1. 1 Combinational Circuits Alexander Titov 10/27/2012

2. 2 Moscow Institute of Physics and Technology uArchSim Project Layers of Abstraction in Computes Science (CS) Application Algorithms Programming Language Operating System Instruction Set Architecture Microarchitecture Gates/Register-Transfer Level (RTL) Circuits Physics Topics of this lecture Less about physics, wires and transistors… More about logical circuits

3. 3 Moscow Institute of Physics and Technology uArchSim Project Boolean Algebra Boolean Algebra (BA) works with object that can take only two values (true and false, or 1 and 0 ). Such object is called a Boolean object (term bit of information is used in CS) BA defines operations on Boolean objects → Boolean operations and functions It is convenient to represent these operations and functions via truth tables: abc b a c Boolean function Truth table abc 00 abc 00 abc 00 01 10 11

4. 4 Moscow Institute of Physics and Technology uArchSim Project Main Boolean operations The simplest Boolean operation is … AND (or Boolean multiplication) inversion (NOT) x!x xx 01 10 x*y x y xy 00 01 10 11 0 0 0 1

5. 5 Moscow Institute of Physics and Technology uArchSim Project Main Boolean operations OR (or Boolean addition) x+y x y xy 00 01 10 11 0 1 1 1 XOR (or exclusive OR, or addition by module 1) xyx + y 00 01 10 11 0 1 1 1 x y

6. 6 Moscow Institute of Physics and Technology uArchSim Project Boolean functions Boolean operations can be combined into functions F(x, y, z) = x + !y*z x y z x + !y*z xyzF 100 110 100 110 0……0 1 0 1 0

7. 7 Moscow Institute of Physics and Technology uArchSim Project Main axioms There about ten axioms that can be used to create new or to simplify existed functions AxiomAND formOR form AxiomAND formOR form Identity law1*x = x0 + x = x AxiomAND formOR form Identity law1*x = x0 + x = x Idempotent lawx*x = xx + x = x AxiomAND formOR form Identity law1*x = x0 + x = x Idempotent lawx*x = xx + x = x Inverse law!x*x = 0!x + x = 1 AxiomAND formOR form Identity law1*x = x0 + x = x Idempotent lawx*x = xx + x = x Inverse law!x*x = 0!x + x = 1 Commutativity lawx*y = y*xx + y = y + x AxiomAND formOR form Identity law1*x = x0 + x = x Idempotent lawx*x = xx + x = x Inverse law!x*x = 0!x + x = 1 Commutativity lawx*y = y*xx + y = y + x Associativity law(x*y)*z = x*(y*z)(x + y) + z = x + (y + z) AxiomAND formOR form Identity law1*x = x0 + x = x Idempotent lawx*x = xx + x = x Inverse law!x*x = 0!x + x = 1 Commutativity lawx*y = y*xx + y = y + x Associativity law(x*y)*z = x*(y*z)(x + y) + z = x + (y + z) … There are a lot of other useful equations: !(xy) = !x + !y or something more complex … n(n(x) + y) + n(n(x) + n(y)) = x (Huntington equation)

8. 8 Moscow Institute of Physics and Technology uArchSim Project Combinational logic If the output of a function is completely defined by the current input then the function is called combinational. F t = F(x t, y t, z t )F = F(x t, y t, z t, F t- 1 ) combinational sequentional Combinational logic (scheme) is an implementation of a combinational function. Ok, but how does this all connect with the microarchitecture?

9. 9 Moscow Institute of Physics and Technology uArchSim Project Half adder scheme x y z0z0 z1z1 xyz0z0 z1z1 10 11 10 11 0 1 1 0 0 0 0 1 It is an adder, but it is not a full adder, because it does not have input carry It is called a half adder half + y x sum carry sum carry It is an adder, but it is not a full adder, because it does not have input carry

10. 10 Moscow Institute of Physics and Technology uArchSim Project Full adder scheme half + ynyn xnxn c n-1 half + snsn Cn xnxn ynyn c n-1 snsn cncn 001 011 101 111 1 0 1 1 0 1 0 1

11. 11 Moscow Institute of Physics and Technology uArchSim Project Wide adder half + ynyn xnxn y n+1 x n+1 c n-1 cncn c n+1 snsn s n+1

12. Thank You 12