Autosave: 2022-12-19 08:00:05

Logic/Propositional_logic/Boolean_function_synthesis.md

@@ -14,3 +14,48 @@ This is an important skill that we will use when constructing [logic circuits](/
 ## The process
 
 The process proceeds as follows:
+
+1. Work out the truth conditions for the circuit we want to construct
+2. Identify the rows where the output is equal to 1
+3. For each of these rows construct a Boolean expression that evaluates to that output
+4. Join each expression with OR
+5. Reduce these expressions to a single expression in its simplest form
+
+## Example
+
+Let's say we have the following truth table:
+
+| Line | $x$ | $y$ | $z$ | $f$ |
+| ---- | --- | --- | --- | --- |
+| 1    | 0   | 0   | 0   | 1   |
+| 2    | 0   | 0   | 1   | 0   |
+| 3    | 0   | 1   | 0   | 1   |
+| 4    | 0   | 1   | 1   | 0   |
+| 5    | 1   | 0   | 0   | 1   |
+| 6    | 1   | 0   | 1   | 0   |
+| 7    | 1   | 1   | 0   | 0   |
+| 8    | 1   | 1   | 1   | 0   |
+
+We only need to focus on lines 1, 3, and 5 since they have the output 1:
+
+| Line | $x$ | $y$ | $z$ | $f$ |
+| ---- | --- | --- | --- | --- |
+| 1    | 0   | 0   | 0   | 1   |
+| 3    | 0   | 1   | 0   | 1   |
+| 5    | 1   | 0   | 0   | 1   |
+
+For each line we construct a Boolean expression that would result in the value in the $f$ column. In other words we construct the function:
+
+| Line | $x$ | $y$ | $z$ | $f$                                         |
+| ---- | --- | --- | --- | ------------------------------------------- |
+| 1    | 0   | 0   | 0   | $ \lnot(x) \land \lnot (y) \land \lnot(z) $ |
+| 3    | 0   | 1   | 0   | $ \lnot(x) \land y \land \lnot(z) $         |
+| 5    | 1   | 0   | 0   | $ x \land \lnot(y) \land \lnot(z) $         |
+
+We can now join each expression to create a complex expression that covers the entire truth table. Since 1 will be output for any one of these sub-expressions we can just join them up with OR:
+
+$$
+(\lnot(x) \land \lnot (y) \land \lnot(z)) \lor \lnot(x) \land y \land \lnot(z) \lor  x \land \lnot(y) \land \lnot(z)
+$$
+
+It's clear that we have transcribed the truth conditions accurately but that we are doing so in a rather verbose way. Let's simplify: