1. CS1022 Computer Programming & Principles Lecture 9.2 Boolean Algebra (2)

2. Plan of lecture Simplifying Boolean expressions Karnaugh maps Logic circuits 2 CS1022

3. Simplifying Boolean expressions (1) Given a Boolean expression, can we find a “simpler” equivalent expression? – “Simpler” = “fewer symbols” Technique applied to disjunctive normal form (DNF) – Even though it might be more complex than original – Process: original expr.  DNF  simplified expr. Technique restricted to at most 3 Boolean variables – Extension to more variables is possible 3 CS1022

4. Simplifying Boolean expressions (2) To simplify, let’s suppress “  ” – Just like “  ” in algebra can be dropped (e.g., 2y = 2  y) – For instance, the expression in disjunctive normal form (p  q  r)  (p  q  r)  (p  q  r) Will be represented as pqr  pqr  pqr We can simplify the above expression as follows: 4 CS1022 pqr  pqr  pqr  (prq  prq)  pqr Comm.& assoc. Laws  pr(q  q)  pqr Distr. Law  pr  pqrSince q  q = 1

5. Simplifying Boolean expressions (3) Compare pr  pqr and pqr  pqr  pqr – They are equivalent – all changes done via a “law” – It is much simpler than original expression Very important: simplification can be automated – Each step in the process is an operation – Sequence of steps terminates at some point Notice: – First step puts together two minterms which differed by one symbol – Second step reduced two terms to one 5 CS1022

6. Karnaugh maps (1) Simplification done via a Karnaugh map – “Device” invented in the 1950s to help circuit design – Uses a visual display to indicate which pairs of minterms can be merged as a simpler expression For Boolean expressions with 3 variables (p, q and r) – Karnaugh map is a table with 2 rows and 4 columns – Columns labelled with all possible disjunctions of p and q and their negations – Rows labelled with r and r 6 CS1022 pq pqpqpqpq r r

7. Karnaugh maps (2) As we move from column to column precisely one change occurs in the column label Cells correspond to 8 different minterms stemming from 3 Boolean variables For a given Boolean expression, we write “1” in each cell whose row/column expression appears For instance, pqr  pqr  pqr has map 7 CS1022 pq pqpqpqpq r11 r 1

8. Karnaugh maps (3) Required simplification suggested by “clusters” of 1s – Shaded in green here There is only one such cluster in the example – Corresponds to terms we combined using the laws 8 CS1022 pq pqpqpqpq r11 r 1

9. Karnaugh maps (3) Relabelling columns preserves adjacency – Some clusters may be “hidden” – NB relabelling must be consistent with requirement that adjacent columns differ by one symbol only Example: Karnaugh map of pqr  pqr  pqr – Relabelling columns produces an alternative Karnaugh map and reveals a pair of adjacent minterms 9 CS1022 pq pqpqpqpq r11 r 1 pqpqpqpq r11 r 1

10. Karnaugh maps (4) Hence, 10 CS1022 pqpqpqpqpq r11 r 1 pqr  pqr  pqr  pqr  (pqr  pqr)  pqr  pr(q  q)  pqr  pr

11. Karnaugh maps (5) Let’s simplify pqr  pqr  pqr  pqr  pqr Karnaugh map: It contains a cluster of 4 (A) and a pair (B) – There are no hidden pairs – We must try relabelling exhaustively 11 CS1022 pq pqpqpqpq r11 r 111 pqr  pqr  pqr  pqr  (pqr  pqr)  pqr  pr(q  q)  pqr  pr

12. Karnaugh maps (6) Cluster (A) corresponds to 12 CS1022 pq pqpqpqpq r11 r 111 pqr  pqr  pqr  pqr  (p  p)qr  (p  p)qr  qr  qr  q(r  r)  q

13. Karnaugh maps (7) Cluster (B) corresponds to So, simplified expression is q  pr 13 CS1022 pq pqpqpqpq r11 r 111 pqr  pqr  pr(q  q)  prpr

14. Logic circuits (1) One of the main application of Boolean algebra is the design of binary devices – They accept a (finite) number of inputs – They produce a (finite) number of outputs – Binary: only two possible values for each input & output Circuits as “black boxes”: 14 CS1022 ?... p1p1 p2p2 p3p3 pnpn q1q1 q2q2 q3q3 qmqm

15. Logic circuits (2) Devices are built of logic gates which perform basic Boolean operations ( , , and ) 15 CS1022 a b a  b a b a  b a a a b (a  b) OR gate AND gate NOT gate NAND gate

16. Interconnecting gates produces a logical circuit A logical circuit evaluates a Boolean expression Example: What is the final output of this circuit? Logic circuits (3) 16 CS1022 4 3 2 1 p q r 5 6 7

17. 4 3 2 1 p q r 5 6 7 Example: What is the final output of this circuit? Logic circuits (4) 17 CS1022 4 3 2 1 p q r 5 6 7 pq 4 3 2 1 p q r 5 6 7 4 3 2 1 p q r 5 6 7 pqr pqr  pqr pqr  pqr  pqr

18. Solution: pqr  pqr  pqr Logic circuits (5) 18 CS1022 GateInputsOutput 1p, qpq 2 p, qpq 3pq, rpqr 4 pq, rpqr 5 pq, rpqr 6 pqr, pqrpqr  pqr 7 pqr  pqr, pqrpqr  pqr  pqr

19. Logic circuits can be simplified if we allow AND and OR gates to have more than 2 inputs More dramatic simplifications w/ Karnaugh maps – 2 clusters (one hidden) We can simplify things: – pqr  pqr  pq(r  r)  pq – pqr  pqr  (q  q)pr  pr That is, pqr  pqr  pqr  pq  pr – We can further simplify this as p(q  r) Logic circuits (6) 19 CS1022 pq pqpqpqpq r11 r 1

20. With p(q  r), we simplify previous circuit to obtain Logic circuits (7) 20 CS1022 p q r

21. Further reading R. Haggarty. “Discrete Mathematics for Computing”. Pearson Education Ltd. 2002. (Chapter 9) Wikipedia’s entry on Boolean algebra Wikibooks entry on Boolean algebra 21 CS1022