CMSC 100 CMSC 100 From the Bottom Up: It's All Just Bits Professor Marie desJardins Thursday, September 10, 2009

Thu 9/10/09 CMSC Just Bits 2 Just Bits  Inside the computer, all information is stored as bits  A “bit” is a single unit of information  Each “bit” is set to either zero or one  How do we get complex systems like Google, Matlab, and our cell phone apps?

Thu 9/10/09 CMSC Just Bits 3 Storing Bits  How are these “bits” stored in the computer?  A bit is just an electrical signal or voltage (by convention: “low voltage” = 0; “high voltage” = 1)  A circuit called a “flip-flop” can store a single bit  A flip-flop can be “set” (using an electrical signal) to either 0 or 1  The flip-flop will hold that value until it receives a new signal telling it to change  Bits can be operated on using gates (which “compute” a function of two or more bits)  Later we’ll talk about how a flip-flop can be built of out gates

Thu 9/10/09 CMSC Just Bits 4 Manipulating Bits  What does a bit value “mean”?  0 = FALSE[off, no]  1 = TRUE[on, yes]  Just as in regular algebra, we can think of “variables” that represent a single bit  If X is a Boolean variable, then the value of X is either: 0[FALSE, off, no] or 1[TRUE, on, yes]  In algebra, we have operations that we can perform on numbers:  Unary operations: Negate, square, square-root, …  Binary operations: Add, subtract, multiply, divide, …  What are the operations you can imagine performing on bits?  Unary: ??  Binary: ??  Thought problem: How many unary operations are there? How many binary operations?

Thu 9/10/09 CMSC Just Bits 5 Remember Function Tables? XX2X …… XYX + Y ………

Thu 9/10/09 CMSC Just Bits 6  Negation (NOT):   Conjunction (AND):  Boolean Operations: “Truth Tables” X XX XY X  Y XY X  Y  Disjunction (OR):  X XX FT TF Or equivalently… XY X  Y FFF FTF TFF TTT XY X  Y FFF FTT TFT TTT

Thu 9/10/09 CMSC Just Bits 7 Logic Gates  In hardware, Boolean operations are implemented using circuits called logic gates XOR: Exclusive or (one input is TRUE, but not both)

Thu 9/10/09 CMSC Just Bits 8 Boolean Expressions  What would a truth table look like for the expression: (A  B)  (  B  C) ABC A  B BB  B  C(A  B)  (  B  C)

Thu 9/10/09 CMSC Just Bits 9 Boolean Expressions  What would a truth table look like for the expression: (A  B)  (  B  C) ABC A  B BB  B  C(A  B)  (  B  C)

Thu 9/10/09 CMSC Just Bits 10 Boolean Expressions  What would a truth table look like for the expression: (A  B)  (  B  C) ABC A  B BB  B  C(A  B)  (  B  C)

Thu 9/10/09 CMSC Just Bits 11 (Boolean) Algebraic Laws  DeMorgan’s Theorem  Analogous to the distributive law in regular algebra  (A  B) =  A   B  (A  B) =  A   B AB A  B  (A  B) AA BB  A   B FFFTTTT FTFTTFT TFFTFTT TTTFFFF

Thu 9/10/09 CMSC Just Bits 12 Logic Circuits  We can implement any logical expression just by assembling the associated logical gates in the right order  What would a logic circuit look like for the expression: (A  B)  (  B  C)

Thu 9/10/09 CMSC Just Bits 13 Implementing a Flip-Flop

Thu 9/10/09 CMSC Just Bits 14 Setting the Output to 1

Thu 9/10/09 CMSC Just Bits 15 Setting the Output to 1 (cont.)

Thu 9/10/09 CMSC Just Bits 16 Setting the Output to 1 (cont.)

Thu 9/10/09 CMSC Just Bits 17 What Else Can We Do? What happens if we put a zero on both inputs? …a one on the upper input and a zero on the lower input? …a zero on the upper input and a one on the lower input? …a one on both inputs?

Thu 9/10/09 CMSC Just Bits 18 Memory & Abstraction  There are other circuits that will also implement a flip-flop  These are sometimes called SRM (Static Random Access Memory)  …meaning that once the circuit is “set” to 1 or 0, it will stay that way until a new signal is used to re-set it  DRAM (Dynamic Random Access Memory):  Use a capacitor to store the charge (has to be refreshed periodically)  BUT…  Abstraction tells us that (for most purposes) it really doesn’t matter how we implement memory -- we just know that we can store (and retrieve) “a bit” at a time

Thu 9/10/09 CMSC Just Bits 19 Storing Information  One bit can’t tell you much… (just 2 possible values)  Usually we group 8 bits together into one “byte”  How many possible values (combinations) are there for one byte?  A byte can just be thought of as an 8-digit binary (base 2) number  Michael Littman's octupus counting video Michael Littman's octupus counting video  Low-order or least significant bit == ones place  Next bit would be “10s place” in base what about base 2?  High-order bit or most significant bite in a byte == ?? place

Thu 9/10/09 CMSC Just Bits 20 Orders of Magnitude  One 0/1 (“no/yes”) “bit” is the basic unit of memory  Eight (2 3 ) bits = one byte  1,024 (2 10 ) bytes = one kilobyte (1K) *  1,024K = 1,048,576 (2 20 bytes) = one megabyte (1M)  1,024K (2 30 bytes) = one gigabyte (1G)  1,024 (2 40 bytes) = one terabyte (1T)  1,024 (2 50 bytes) = one petabyte (1P)  bytes = one yottabyte (1Y?)

Thu 9/10/09 CMSC Just Bits 21 Scaling Up Memory  Computer chip:  Many (millions) of circuits  Etched onto a silicon wafer using VLSI (Very Large-Scale Integration) technology  Lots of flip-flops or DRAM devices == memory chip  Each byte has an address (and we use binary numbers to represent those addresses…)  An address is represented using a word, which is typically either; 2 bytes (16 bits) -- earliest PCs  Only 64K combinations  memory is limited to 64K (65,535) bytes! 4 bytes (32 bits) -- first Pentium chips  This brings us up to 4G (4,294,967,295) bytes of memory! 8 bytes (64 bits) -- modern Pentium chips  Up to 16.8 million terabytes (that’s 18,446,744,073,709,551,615 bytes!) 

Thu 9/10/09 CMSC Just Bits 22 Hexadecimal  It would be very inconvenient to write out a 64-bit address in binary:  Instead, we group each set of 4 bits together into a hexadecimal (base 16) digit:  The digits are 0, 1, 2, …, 9, A (10), B (11), …, E (14), F (15) D 6 F C D E 0 E 0  …which we write, by convention, with a “0x” preceding the number to indicate it’s heXadecimal:  0x29D6F89611CDE0E0

Thu 9/10/09 CMSC Just Bits 23 Other Memory Concepts (read the book!!)  Mass storage: hard disks, CDs, USB/flash drives…  Stores information without a constant supply of electricity  Larger than RAM   Slower than RAM  Often removable   Physically often more fragile than RAM  CDs, hard drives, etc. actually spin and have tracks divided into sectors, read by a read/write head  Seek time: Time to move head to the proper track  Latency: Time to wait for the disk to rotate into place  Access time: Seek + latency  Transfer rate: How many bits/second can be read/written once you’ve found the right spot  Flash memory: high capacity, no moving parts, but less reliable for long-term storage

Thu 9/10/09 CMSC Just Bits 24 Representing Information  Positive integers: Just use the binary number system  Negative integers, letters, images, … not so easy!  There are many different ways to represent information  Some are more efficient than others  … but once we’ve solved the representation problem, we can use that information without considering how it’s represented… via

Thu 9/10/09 CMSC Just Bits 25 Representing Characters  ASCII representation: one byte [actually 7 bits…] == one letter == an integer from  No specific reason for this assignment of letters to integers!  UNICODE is a popular 16-bit representation that supports accented characters like é [Chart borrowed from ha.ckers.org]

Thu 9/10/09 CMSC Just Bits 26 Representing Integers  Simplest idea (“ones’ complement”):  Use one bit for a “sign bit”: 1 means negative, 0 means positive  The other bits are “complemented” (flipped) in a negative number  So, for example, +23 (in a 16-bit word) is represented as: and -23 is represented as:  But there are two different ways to say “zero” (0000… and 1111…)  It’s tricky to do simple arithmetic operations like addition in the ones’ complement notation

Thu 9/10/09 CMSC Just Bits 27 Two’s Complement  Two’s complement is a clever representation that allows binary addition to be performed in an elegant way

Thu 9/10/09 CMSC Just Bits 28 Two’s Complement cont.

Thu 9/10/09 CMSC Just Bits 29 Floating Point Numbers  Non-integers are a problem…  Remember that any rational number can be represented as a fraction  …but we probably don’t want to do this, since (a) we’d need to use two words for each number (i.e., the numerator and the denominator) (b) fractions are hard to manipulate (add, subtract, etc.)  Irrational numbers can’t be written down at all, of course  Notice that any representation we choose will by definition have limited precision, since we can only represent 2 32 different values in a 32-bit word  1/3 isn’t exactly 1/3 (let’s try it on a calculator!)  In general, we also lose precision (introduce errors) when we operate on floating point numbers  You don’t need to know the details of how “floating point” numbers are represented

Thu 9/10/09 CMSC Just Bits 30 Summary: Main Ideas  It’s all just bits  Abstraction  Boolean algebra  TRUE/FALSE  Truth tables  Logic gates  Representing numbers  Hexadecimal representation  Ones’ (and two’s) complement  Floating point numbers (main issues)  ASCII representation (main idea)  Types and properties of RAM and mass storage

Thu 9/10/09 CMSC Just Bits 31 ACTIVITY (if time)  Design a one-bit adder (i.e., a logic circuit that adds two 1-bit numbers together)  X + Y  Z 2 Z 1  Z 2 is needed since the result may be two binary digits long  First let’s figure out the Boolean expression for each output…  Then we’ll draw the logic circuit