CMSC 100 From the Bottom Up: It's All Just Bits Professor Marie desJardins mariedj@cs.umbc.edu Thursday, September 10, 2009

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? Abstraction! Thu 9/10/09 CMSC 100 -- Just Bits

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 100 -- Just Bits

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 100 -- Just Bits

Remember Function Tables? X X2 1 2 4 3 9 16 … X Y X + Y 1 2 3 … Thu 9/10/09 CMSC 100 -- Just Bits

Boolean Operations: “Truth Tables” Negation (NOT):  Conjunction (AND):  Disjunction (OR):  X X F T X X 1 Or equivalently… X Y X  Y 1 X Y X  Y F T X Y X  Y 1 X Y X  Y F T Thu 9/10/09 CMSC 100 -- Just Bits

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 100 -- Just Bits

Boolean Expressions What would a truth table look like for the expression: (A  B)  (B  C) A B C A  B B B  C (A  B)  (B  C) Thu 9/10/09 CMSC 100 -- Just Bits

Boolean Expressions What would a truth table look like for the expression: (A  B)  (B  C) A B C A  B B B  C (A  B)  (B  C) 1 Thu 9/10/09 CMSC 100 -- Just Bits

Boolean Expressions What would a truth table look like for the expression: (A  B)  (B  C) A B C A  B B B  C (A  B)  (B  C) 1 Thu 9/10/09 CMSC 100 -- Just Bits

(Boolean) Algebraic Laws EC DeMorgan’s Theorem Analogous to the distributive law in regular algebra (A  B) = A  B (A  B) = A  B A B A B (A B) A B A  B F T Thu 9/10/09 CMSC 100 -- Just Bits

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 100 -- Just Bits

Implementing a Flip-Flop Thu 9/10/09 CMSC 100 -- Just Bits

Setting the Output to 1 Thu 9/10/09 CMSC 100 -- Just Bits

Setting the Output to 1 (cont.) Thu 9/10/09 CMSC 100 -- Just Bits

Setting the Output to 1 (cont.) Thu 9/10/09 CMSC 100 -- Just Bits

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 100 -- Just Bits

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 100 -- Just Bits

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 Low-order or least significant bit == ones place Next bit would be “10s place” in base 10 -- what about base 2? High-order bit or most significant bite in a byte == ?? place Thu 9/10/09 CMSC 100 -- Just Bits

Orders of Magnitude One 0/1 (“no/yes”) “bit” is the basic unit of memory Eight (23) bits = one byte 1,024 (210) bytes = one kilobyte (1K)* 1,024K = 1,048,576 (220 bytes) = one megabyte (1M) 1,024K (230 bytes) = one gigabyte (1G) 1,024 (240 bytes) = one terabyte (1T) 1,024 (250 bytes) = one petabyte (1P) ... 280 bytes = one yottabyte (1Y?) Thu 9/10/09 CMSC 100 -- Just Bits

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!) http://cnx.org/content/m13082/latest/ Thu 9/10/09 CMSC 100 -- Just Bits

Hexadecimal It would be very inconvenient to write out a 64-bit address in binary: 0010100111010110111110001001011000010001110011011110000011100000 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) 0010 1001 1101 0110 1111 1000 1001 0110 0001 0001 1100 1101 1110 0000 1110 0000 2 9 D 6 F 8 9 6 1 1 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 100 -- Just Bits

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 100 -- Just Bits

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 Abstraction! Thu 9/10/09 CMSC 100 -- Just Bits

Representing Characters ASCII representation: one byte [actually 7 bits…] == one letter == an integer from 0-128 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 100 -- Just Bits

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: 0000000000010111 and -23 is represented as: 1111111111101000 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 100 -- Just Bits

Two’s Complement Two’s complement is a clever representation that allows binary addition to be performed in an elegant way EC Thu 9/10/09 CMSC 100 -- Just Bits

Two’s Complement cont. EC Thu 9/10/09 CMSC 100 -- Just Bits

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 232 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 100 -- Just Bits

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 100 -- Just Bits

ACTIVITY (if time) Design a one-bit adder (i.e., a logic circuit that adds two 1-bit numbers together) X + Y  Z2 Z1 Z2 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 Thu 9/10/09 CMSC 100 -- Just Bits