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

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

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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

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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

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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

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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

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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

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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

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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

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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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

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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

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