Due: **Friday, Nov 8th** by 11:59 PM

## Boolean Functions

A *boolean operator* operates on one or more boolean values (true/false)
to produce a boolean result. A binary boolean operator has two boolean operands.

We can describe a binary boolean operator using a *truth table*,
which shows the result of the operator for each of the four possible
combinations of operator values. For example, here is the truth
table for the **and** operator, denoted **^**:

L R L ^ R t t t t f f f t f f f f

For operands *L* and *R*, *L* **^** *R* is true when *L* and *R* are
both true, false otherwise. Here are the truth tables for four
additional binary boolean operators, **or** (denoted **+**),
**nand**, **nor**, and **xor**.

L R L + R t t t t f t f t t f f f

L R L nand R t t f t f t f t t f f t

L R L nor R t t f t f f f t f f f t

L R L xor R t t f t f t f t t f f f

We can think of boolean operators as being mathematical functions with two
parameters. By combining two boolean operators with two operands,
we can create a boolean function with three operands. For example,
*A* **^** (*B* **+** *C*) is a boolean function with the following
truth table:

A B C A ^ (B + C) t t t t t t f t t f t t t f f f f t t f f t f f f f t f f f f f

### Synthesizing an arbitrary boolean function

In digital circuit design, it is sometimes helpful to be able to create
an arbitrary boolean functions out of binary operators.
For example, *A* *op1* (*B* *op2* *C*) is a "template" for a boolean
function with three parameters. By substituting different operators
for *op1* and *op2*, we can create various functions.

## Your Task

Using Prolog, determine whether it is possible to synthesize the following
binary functions by substituting the **and**, **or**, **nand**,
**nor**, and **xor** operators in

Aop1(Bop2C)

If it is possible to synthesize a function, determine which operands
to use for *op1* and *op2*.

### Functions 1-3 (left to right)

A B C Aop1(Bop2C)t t t t t t f t t f t t t f f f f t t f f t f f f f t f f f f t

A B C Aop1(Bop2C)t t t f t t f f t f t f t f f f f t t f f t f t f f t t f f f t

A B C Aop1(Bop2C)t t t t t t f t t f t t t f f t f t t f f t f t f f t t f f f t

### Functions 4-6 (left to right)

A B C Aop1(Bop2C)t t t t t t f t t f t f t f f t f t t t f t f t f f t t f f f f

A B C Aop1(Bop2C)t t t f t t f f t f t f t f f f f t t t f t f f f f t f f f f t

A B C Aop1(Bop2C)t t t f t t f t t f t t t f f t f t t t f t f f f f t f f f f f

## Hints

Here is a suggestion for how to model the binary boolean operators:

eval(and,t,t,t). eval(and,t,f,f). eval(and,f,t,f). eval(and,f,f,f).

This establishes the ground truths for the **and** operator.
The **eval** predicate specifies what the result of a given boolean operator
is when given particular input values. A query with a free variable can
be used to determine the result value for particular inputs: for example,
for the query

eval(and,t,f,What).

Prolog will infer that **What** is **f**.

The key will be finding a way to evaluate two boolean operators for three input values. Suggestion: define a predicate of the following form:

composeTwo(X, Y, A, B, C, Z) :-something.

This predicate asserts that boolean operators *X* and *Y*,
when evaluated on the expression

AX(BYC)

will produce the result *Z*. This is useful for specifying one row
of the synthesized function's truth table. For example:

composeTwo(X, Y, t, t, t, f)

would assert that the result of the function is **f** when
*A* =**t**, *B* =**t**, and *C* =**t**.

## Deliverables

There are two deliverables.

The first deliverable is a text file which specifies, for functions 1-6, either

- which operators
*op1*and*op2*can be used to define the function, or - that there is no way to substitute the 5 binary operators for
*op1*and*op2*to define the function

The text file should also explain briefly how you used your Prolog code to find the operators (or prove their nonexistence) for each function.

The second deliverable is a text file containing your Prolog code.

## Submitting

Create a zip file with both deliverables described above
and submit it to Marmoset as **assign06**:

https://cs.ycp.edu/marmoset/