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Understanding Binary Arithmetic: Half and Full Adders in Digital Logic Circuits

Dive into the fundamentals of binary arithmetic, focusing on how digital computers perform addition using only two bits: 0 and 1. This guide covers crucial concepts such as half adders and full adders, detailing how they process inputs (A, B, and Cin) to produce outputs (S, Cout). Learn about key operations like sum and carry using logic gates, including AND, OR, and XOR. We explore how to extend these principles for multi-bit addition and the impact of circuit design on performance. Join us in simplifying circuits and examining the P=NP challenge in logic.

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Understanding Binary Arithmetic: Half and Full Adders in Digital Logic Circuits

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Presentation Transcript

  1. Logic and computers

  2. Binary Arithmetic Only two digits: the bits 0 and 1 (Think: 0 = F, 1 = T) 0 + 1 ---- 1 1 + 0 ---- 1 1 + 1 ---- 10 0 + 0 ---- 0

  3. Logic and Computers • A half adder: • Two bits in (A, B: to be added together) • Two bits out (S, C: sum and carry) • 0+0=0, carry 0 • 0+1=1, carry 0 • 1+0=1, carry 0 • 1+1=0, carry 1 • S := A⊕B • C := A∧B

  4. NOT OR NOR AND NAND XOR NXOR (EQUIV)

  5. Logic and Computers A S B C S := A⊕B C := A∧B

  6. Half Adder A S B C A S B C HA

  7. A Longer Addition 1 1 1 1 0 11 + 11

  8. Full Adder Need a third input to create a component of a ripple-carry adder: the carry from the previous bit position Inputs: A, B, Cin Outputs: S, Cout

  9. Full Adder Cin S HA A B HA Cout

  10. Full Adder Cin S A B Cout FA Cin S HA A B HA Cout

  11. Ripple carry adder c2 c1 carryout 0 a2 b2 a1 b1 FA FA 2-bit adder: a1a2+b1b2 = c1c2 with carryout Generalizes to n-bit addition How does the time delay through the circuit depend on n, the number of bits to be added?

  12. Simplifying Circuits • Simpler formulas turn into circuits that use less hardware! • E.g. p ⋁ q ⋁ (p⋀q) is equivalent to p ⋁ q but would use more logic gates • But the P=NP? question means that it may be hard to simplify formulas as much as possible • Any tautology is equivalent to p ⋁ ¬p so if we could easily simplify formulas we could easily determine whether a formula is a tautology

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