalize=, write the template corresponding to the inductive data definition of =norm-ifExp=.

----+Symbolic Evaluation

The symbolic evaluation process, implemented by the function =eval: norm-if-form environment -> norm-if-form=, reduces a =norm-if-form=
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=.
<p>
Symbolic evaluation applies the following rewrite rules to
an expression until
none is applicable (with one exception
discussed below):

...

(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\[X <\- false\])

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The

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notation

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M[X

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<-

...

N]

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means

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M

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with

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all

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occurrences

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of

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the

...

symbol

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X

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replaced

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by

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the

...

expression

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N

...

.

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It

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is

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very

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costly

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to

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actually

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perform

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these

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subtitutions

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on

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=norm-if-form=

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data.

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To

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avoid

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this

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computational

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expense,

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we

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simply

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maintain

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a

...

list

...

of

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bindings

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which

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are

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pairs

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consisting

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of

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symbols

...

(variable

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names)

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and

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boolean

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values

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{

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true

...

,

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false

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.

...

The

...

following

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data

...

definition

...

definition

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formally

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defines

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the

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binding

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type.

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A

...

binding

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is

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a

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pair

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(make-binding

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s

...

v)

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where

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s

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is

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a

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symbol

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(a

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variable)

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and

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v

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is

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a

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boolean

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value

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(an

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element

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of

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{

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true

...

,

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false

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}.

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An

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environment

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is

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a

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(list-of binding)

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.

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When

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the

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eval

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function

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encounters

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a

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variable

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(symbol),

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it

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looks

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up

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the

...

symbol

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in

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the

...

environment

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and

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replaces

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the

...

symbol

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it's

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boolean

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value

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if

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it

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exists.

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These

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rewrite

...

rules

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do

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not

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have

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the

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Church-Rosser

...

property.

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The

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last

...

two

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rewrite

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rules

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are

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the

...

spoilers;

...

the

...

relative

...

order

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in

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which

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they

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are

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applied

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can

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affect

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the

...

result

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in

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some

...

cases.

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However,

...

the

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rewrite

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rules

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do

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have

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the

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Church-Rosser

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property

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on

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expressions

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which

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are

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tautologies

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or

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contradictions.

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If

...

the

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final

...

rule

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is

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applied

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only

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when

...

X

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actually

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occurs

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in

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either

...

Y

...

or

...

Z

...

,

...

then

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the

...

symbolic

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evaluation

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process

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is

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guaranteed

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to

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terminate.

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In

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this

...

case,

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every

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rule

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either

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reduces

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the

...

size

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of

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the

...

expression

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or

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the

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number

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of

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variable

...

occurrences

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in

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it.

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We

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recommend

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applying

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the

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rules

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in

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the

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order

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shown

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from

...

the

...

top

...

down

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until

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no

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more

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reductions

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are

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possible

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(using

...

the

...

constraint

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on

...

the

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final

...

rule).

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Note

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that

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the

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last

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rule

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should

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only

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be

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applied

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once

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to

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a

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given

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sub-expression.

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Conversion to Boolean Form

The final phase converts an expression in (not necessarily reduced) 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

(make-If X Y Z)

where one of the arguments {X, Y, Z is a constant { true, false }.

Use the following set of reduction rules to perform this conversion

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

(make-If X Y Z)

where one of the arguments {=X=, =Y=, =Z=} is a constant
{ =true=, =false=}.

Use the following set of reduction rules to perform this conversion

...

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%)