1. 1 COMP541 Combinational Logic - II Montek Singh Aug 27, 2014

2. 2Today  Digital Circuits (review)  Basics of Boolean Algebra (review) Identities and Simplification Identities and Simplification  Basics of Logic Implementation Minterms and maxterms Minterms and maxterms Going from truth table to logic implementation Going from truth table to logic implementation

3. Digital Circuits  Digital Circuit = network that processes binary variables one or more binary inputs one or more binary inputs one or more binary outputs one or more binary outputs  inputs and outputs are called “terminals” a functional specification a functional specification  relationship between inputs and outputs a timing specification a timing specification  describes delay from inputs changing to outputs responding 3

4. Digital Circuits  Inside the black box subcircuits or components or elements subcircuits or components or elements connected by wires connected by wires wires and terminals often called “nodes” wires and terminals often called “nodes”  each node has a binary value  each node is an input, an output, or “internal”  Example: E1, E2, E3 are elements E1, E2, E3 are elements A, B, C are input nodes A, B, C are input nodes Y, Z are output nodes Y, Z are output nodes n1 is an internal node n1 is an internal node 4

5. Types of circuits  Two types: with memory and without Combinational Circuit Combinational Circuit  output depends only on the current values of the inputs –provided enough time is given for output to respond  output does not depend on past inputs or outputs  called “memoryless”  example: AND gate Sequential Circuit Sequential Circuit  anything not combinational is sequential  output depends on not only current inputs, but also past behavior –previous inputs and/or outputs affect behavior  has “memory”, or is “stateful”  example: counter 5

6. Combinational Circuits: Examples 6 Adder OR Multi-output example Slash notation

7. Combinational Circuits  Theorem: A circuit is combinational if: every element is itself combinational every element is itself combinational every node is either designated as an input, or connects to exactly one output terminal of an element every node is either designated as an input, or connects to exactly one output terminal of an element  outputs of two elements are never “shorted together”  ensures that each node has a unique/unambiguous value contains no cyclic paths contains no cyclic paths  every path through the circuit visits each node at most once  no “feedback”  Conditions above ensure that output is only a function of inputs Proof: By induction Proof: By induction 7

8. Combinational Circuits: Examples  Which meet the conditions for combinational logic? 8

9. 9 Identities in Boolean Algebra  Use identities to manipulate functions  You can use distributive law … … to transform … to transform to

10. 10 Table of Identities

11. 11Duals  Left and right columns are duals  Replace ANDs and ORs, 0s and 1s

12. 12 Single Variable Identities

13. 13Commutativity  Operation is independent of order of variables

14. 14Associativity  Independent of order in which we group  So can also be written as and

15. 15Distributivity

16. Substitution  Can substitute arbitrarily large algebraic expressions for the variables Distribute an operation over the entire expression Distribute an operation over the entire expression Example: Example: X + YZ = (X+Y)(X+Z) Substitute ABC for X ABC + YZ = (ABC + Y)(ABC + Z) 16

17. 17 DeMorgan’s Theorem  Used a lot NOR  invert, then AND NOR  invert, then AND NAND  invert, then OR NAND  invert, then OR

18. 18 Truth Tables for DeMorgan’s

19. 19 Algebraic/Boolean Manipulation  Apply algebraic and Boolean identities to simplify expression example: example:

20. 20 Simplification Example Apply

21. 21 Fewer Gates

22. 22 Consensus Theorem  The third term is redundant Can just drop third term (consensus term) Can just drop third term (consensus term)  Proof summary (for first version): For third term to be true, Y & Z both must be 1 For third term to be true, Y & Z both must be 1 Then one of the first two terms is already 1! Then one of the first two terms is already 1!  Exercise: Provide a similar proof for the 2 nd version

23. 23 Complement of a Function  Definition: 1s & 0s swapped in truth table  Mechanical way to derive algebraic form Take the dual Take the dual  Recall: Interchange AND and OR, and 1s & 0s Complement each literal Complement each literal  x becomes x’ (x’ means complement of x)

24. 24 Next Lecture  Next Class: More on combinational logic Commonly-used combinational building blocks Commonly-used combinational building blocks