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Logic Circuits and Computer Architecture

Logic Circuits and Computer Architecture. Appendix A Digital Logic Circuits Part 1: Combinational Circuits and Minimization. Structured organization. Problem-oriented language level Assembly language level Operating system machine level Instruction set architecture level

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Logic Circuits and Computer Architecture

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  1. Logic Circuits and Computer Architecture Appendix A Digital Logic Circuits Part 1: Combinational Circuits and Minimization Rev. 4.1 (2006-07) by Enrico Nardelli

  2. Structured organization • Problem-oriented language level • Assembly language level • Operating system machine level • Instruction set architecture level • Microarchitecture level • Digital logic level Abstraction lelvel Rev. 4.1 (2006-07) by Enrico Nardelli

  3. Digital Logic level • Digital circuits • Only two logical levels present (i.e., binary) • low/high voltage • Basic gates • AND, OR, NOT • Basic circuits • Combinational (without memory, stateless) • Sequential (with memory, state dependent behaviour) Rev. 4.1 (2006-07) by Enrico Nardelli

  4. Boolean Algebra • Variables: A, B, … • Domain of variables: 2 values • 1 or 0; Y or N; true or false; … • Fundamental Operations • AND, OR, NOT • Intended meaning (for humans - Laws of Thought) • AND: both inputs are true • OR: at least one input is true • NOT: negate the input • Named from George Boole Rev. 4.1 (2006-07) by Enrico Nardelli

  5. George Boole (1815-1864) An Investigation of the Laws of Thought, on Which are founded the Mathematical Theories of Logic and Probabilities (1854) Rev. 4.1 (2006-07) by Enrico Nardelli

  6. Claude Shannon (1916-2001) A Symbolic Analysis of Relay and Switching Circuits (1938) ENIAC (Electronic Numerical Integrator And Calculator) (1946) Rev. 4.1 (2006-07) by Enrico Nardelli

  7. Formal definition of functions (1) • By means of “truth tables” • Explicit representation of the output for all possible inputs Rev. 4.1 (2006-07) by Enrico Nardelli

  8. …truth table for a 3-variable function… f(A,B,C)= 1 if and only if at least 2 variables are equal to 1 Rev. 4.1 (2006-07) by Enrico Nardelli

  9. Formal definition of functions (2) • By means of “boolean equation” • boolean equation consists of: • variables • constants 0 and 1 • boolean operations (AND, OR, NOT) • parentheses M = OR( AND(NOT(A),NOT(B)), AND(A,B) ) Rev. 4.1 (2006-07) by Enrico Nardelli

  10. Boolean functions • Conventions • NOT (negation): NOT(A) = A’ = A • AND (conjunction): AND(A,B) = AB = A.B • OR (disjunction): OR(A,B) = A+B M = OR( AND(NOT(A),NOT(B)), AND(A,B) ) M = (((A)’(B)’) + (AB)) Rev. 4.1 (2006-07) by Enrico Nardelli

  11. Boolean Operator Precedence • The order of evaluation in boolean expression is: • Parentheses • NOT • AND • OR • Consequence: parantheses appear around OR expressions • Example: F=A(B+C)(C+D’) M = (((A)’(B)’) + (AB)) M = A’B’ + AB Rev. 4.1 (2006-07) by Enrico Nardelli

  12. NOT gate - the simplest one • NOT gate - inverts the signal If A is 0, X is 1 If A is 1, X is 0 • A NOT gate is also called an inverter Rev. 4.1 (2006-07) by Enrico Nardelli

  13. AND gate • Output is 1 if all inputs are 1 • In general, if the AND gate has N inputs, both input 1 AND input 2 AND … AND input N must be 1 for the output to be 1 • 2-input AND gate Rev. 4.1 (2006-07) by Enrico Nardelli

  14. OR gate • Output is 1 if at least one input is 1 • In general, if the OR gate has N inputs, input 1 OR input 2 OR … OR input N must be 1 for the output to be 1 • 2-input OR gate Rev. 4.1 (2006-07) by Enrico Nardelli

  15. A more complex example • 2-input “equivalence” circuit • The output is 1 if the inputs are the same • (i.e., both 0 or both 1) Truth table • Boolean function: M = A’B’ + AB Rev. 4.1 (2006-07) by Enrico Nardelli

  16. Formal definition of functions (3) • By means of logic circuits • Combination of logic gates joined by wires Rev. 4.1 (2006-07) by Enrico Nardelli

  17. Conventions for logic circuits Rev. 4.1 (2006-07) by Enrico Nardelli

  18. Exercise (1) • Write the truth table and the logic circuit for F = X + Y’Z Rev. 4.1 (2006-07) by Enrico Nardelli

  19. Truth table Rev. 4.1 (2006-07) by Enrico Nardelli

  20. Logic Circuit Rev. 4.1 (2006-07) by Enrico Nardelli

  21. Exercise (2) • Write the boolean function and its truth table for the following logic circuit Rev. 4.1 (2006-07) by Enrico Nardelli

  22. Function and Truth Table • F = Y’ + X’YZ’ + XY Rev. 4.1 (2006-07) by Enrico Nardelli

  23. Exercise (3) • Write the boolean function and its truth table for the following logic circuit Rev. 4.1 (2006-07) by Enrico Nardelli

  24. Function and Truth Table • F = X’YZ + X’YZ’ + XZ Rev. 4.1 (2006-07) by Enrico Nardelli

  25. Conversion between represent. • Circuit -> -> Boolean formula (left-to-right inspection) -> Truth table (explicit case-by-case computation) • Boolean formula -> -> Circuit (bottom-up construction) -> Truth table (explicit case-by-case evaluation) • Truth table -> -> Circuit (through boolean formula) -> Boolean formula (through canonical form – see later) Rev. 4.1 (2006-07) by Enrico Nardelli

  26. duality principle: any algebraic equality remains true when the operators OR and AND, and the elements 0 and 1 are interchanged Boolean Identities 1A = A 0+A = A Identity 0A = 0 1+A = 1 Null AA = A A+A = A Idempotent AA’ = 0 A+A’ = 1 Inverse AB = BA A+B = B+A Commutative (AB)C = A(BC) (A+B)+C = A+(B+C) Associative A+BC = (A+B)(A+C) A(B+C) = AB+AC Distributive A(A+B) = A A+AB = A Absorption (AB)’ = A’+B’ (A+B)’ = A’B’ De Morgan Rev. 4.1 (2006-07) by Enrico Nardelli

  27. Truth tables to verify De Morgan’s theorem Rev. 4.1 (2006-07) by Enrico Nardelli

  28. Remark Each equality remains true if you sobstitute any variable with any expression Examples (A+B)(A+CD’) = A + BCD’ (distributive) ((A+BC)(D+A))’ = (A+BC)’ + (D+A)’ (De Morgan) = A’ (BC)’ + D’A’ (De Morgan) = A’(B’+C’) + D’A’ Rev. 4.1 (2006-07) by Enrico Nardelli

  29. … algebraic manipulation… F = X’YZ + X’YZ’ + XZ (distributive) = X’Y(Z + Z’) + XZ (inverse) = X’Y 1 + XZ (identity) = X’Y + XZ Rev. 4.1 (2006-07) by Enrico Nardelli

  30. Boolean Algebra Vs Switching Algebra (1) A Boolean Algebra is a structure A = <A, +, · , ’, 0, 1> where • A is a set • + and · are binary operations • ‘ is a unary operation • 0, 1  A satisfying the following axioms (i) + and · are commutative (ii) 0 and 1 satisfy: a·1=a and a+0=a,  a A (iii) + and · distribute over each other (iv) for each element a A, there exists an element a’ A such that a + a’= 1 and a·a’=0 Rev. 4.1 (2006-07) by Enrico Nardelli

  31. Boolean Algebra Vs Switching algebra (2) Switching Algebra is the following boolean algebra A = <{0,1}, +, · , ’, 0, 1> Rev. 4.1 (2006-07) by Enrico Nardelli

  32. Observation Axioms (i)-(iv) can be used to prove all the other identities An example: Idempotent X + X = X X + X = (X + X)·1 (ii) = (X + X)(X + X’) (iv) = X + (X·X’) (iii) = X + 0 (iv) = X (ii) Rev. 4.1 (2006-07) by Enrico Nardelli

  33. De Morgan circuit equivalents • AND/OR can be interchanged if you invert the inputs and outputs bubble means inversion Rev. 4.1 (2006-07) by Enrico Nardelli

  34. NAND gate - the negation of AND • The opposite of the AND gate is the NAND gate (output is 0 if all inputs are 1) Truth table • Logic diagram Rev. 4.1 (2006-07) by Enrico Nardelli

  35. NOR gate - the negation of OR • The opposite of the OR gate is the NOR gate (output is 0 if any input is 1) Truth table • Logic diagram Rev. 4.1 (2006-07) by Enrico Nardelli

  36. Exercise • Write the truth table for: • a 3 input NAND gate • a 3 input NOR gate Rev. 4.1 (2006-07) by Enrico Nardelli

  37. XOR gate - the exclusive OR • For a 2-input gate • Output is 1 if exactly one of the inputs is 1 Truth table • Logic diagram • For > 2 inputs: output is 1 if an odd number of inputs is 1 Rev. 4.1 (2006-07) by Enrico Nardelli

  38. Universal Gates • How many logical functions there are with n input? • With n inputs there are 2(2n ) possible logical functions Rev. 4.1 (2006-07) by Enrico Nardelli

  39. Universal Gates (2) • AND, OR, NOT can generate all possible boolean functions (see later) • Is it possible to use fewer basic operations? • Universal gate: a gate type that can implement any Boolean function Rev. 4.1 (2006-07) by Enrico Nardelli

  40. Universal Gates (3) • AND, NOT are enough ! • OR, NOT are enough ! • Even NAND alone or NOR alone are enough ! Rev. 4.1 (2006-07) by Enrico Nardelli

  41. How NAND simulates AND, OR • Simulation of NOT ??? Rev. 4.1 (2006-07) by Enrico Nardelli

  42. Alternative NAND representations Rev. 4.1 (2006-07) by Enrico Nardelli

  43. How NOR simulates AND, OR • Simulation of NOT ??? Rev. 4.1 (2006-07) by Enrico Nardelli

  44. Alternative NOR representations Rev. 4.1 (2006-07) by Enrico Nardelli

  45. Gate equivalence • Any AND, OR, NOT gate can be obtained using just NAND gates or just NOR gates • Consequence: any circuit can be constructed using just NAND gates or just NOR gates (easier to build) Rev. 4.1 (2006-07) by Enrico Nardelli

  46. Equivalence modifications (1) Rev. 4.1 (2006-07) by Enrico Nardelli

  47. Equivalence modifications (2) • Substitute equivalent gates Rev. 4.1 (2006-07) by Enrico Nardelli

  48. Transforming OR, AND to NAND • Transform the following circuit Rev. 4.1 (2006-07) by Enrico Nardelli

  49. Solution Rev. 4.1 (2006-07) by Enrico Nardelli

  50. Exercise • Write a NAND only logic circuit for F = XY’ + X’Y + Z Rev. 4.1 (2006-07) by Enrico Nardelli

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