ece124: add multilevel synthesis

docs/1b/ece124.md

@@ -524,6 +524,47 @@ Alternatively, each state can be assigned exactly one `1` bit, and transitions f
 
 **Dynamic hazards** occur when outputs flip multiple times before stabilising. These can be avoided by switching everything to 2-term POS or SOP and fixing static hazards.
 
+## Multilevel synthesis
+
+!!! definition
+    - A **literal** is an input character.
+    - An **implicant** is a collection of inputs that results in a true output.
+    - A **prime implicant** is such that no literals can be removed while remaining an implicant.
+    - A **cover** is a set of implicants that cover every possible way $f=1$.
+    - An **essential prime implicant** is such that there is no other prime implicant that fulfill a necessary condition to make $f$ true.
+
+TO reduce fan-in, multi-input ANDs and ORs can be broken up to multiple versions of their nested form via **factoring**.
+
+$$abcde\to(abc)(de)$$
+
+**Functional decomposition** takes common terms and only calculates them first before feeding that input into the rest of the circuit.
+
+### Tabular method
+
+Cost is minimised when all essential prime implicants are present and the fewest number of prime implicants for the remaining terms.
+
+1. List minterms, group by the number of ones in binary (don't cares can be treated as minterms)
+2. Write the implicant for each
+3. For each group, if an implicant differs by one bit from an implicant in the group above, merge them (replacing the distinctive term with $x$) and check that minterm / implicant off
+4. Repeat, ensuring that $x$ only merges with $x$ in the same columns
+
+Implicants not checked off are prime implicants.
+
+1. List all primes and the minterms they cover as a table, excluding don't cares
+  - Minterms with only one prime have that as an essential prime
+2. Primes that cover the same minterms as another but also more are objectively better (**row dominance**)
+3. Make educated guesses to minimise prime implicants
+
+Alternatively, instead of removing **dominated rows**, **dominatING columns** can be removed instead.
+
+### Petrick method
+
+Once reduced to tablular form:
+
+1. For each column, sum all the possible ways a minterm can be covered, then product those sums
+2. Expand and simplify, then choose the products with the least number of literals
+3. Each product is a solution if you replace the product with a sum of the multiplied literals instead
+
 ## VHDL
 
 VHDL is a hardware schematic language.