WEEK 2.1 BOOLEAN ALGEBRA

Transcription

1 WEEK 2.1 BOOLEAN ALGEBRA 1

2 Boolean Algebra Boolean algebra was introduced in 1854 by George Boole and in 1938 was shown by C. E. Shannon to be useful for manipulating Boolean logic functions. The postulates and theorems of Boolean algebra are useful to simplify expressions, to prove equivalence of expressions, etc. ECE 124 Digital Circuits and Systems Page 2

3 Axioms/Postulates of Boolean Algebra (1) Boolean algebra is an algebraic structure defined by a set of elements, B, together with two binary operators + and * that satisfy the following postulates: 1. Postulate 1: a) Closure with respect to +. b) Closure with respect to *. 2. Postulate 2: a) An identity element with respect to +, designated by 0. b) An identity element with respect to *, designated by Postulate 3: a) Commutative with respect to +. b) Commutative with respect to *. 4. Postulate 4: a) Distributive * over +. b) Distributive + over *. 5. Postulate 5: For each element x in B, there is an element : x in B (the complement) such that (a) x+!x = 1 and (b) x *!x = Postulate 6: There exists at least two elements x,y in B, such that x y. ECE 124 Digital Circuits and Systems Page 3

4 Axioms/Postulates of Boolean Algebra (2) Using the definitions of AND/OR and NOT functions, we can show all the postulates are satisfied. Note: Postulates are facts that can be taken as true; they do not require proof. ECE 124 Digital Circuits and Systems Page 4

5 Theorems and Properties of Boolean Algebra Theorems help us out in manipulating Boolean expressions. They must be proven from the postulates and/or other theorems known to be proven correct. Note the duality in relationships if we interchange + with * and 0 with 1. ECE 124 Digital Circuits and Systems Page 5

6 DeMorgan: U?A 2 05 U?A U?A U?A 1 05 U?A 2 05 U?A U?A 3 3 x + y = x. y (x. y ) = x + y (x. y ) = x + y (x + y ) = x. y 6

7 Synthesis While Boolean algebra is good and truth tables tell us what a function does, how do we actually make or implement a function? The implementation of a circuit is known as synthesis and we can draw circuits implementing functions using the logic gate symbols. Consider two functions and how we can implement them with gates. How can we use three input and gate and inverters to produce a desirable function output? REMEMBER: AND produces o/p of 1 only if all i/p s are 1. f 1 = x. y. z + x. y. z + x. y. z + x. y. z + x. y. z f 2 = x. y. z + x. y. z + x. y. z + x. y. z ECE 124 Digital Circuits and Systems Page 7

8 Synthesis Example for f 1 and f 2 Circuit for f1: Circuit for f2: ECE 124 Digital Circuits and Systems Page 8

9 Circuit cost for f 1 and f 2 For a circuit, we will define a COST. We will ignore NOT gates at the inputs of a circuit; all other gates count as 1 and every gate input will count as 1. Cost of f1 is 1 AND + 1 OR + 4 GATE INPUTS = 6. Cost of f2 is 3 AND + 1 OR + 11 GATE INPUTS = 15. ECE 124 Digital Circuits and Systems Page 9

10 Synthesis Example for f 2 (Again) The cost of the circuit is a measure of how expensive it will be to implement (in this case, we compute cost that measures the area of the circuit). So, why is Boolean algebra important? Well, if we can get a simpler expression for a function, we can get a cheaper and simpler circuit (this is good!!!) This new circuit also implements f 2 and its cost is 2 AND + 1 OR + 6 GATE INPUTS = 9. ECE 124 Digital Circuits and Systems Page 10

11 Terminology Basic terminology We have seen that Boolean algebra, logic, etc. works with binary variables. The variables have values in {0,1}; E.g., x in {0,1} and x is a binary variable. It is sometimes convenient to refer to the un-complemented (i.e., x) and complemented (i.e.,!x) versions of a variable x. These two terms are called literals. The un-complemented version x is called the positive literal of variable x. The complemented version!x is called the negative literal of variable x. So, for each binary variable, there are two associated literals. ECE 124 Digital Circuits and Systems Page 11

12 Minterms and Maxterms A truth table for an n-input function will have 2 n rows. Minterms: For each row of the truth table, create an AND of the literals according to the following rule: If a variable has value 1 in the row, include its +ve literal. If a variable x has value 0 in the row, include its ve literal. The resulting AND of each literal is called a MINTERM. Maxterms For each row of the truth table, create an OR of the literals according to the fllowing rule: If a variable x has value 0 in the row, include its +ve literal. If a variable x has value 1 in the row, include its ve literal. The resulting OR of each literal is called a MAXTERM. So minterms and maxterms are created opposite of each other. ECE 124 Digital Circuits and Systems Page 12

13 Illustration of Minterms and Maxterms. Minterms are denoted by a lower case m while Maxterms are denoted by an upper case M. When a particular input pattern appears, its associated minterm will be 1 while all other minterms will evaluate to 0. When a particular input pattern appears, its associated maxterm will be 0 while all other maxterms will evaluate to 1. Minterms and maxterms are duals of each other;!mi = Mi and!mi = mi ECE 124 Digital Circuits and Systems Page 13

14 Canonical Sum-of-Products (SOP) Given a truth table, we can ALWAYS write a logic expression for the function by taking the OR of the minterms for which the function is a 1. This representation of a function is a sum of minterms and is called a canonical sum-of-products (SOP) representation of the function. Examples: Shortcut notation: ECE 124 Digital Circuits and Systems Page 14

15 Sum-of-Products Implementations If implemented with gates, a SOP will always have the following form. A plane of NOT gates (inverters to generate all literals), followed by A plane of AND gates (to implement the minterms), followed by A single OR gate (to take the sum ). ECE 124 Digital Circuits and Systems Page 15

16 Canonical Product-of-Sums (POS) Given a truth table, we can ALWAYS write a logic expression for the function by taking the AND of the maxterms for which the function is a 0. This representation of a function is a product of maxterms and is called a canonical product-of-sums (POS) representation of the function. Examples: Shortcut notation: ECE 124 Digital Circuits and Systems Page 16

17 Product-Of-Sums Gate Implementations If implemented with gates, a POS will always have the form: A plane of NOT gates (inverters to generate all literals), followed by A plane of OR gates (to implement the maxterms), followed by A single AND gate (to take the product ). ECE 124 Digital Circuits and Systems Page 17

18 General Comments There are always two canonical representations for a function, the SOP or the POS. Sometimes, one implementation is simpler than the other implementation (in terms of its cost). SOP and POS implementations are often referred to as 2-level logic implementations. This is because we assume NOT gates at the input are free, so we see that there are two levels of gates (AND-OR for SOP and OR-AND for POS) required to implement the function. ECE 124 Digital Circuits and Systems Page 18

19 Conversion between SOP and POS It is always possible to convert between a POS and SOP representation for a functin. Consider f1= (1,4,7) which can also be expressed as f1 = (0,2,3,5,6). f1 = (1,4,7) = m1+m4+m7 =!(!f1) // double inversion is okay =!(m0 + m2 + m3 + m5 + m6) //!f1 is those minterms not in f1 = (!m0)(!m2)(!m3)(!m5)(!m6) = (M0)(M2)(M3)(M5)(M6) = (0,2,3,5,6). Note: Quickly, we can from minterms (maxterms) to maxterms (minterms) by changing ( ) to ( ) and list those indices of terms missing from the original list. ECE 124 Digital Circuits and Systems Page 19

20 Standard Sum-Of-Products (1) A function described using a canonical SOP (minterms) is by no means minimal. It might require more gates/literals than required. Let us call any AND of literals a product term. We can then express logic functions in Standard Sum-Of-Products form where, instead of minterms, the AND terms are simply product terms. We can start with a canonical SOP and use Boolean algebra to simply the expression into something simpler. ECE 124 Digital Circuits and Systems Page 20

21 Standard Sum-Of-Products (2) Let s consider one of our previous functions in Canonical SOP. If we used Boolean algebra to simplify, we would find that f 2 can also be written as a Standard SOP using a sum of product terms: ECE 124 Digital Circuits and Systems Page 21

22 Standard Product-Of-Sums (1) A function described using a canonical POS (maxterms) is by no means minimal. It might require more gates/literals than required. Let us call any OR of literals a sum term. We can then express logic functions in Standard Product-Of-Sums form where, instead of maxterms, the OR terms are simply sum terms. We can start with a canonical POS and use Boolean algebra to simply the expression into something simpler. ECE 124 Digital Circuits and Systems Page 22

23 Example of Standard Product-Of- Sums Forms Let s consider one of our previous functions in Canonical POS. If we used Boolean algebra to simplify, we would find that f 1 can also be written as a Standard POS using a product-of-sum terms: ECE 124 Digital Circuits and Systems Page 23

24 Other Logic Gates Although we can always implement any function we want using AND/OR/NOT, there are other types of logic gates that prove useful. ECE 124 Digital Circuits and Systems Page 24

25 NAND and NOR gates (2-inputs) NAND gate performs a NOT-AND operation. NOR gate performs a NOT-OR operation. NAND/NOR gates can be extended to multiple inputs, but the NAND/NOR gates are not associative (explained later). We should always think of NAND as NOT-AND and NOR as NOT-OR. ECE 124 Digital Circuits and Systems Page 25

26 NAND and NOR gates (n-inputs) Think of multiple input NAND/NOR gates in terms of the operations they perform; i.e., NOT-AND (for a NAND) and NOT-OR (for a NOR). Example: 3-input versions: ECE 124 Digital Circuits and Systems Page 26

27 XOR and NXOR gates (2-inputs) XOR gate (with 2-inputs performs a difference operation ): NXOR gate (with 2-inputs performs a equivalence operation ): XOR/NXOR gates are incredibly useful for arithmetic operations like addition/subtraction/multiplication. These gates can also be extended to multiple inputs, but we need to be clear on their definitions with multiple inputs. ECE 124 Digital Circuits and Systems Page 27

28 XOR gates with multiple inputs. A XOR gate with > 2 inputs performs the odd operation ; the output is a 1 whenever an odd number of inputs are 1. Example: 3-input versions: XOR gates are associative (explained later). ECE 124 Digital Circuits and Systems Page 28

29 NXOR gates with multiple inputs. A NXOR gate with > 2 inputs performs the odd function ; the output is a 1 whenver an odd number of inputs are 1. Example: 3-input versions: NXOR gates are non-associative (explained later). ECE 124 Digital Circuits and Systems Page 29

30 Buffer (1-input) Does nothing logically; Used in implementation to boost a signal s strength. ECE 124 Digital Circuits and Systems Page 30

31 Associative and Non-Associative Gates AND/OR gates are associative gates. This means that we can collapse many smaller AND (OR) gates into a single AND (OR) gate with multiple inputs. Example: XOR gates are also associative. Not all types of logic gates are associative. ECE 124 Digital Circuits and Systems Page 31

32 Non-Associative Gates NAND/NOR and NXOR gates are not associative. ECE 124 Digital Circuits and Systems Page 32

33 Example 1: Proof of a Theorem Prove Absorption Theorem 6(a) Can prove using postulates, or using truth tables. Solution: ECE 124 Digital Circuits and Systems Page 33

34 Example 2: Expression Simplification Simply the following logic expression: Solution Note: Could have also asked Show that (i.e., LHS = RHS): ECE 124 Digital Circuits and Systems Page 34