CS1800: Hex & Logic. Professor Kevin Gold
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1 CS1800: Hex & Logic Professor Kevin Gold
2 Reviewing Last Time: Binary Last time, we saw that arbitrary numbers can be represented in binary. Each place in a binary number stands for a different power of 2. If you re familiar with the powers of 2, a reasonable way to convert from decimal to binary is to subtract powers of 2 until you get is too big, so =65 that s close to 64, so = 1, subtract 1 and done
3 Powers of 2 Will Become Familiar Because of Gadgets iphone XR memory possibilities 512 GB Hard Drive (sometimes rounded to 500) This often happens because memory addresses are binary numbers - so the maximum n-bit address is 2 n -1
4 Powers of 2 You Should Become Familiar With 2 8 = 256. A chunk of 8 bits is a byte, and 256 is the number of values a byte can take on (0-255). This makes 255 come up as a max value for things, from IP address values to Zelda rupees 2 10 = Kilobytes don t come up much anymore, but this is still a handy way of estimating how big binary values are = 2 10 * 2 10 * 2 10 = about 1,000,000,000 Megabyte = 2 20 bytes, Gigabyte = 2 30 bytes, etc. 2 4 = 16. This is the number of values possible in half a byte, or a nibble (0-15), which will be useful for our study of hex.
5 Hexadecimal Hexadecimal is base 16, and also a kind of shorthand for binary. As a base 16 number system, each place has 16 possible digits: A B C D E F Places are worth 1, 16, 16 2 = 256, 16 3 = 4096 so A001 is (10* *1) The number may be preceded by 0x or have subscript 16 to signal that it is hex and not binary or decimal: 0x20 = = 32. The evaluation of a particular hex string as a number is therefore a sum of powers of 16. 0xA9 = (16*10 + 9*1) = 169 0x103 = ( ) = ( ) = 259
6 The Trick : One Hex Digit is a Nibble (4 bits) The convenient thing for CS is that we can very easily convert between binary and hex. Every 4 bits can be directly read as a hex digit, and this will be the correct hexadecimal number. For example, can be read as 0101 = 5 and 0010 = 2 so the hex should be 0x52. (Check: = = 82. 5*16+2 = 82.) Or, can be read as = (15,12) = 0xFC. (Check: = 252; 15*16+12=252.)
7 The Trick : One Hex Digit is A Nibble (4 bits) The binary values and the hex digits 0 F serve the same purpose In both cases, they can represent a value between 0 and 15 The rest of the number is worth 16*whatever other value is written down Another way of putting it is, we could just see our binary nibbles as a wordy way of doing base 16 x16 x x16+2= F C 15x16+12=252
8 Some Examples of Hex Values in the Wild IPv6 IP addresses (network addresses): 128 bits in hex 2001:db8:85a3:8d3:1319:8a2e:370:7348 Windows blue screen of death errors Programming languages often refer to memory addresses using hex A Hex Editor lets you change files on a bit-by-bit level
9 Other Bases Decimal, hex, and binary all have features that make them useful to us but other bases work too We add a subscript to indicate the base in base 3: 1* *3 + 1*1 = 10 10, so Interpreting AAA 11 : 10* * *1 = = 1330 Value of 2222 b : 2b 3 + 2b 2 + 2b + 2 The same techniques for converting binary or hex work for other bases We mention this mostly so that you understand the bases we do care about better
10 CS Logic is Real Logic When someone says that s illogical or Sherlock Holmes makes a logical deduction, that is the same kind of logic we are talking about We ll see over the next two lectures how the logic in circuits and programming can also be used in mathematical proofs While different flavors of logic exist, depending on what you re trying to do, they generally share the rules we ll describe Cumberbatch as Sherlock Holmes
11 When Do We Use Logic in CS? Most commonly in writing conditional statements, like If A and B but not C, execute this code If we can simplify conditional statements like that, sometimes there is a modest gain in speed In showing computational problems are hard Logic is often a component to proofs that show problems are unlikely to have quick algorithms In designing circuits The course textbook shows how we can go all the way from simple logic circuits to a basic CPU!
12 Primary Logical Operators and Their Symbols AND: ^ True iff both arguments true It is raining AND I don t have an umbrella OR: v True iff at least one argument is true ( inclusive OR, our default assumption) It is raining OR the sprinklers are on NOT: True iff argument is false It is NOT raining
13 We Will Talk About Implication Later A B means A implies B, or If A, then B. It s similar to the kind of logical deduction you might be familiar with outside math. But if it wasn t Colonel Mustard, then it must be We ll pick it up later and see how we can express using our other operators
14 Relation to Sets A U B = {x : (x A) v (x B)} (or) A B = {x : (x A) ^ (x B)} (and) A = {x : (x A)} (not) Union Intersection Complement A B A B A Universe A U B A B A
15 Truth Tables Truth tables are a way of understanding particular logical formulas. We list all possible combinations of values of true or false for the logical formula variable, and list the outputs in the rightmost column. There will generally be 2 N possible combinations for N variables. A B A ^ B A B A v B F F F F F F A A F T F F T T F T T F F T F T T F T T T T T T
16 Truth Tables And Scratch Columns It can be helpful to add extra columns to a truth table to help figure out the final values. a b c (a^b) (a^ b) (c^ a) (a ^ b) v (a ^ b) v (c ^ a) F F F F F T F T F F T T T F F T F T T T F T T T
17 Truth Tables And Scratch Columns It can be helpful to add extra columns to a truth table to help figure out the final values. a b c (a^b) (a^ b) (c^ a) (a ^ b) v (a ^ b) v (c ^ a) F F F F F F T F F T F F F T T F T F F F T F T F T T F T T T T T
18 Truth Tables And Scratch Columns It can be helpful to add extra columns to a truth table to help figure out the final values. a b c (a^b) (a^ b) (c^ a) (a ^ b) v (a ^ b) v (c ^ a) F F F F F F F T F F F T F F F F T T F F T F F F T T F T F T T T F T F T T T T F
19 Truth Tables And Scratch Columns It can be helpful to add extra columns to a truth table to help figure out the final values. a b c (a^b) (a^ b) (c^ a) (a ^ b) v (a ^ b) v (c ^ a) F F F F F F F F T F F T F T F F F F F T T F F T T F F F T F T F T F T F T T F T F F T T T T F F
20 Truth Tables And Scratch Columns It can be helpful to add extra columns to a truth table to help figure out the final values. a b c (a^b) (a^ b) (c^ a) (a ^ b) v (a ^ b) v (c ^ a) F F F F F F F F F T F F T T F T F F F F F F T T F F T T T F F F T F T T F T F T F T T T F T F F T T T T T F F T
21 Truth Tables and Logical Equivalence If two logical formulas have the same truth table outcomes for all inputs, they are equivalent. a b c (a^b) v (a^ b) v (c^ a) F F F F F F T T F T F F F T T T T F F T T F T T T T F T T T T T a b c (a v c) F F F F F F T T F T F F F T T T T F F T T F T T T T F T T T T T
22 Creating a Formula From a Desired Truth Table Suppose we have desired outputs, and just want a logical formula that behaves exactly that way a b c??? F F F F F F T T F T F F F T T T T F F T T F T T T T F T T T T T
23 Creating a Formula From a Desired Truth Table Suppose we have desired outputs, and just want a logical formula that behaves exactly that way a b c??? F F F F F F T T F T F F F T T T T F F T T F T T T T F T T T T T A brute force approach is to explicitly list all the true possibilities, and join them with OR. If it s one of those possibilities, it s true. If not, it s false. ( a ^ b ^ c) v ( a ^ b ^ c) v (a ^ b ^ c) v (a ^ b ^ c) v (a ^ b ^ c) v (a ^ b ^ c)
24 Disjunctive Normal Form (DNF) A clause is a part of a logical formula in parentheses. A formula in disjunctive normal form consists of clauses where the clauses contain only AND, and the clauses are only joined by OR. The method we just suggested produces disjunctive normal form. ( a ^ b ^ c) v ( a ^ b ^ c) v (a ^ b ^ c) v (a ^ b ^ c) v (a ^ b ^ c) v (a ^ b ^ c) The formula we were describing earlier is, too, despite not mentioning all its variables in all clauses: (a^b) v (a^ b) v (c^ a) disjunctive - having to do with OR (the formula is one big OR) normal form - as in, standardized
25 Simplifying Logical Formulas Simplifying formulas can make them easier to read and quicker to compute. The rules for simplifying logical formulas are mostly commonsense. Example: A ^ A simplifies to F. Something can t be true and false at the same time. There can be more than one path to simplification - different rules applied in different orders. It is sometimes ambiguous what counts as simplified ( as short as possible is generally what s desired, but sometimes multiple formulas are equally short)
26 Memorizing names not necessary except De Morgan s equivalent to
27 A Logical Simplification (a ^ b) v (a ^ b) v (c ^ a) (a ^ (b v b)) v (c ^ a) (distributive) (a ^ T) v (c ^ a) (complement) a v (c ^ a) (identity) (a v c) ^ (a v a) (distributive) (a v c) ^ T (complement) a v c (identity)
28 Another Logical Simplification ((a ^ b ^ c) v (a ^ b ^ c)) v (a v b) ((a ^ b) ^(c v c)) v (a v b) (distributive) (a ^ b ^ T) v (a v b) (complement) (a ^ b) v (a v b) (identity) (a ^ b) v ( a ^ b) (de Morgan s) (a ^ b) v ( a ^ b) (double negation) (a v a) ^ b (distributive) T ^ b (complement) b (identity)
29 De Morgan s Observations Notice that because we can always use De Morgan s law to flip a sign, we only need either AND or OR. We could always write one in terms of the other. a ^ b ^ c <=> ( a v b v c) a v b v c <=> ( a ^ b ^ c)
30 De Morgan s Observations All of the logical laws have their equivalents when talking about sets. We can have a pictorial version of De Morgan s Laws. A B = A U B A B U A B = A B
31 Using De Morgan s to Create Formulas from False TT Entries Before, we saw how it was possible to create formulas directly from truth tables by using just the true entries. We can read off only the false entries, too. a b c??? F F F F F F T T F T F F F T T T T F F T T F T T T T F T T T T T As long as a formula is none of the false entries, it must be true. So we can say, Not this AND not that. ( a ^ b ^ c) ^ ( a ^ b ^ c) Applying De Morgan s law lets us distribute the s: (a v b v c) ^ (a v b v c)
32 Conjunctive Normal Form (CNF) Conjunctive Normal Form is a bunch of clauses joined by AND, where each clause contains only OR. The process described on the previous slide is a way to get CNF for any desired truth table. (Thus, any formula can be converted to CNF.) CNF comes up in theoretical computer science in proofs that problems are hard to compute. (a v b v c) ^ (a v b v c) ^ conjunctive - having to do with AND (the formula is one big AND)
33 Logic Gates Circuit elements called logic gates make up the important hardware in your computer, including the central processing unit (CPU) In principle, it s possible to create any pattern of outputs from any pattern of inputs using just AND, OR, and NOT. AND gate OR gate NOT gate
34 Implementation of Logic Gates (Very Optional) Logic gates rely on semiconductors to act as switches that are either normally open or normally shut If a switch is closed, then current can pass The line that isn t a switch is hidden in the logic diagram AND needs both switches OR can have either shut
35 An Example of a Logical Circuit a b c ((a ^ b) ^ (a v c)) Notice the difference between a connection and a hop
36 Can We Simplify This? ((a ^ b) ^ (a v c)) Distributive: ((a ^ b) ^ a) v ((a ^ b) ^ c)) Commutative, associative: ((a ^ a ^ b) v ((a ^ b) ^ c)) Idempotence: ((a ^ b) v ((a ^ b) ^ c)) Absorption: (a ^ b) a b c (We can actually do this with 1 gate on the next slide )
37 Other Logic Gates, in Decreasing Importance XOR: Exclusive OR NAND: Not And NOR: Not Or XNOR: Not XOR (responds iff inputs same!)
38 Bitwise AND a3 b3 a2 b2 a1 b1 a0 b0 out3 out2 out1 out & Recall that this performs intersection on sets represented with bits
39 Doing Other Computations With Logic The CPU does lots of things. How could it, for instance, add numbers with logic gates? Many things are represented in binary in computers, and T or F can be mapped to 1 and 0. Consider a piece of the machinery that handles just two input bits in the addition. It needs to take two bits as input and have the behavior at right for the sum and the carry
40 The Half-Adder as Example of Logic Circuits Doing Computation Now the above circuit is doing simple addition using just logical circuits How could we add entire multibit numbers? We just need a little more complexity
41 The Full-Adder and Multibit Computations A1 B1 A0 B0 Cout Full adder Cout Full adder S1 S0 The full adder essentially adds A and B, then adds the carry-in bit from the previous place. Stringing these together gives something that can add multibit numbers.
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