Digital Logic. Software. Digital Logic. Boolean value (bit): 0 or 1. Transistors (more in lab) 1/26/16. Program, Application. Programming Language
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1 /26/6 S 24, Fall 24 S 24, Fall 24 Program, pplication Digital Logic Software Programming Language ompiler/interpreter Operating System Instruction Set rchitecture Gateway to computer science Hardware Microarchitecture Digital Logic Devices (transistors, etc.) Solid- State Physics S 24, Fall 24 S 24, Fall 24 oolean value (bit): or basis of all digital data representations oolean functions (ND, OR, NOT, ): basis of all digital computations Electronically: bit high voltage vs. low voltage 3.3V 2.8V.5V.V oolean functions logic gates, built from transistors Transistors (more in lab) If ase voltage is high: urrent may flow freely from ollector to Emitter. If ase voltage is low: urrent may not flow from ollector to Emitter. V in V out low high high low Truth table in out V in in F T ase out T F resistor +V cc (Supply Voltage) ollector Emitter (Ground) NOT gate V out 3
2 /26/6 S 24, Fall 24 Digital Logic Gates Tiny electronic devices t hat compute basic oolean functions. NOT S 24, Fall 24 Integrated ircuits (invented 95s) hip Gates are manufactured in units called integrated circuits. From S SI (tens) t o V LSI (hundreds of t housands t o billions) NND V2 Vin Vout V +Vcc Wafer +Vcc V out V Vout V2 Vin S 24, Fall 24 Five basic gates: define with truth tables oolean lgebra for combinational logic NOT NND NOR ND OR ( ) ND oolean product inputs wires gates circuits S 24, Fall 24 variables pressions operators/functions functions + OR oolean sum + NOT inverse or complement wire identity 2
3 /26/6 S 24, Fall 24 ircuits S 24, Fall 24 Translation onnect inputs and outputs of gates with wires. rossed wires touch only if there is a dot. onnect gates to implement these functions. heck with truth tables. one- to- one translation is straightforward and bidirectional. F ( + )D What is the output if,,? What is the truth table of this circuit? What is an equivalent oolean pression? Z W + (X + WY) S 24, Fall 24 Note on notation: bubble inverse/complement + + S 24, Fall 24 ommutativity, ssociativity Identity law, inverse law () () 3
4 /26/6 S 24, Fall 24 Idempotent law, Null/Zero law S 24, Fall 24 Note on notation: bubble inverse/complement DeMorgan's Law (double bubble, toil and trouble, in Randy's words...) + S 24, Fall 24 One law, bsorption law Write truth tables. Do they correspond to simpler circuits? + S 24, Fall 24 NND is universal. ll oolean functions can be implemented using only NNDs. Let's prove it! uild NOT, ND, OR, NOR, using only NND gates. + 4
5 /26/6 S 24, Fall 24 XOR: Exclusive OR Output if actly one input. S 24, Gates Fall 24 of XOR Truth table: uild from earlier gates: Often used as a one- bit comparator. Video game designers, Halloween costumers traordinaire, sci- fi/fantasy screenwriters, I have an idea 25 en Wood Exclusive preview or something. S 24, Fall 24 Larger gates S 24, Fall 24 ircuit simplification Why simplify? Smaller cheaper, faster, cooler, easier to design/build. uild a 4- input ND gate using any number of 2- input gates. an we find a simpler circuit that performs the same function? Start with an equivalent oolean pression, then simplify with algebra. F(,, ) heck the answer with a truth table. 5
6 /26/6 S 24, Fall 24 ircuit derivation: code detectors S 24, Fall 24 ircuit derivation: sum- of- products form ND gate + NOT gates code detector, recognizes actly one input code. Design a 4- input code detector to output if D, and otherwise. D Design a 4- input code detector to accept two codes (D, D) and reject all others. (accept, reject ) logical sum (OR) of products (ND) of inputs or their complements (NOT) Draw the truth table and design a sum- of- products circuit for a 4- input code detector to accept two codes (D, D) and reject all others. How are the truth table and the sum- of- products circuit related? S 24, Fall 24 Voting machines majority circuit outputs if and only if a majority of its inputs equal. Design a majority circuit for three inputs. Use a sum of products. Majority Triply redundant computers in spacecraft ompute everything on three separate devices and compare answers. Majority wins. osmic rays can flip bits on wires, especially in space. Triple redundancy survives failure in one device. Space program also hastened Integrated ircuits. S 24, Fall 24 Margaret Hamilton (speaking of space and reliability) Led software team for pollo Guidance omputer. Developed software engineering techniques for correctness and reliability. oined "software engineering". Software avoided mission abort on first moon landing! pollo code print- out 6
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