⬅ Exam 1 Review Day Notes

Please read this page about taking my exams!

Exam format

• When/where
  • During class, here, like normal
  • 75 minutes
  • it is not going to be “too long to finish”
  • no calculator
• Closed-note
  • You may not have any notes, cheat sheets etc. to take the exam
  • The open-note thing was just for when we were remote
• Length
  • 3 sheets of paper, double-sided
  • there are A Number of Questions and I cannot tell you how many because it is not a useful thing to tell you because they are all different kinds and sizes.
    • But I will say that I tend to give many, small questions instead of a few huge ones.
• Topic point distribution
  • More credit for earlier topics (e.g. numerical representation, memory addresses, arrays, MIPS programming)
  • Less credit for more recent ones (e.g. overflow)
  • More credit for things I expect you to know because of your experience (labs, project)
  • VERY ROUGHLY:
    • ~15% “pick n” and fill-in-the-blank
    • ~20% short answer (just writing stuff)
    • ~20% math (bases, ranges, +/-)
    • ~20% MIPS ASM (tracing, interpreting, filling in blanks)
    • ~15% understanding memory
    • ~10% bitfields/bitsets
• Kinds of questions
  • No multiple choice
  • A few “pick n“ (but not many)
  • Some fill in the blanks
    • mostly for vocabulary
    • or things that I want you to be able to recognize, even if you don’t know the details
  • Application questions about numbers and arithmetic (i.e. math problems, basically)
    • Base conversion
    • Interpreting patterns of bits in different ways (signed, unsigned, bitfields, floats etc)
    • Unsigned and signed (2’s complement) addition
  • Several short answer questions
    • again, read that page above about answering short answer questions!!
  • No writing code from scratch, but:
    • tracing (reading code and saying what it does)
    • debugging (spot the mistake)
    • interpreting asm as HLL code (identifying common asm patterns)
    • fill in the blanks (e.g. picking right registers, right branch instructions)
    • identifying loads and stores in HLL code

Things people asked about in the reviews

This is a list of what people asked about. The exam may have other topics not listed, and some of these topics may not appear on the exam.

• CISC vs RISC
  • CISC: Complex Instruction Set Computer
    • made for humans to write programs directly in assembly
  • RISC: Reduced Instruction Set Computer
    • made for compilers to produce assembly/machine code from high-level languages
  • The differences are really in the name:
    • CISC has complex, flexible, multi-step instructions that are great for humans (do more stuff with fewer instructions!) but terrible for performance
      • x86 is really the only CISC still in widespread use
    • RISC has reduced, simple, single-step instructions that are great for compilers (so easy to write algorithms to write RISC code!) but more awkward for humans to write
      • MIPS and Berkeley RISC were the first mainstream RISC architectures (and where the name RISC came from)
      • most architectures designed after MIPS works like MIPS (e.g. ARM)
• Conversion from decimal to binary
  • I presented one way on the slides, the “long division” method:
    • You have to know the binary place values
    • From MSB to LSB (left to right):
      • If the place value fits into the remainder, put a 1 and subtract it off the remainder
      • Otherwise put a 0
  • There’s another method that involves repeatedly dividing by 2 until you get a quotient of 0, and you write every remainder even if it’s a 0, and the binary representation is the remainders read from top to bottom.
    • Try doing both methods to see what I mean, if you’re curious.
• Conversion between hex and binary
  • 4 bits = 1 hex digit (nybble)
  • The table is simple - count up in binary from 0000 to 1111, and count up in hex from 0 to F next to it.
  • When going from binary to hexadecimal, group the bits into 4 starting from the right (LSB)
    • add 0s to the left side as needed to make a group of 4 bits
    • then each group of 4 bits is 1 hex digit
• Unsigned integers
  • There are no negatives. It’s in the name: unsigned = NO SIGN.
  • To convert to decimal, add up the place values for each 1 bit.
  • You, the programmer, decide when an integer is unsigned. It’s then up to you to use the appropriate unsigned versions of things like addu, bltu, lbu/lhu, “print unsigned integer” syscall, etc.
• Sign-magnitude
  • Is NOT used for integers, it’s used for floats
  • Is also how we write numbers on paper. +123 and -123: same digits, different sign.
  • The MSB is the sign, 0 for positive, 1 for negative, and is totally separate from the rest of the bits
  • Downsides: two “versions” of 0 (+0 and -0); arithmetic is more complicated (special cases)
  • To negate: just flip the sign bit. The rest of the digits are unchanged.
• 2’s complement integers
  • The one and only system used to represent signed integers on computers today
  • It works by making the MSB the negative version of its place value
    • The MSB also represents the sign - 0 for positive, 1 for negative
  • This representation is great because it makes arithmetic super simple, no special cases
    • You can add any two numbers of any signs and it will Just Work (unless there’s overflow lol)
  • Downside: there is one more negative number than positives, and it is A Bit Weird (it has no positive counterpart, so if you negate it, you get the same value back out).
  • To convert to decimal, you still just add the place values up. e.g. 1001 is -8 + 1 = -7.
  • To negate: -x == ~x + 1, or, “flip the bits, then add 1.”
    • The negative of a number is also called its “2’s complement.”
• Addition and subtraction
  • Binary addition works just like in base 10, but you carry at 2 instead of 10.
  • The same addition algorithm is used for both unsigned and signed integers.
  • Remember, when adding 2’s complement numbers, nothing special happens. You just add the bits and you will get the correct value/sign at the end.
  • Subtraction is defined in terms of addition:
    • x - y == x + (-y)… and because of how 2’s complement works…
    • x + (-y) == x + (~y + 1)
    • amazingly, this works for signed and unsigned subtraction! the 2’s complement of x (~x + 1) “behaves like” its negative, even in a number system that has no negative numbers. remember: two ways around the number circle.
• Extension
  • Going from a smaller number of bits to a bigger number of bits while preserving the value
    • e.g. the number 5 can be represented as 0101 binary, or as 0000 0101 binary - same value, but the second one has more bits
  • There are two flavors of extension:
    • Zero-extension is for unsigned numbers and puts 0 bits on the left side of the number
    • Sign-extension is for signed numbers and puts copies of the sign bit (MSB) on the left side of the number
• Truncation
  • Going from a larger number of bits to a smaller number
    • You can also look at it as “erasing bits on the left side of the number”
  • There is only one kind of truncation, doesn’t matter if it’s signed or unsigned.
  • However, truncating too far can change the value
    • e.g. If you have 0001 0010 (decimal 18), and truncate it to 5 bits, you get 1 0010 (still decimal 18)… but if you keep going and truncate to 4 bits, you get 0010 (decimal 2)!
    • This is actually performing modulo: truncating to n bits gives you the value modulo 2ⁿ. If the number is less than that, it’ll be preserved; if it’s bigger, it’ll be changed.
• Control flow
  • Conditional branches go to the label if their condition is true (satisfied)
    • e.g. beq t0, 10, _label says “if t0 == 10, then go to _label”
  • So there is a mismatch between the way we write conditions in ifs in Java vs. how they work in asm
    • When writing ifs in asm, you usually have to invert the condition
    • Because in asm, the condition is really testing “when do we skip the contents of the if”
  • However, you don’t always invert the condition. E.g. do-while, “cheesy” for loops
    • These test the condition at the end of the loop, so we do want to go backwards when the condition is true
• MEMORY
  • Alignment
    • The address of an n-byte value must be a multiple of n.
      • e.g. words are 4 bytes. so, their addresses are multiples of 4 (0x00, 0x04, 0x08, 0x0C, 0x10, 0x14, 0x18, 0x1C,...)
    • There are underlying hardware design reasons for this, but some architectures (like MIPS) will crash your program if you don’t respect this rule.
  • Zero/sign extension
    • when you load a value < 32 bits (byte, half) into a 32-bit register, have to extend it
      • want to preserve the same value, just represent it with more bits
    • lb/lh does sign extension (copies sign bit (0 OR 1) to left)
    • lbu/lhu does zero extension (fills extra bits with 0s)
    • No extension happens with lw because you’re loading a 32-bit value into a 32-bit register - same size
  • Truncation
    • When you store into a half/byte variable, only the least significant bits (rightmost bits) of the register are stored
    • The rest are truncated (cut off)
    • sb stores the 8 least significant bits of the register in memory and leaves 24 behind
    • sh stores the 16 least significant bits of the register in memory and leaves 16 behind
  • Endianness, discussed below
  • Does the CPU crash if you lw a byte or lb a word?
    • NO! the only thing it will crash for is address misalignment.
    • Otherwise it assumes you know what you’re doing and does exactly what you tell it to do.
• Endianness
  • it is a rule which is used to decide the order of BYTES
    • when going from things bigger than a byte to bytes
    • or vice versa.
  • it comes up in…
    • memory (cause it’s an array of bytes)
    • files (also arrays of bytes)
    • networking
  • big endian stores the big end (most significant byte) first.
    • “read it in order”
    • 0xDEADBEEF is stored in memory as 0xDE, 0xAD, 0xBE, 0xEF
  • little endian does the opposite, stores the least significant byte first.
    • “swap the order”
    • 0xDEADBEEF is stored in memory as 0xEF, 0xBE, 0xAD, 0xDE
    • Notice that we don’t swap the hex digits or the bits, we swap the order of entire bytes
  • but 1-byte values and arrays of 1-byte values are immune to endianness
    • because they aren’t chopped up when loading or storing
• Accessing arrays in MIPS
  • An array is multiple variables of the same type and size, equidistantly spaced apart in memory
    • E.g. arr: .word 1, 2, 3 is 3 words/12 bytes of memory; each item of the array is 4 bytes apart because a word is 4 bytes.
  • The address of A[i] is A + S×i where:
    • A is the address of the array (in asm, the label is the address)
    • S is the size of one item in bytes (so for .word it’s 4, .byte it’s 1, etc)
    • i is the index you want to access (in asm, typically a register)
  • The “long form” of array access looks like:

    # ASSUMING that s0 is the index (maybe we're in a for loop and s0 is the loop counter):
    la  t0, arr    # t0 = address of arr
    mul t1, s0, 4  # t1 = s0 * 4
    add t0, t0, t1 # t0 = address of arr + (s0 * 4)
  
    # Now you can load/store using (t0) as the address
    lw  a0, (t0)   # a0 = arr[s0]
  

  • The “short form” folds the la and add into the lw instruction, but you still have to multiply the index:

    mul t1, s0, 4   # t1 = s0 * 4
    lw  a0, arr(t1) # a0 = arr[s0]
  

  • The stupid arr(t1) syntax means “add the address of arr and t1 together, and use that as the address to load from”
• ATV Rule
  • Any function is allowed to change the A, T, V registers at any time for any reason! :))))))
  • but the consequence is that a caller cannot assume that the a, t, or v registers have the same values after a jal as they did before it.
  • So every time you jal, on the line after jal, you have no clue what is in any of the a, t, or v registers - only the s registers’ values will be the same as they were before the call.
    • you just can’t read the value out of those registers anymore. they’re not like, poisoned or something. you can use the same register before and after a jal for different purposes.
  • This is a scary-sounding rule, but it gives you the freedom to:
    • Use any a, t, or v register at any time for any purpose
      • yes! go ahead and use t0 everywhere! everyone is allowed to use it! :DDDDDD
    • Use the a, t, and v registers without having to “ask permission” or “put them back the way they were” by pushing and popping them
      • you never have to push or pop any of them.
• Function call mechanism in MIPS
  • jal func does two things:
    • sets ra = pc + 4 (pc is pointing at the jal, so pc + 4 is the instruction after it)
    • sets pc = func (whatever its address is)
  • jr ra does one thing:
    • sets pc = ra (where ra was the address of the instruction after the jal that jal set up for us)
  • this is all these instructions do!
  • there is also only one ra register. what this means is: you can only go one function call deep.
    • if main calls fork, and fork calls knife…
    • then the jal knife overwrites the value that was in ra
    • meaning we will be able to get back to fork, but we will get stuck in an infinite loop when we try to return from fork
  • to solve this, we make every function push ra at the beginning, and pop ra at the end
    • this way, every function’s return address goes on the stack, where it’s safe (because there are lots of stack slots)
• The stack
  • A region of memory that contains information about function calls
  • Pushing puts a value on top of the stack, popping removes a value from the top of the stack
  • A stack is a perfect match for the way function calls work:
    • Whenever a function is called, it pushes its activation record (AR) -saved registers, local variables in HLLs
    • Whenever a function is about to return, it pops the AR
    • ARs are removed in the opposite order from when they are created, so stack is exactly what is needed
    • In-progress functions’ data is safe on the stack (in memory)
  • The stack is necessary to make recursive functions work:
    • Every time a recursive function calls itself, a new copy of its local variables is pushed
    • So there can be multiple activation records for the same function on the stack at the same time, each with different values for the local variables
• Calling Convention
  • Honor system used to let multiple functions work together
    • Remember that all functions share the registers so this is important!
  • Makes them agree on:
    • How arguments are passed from caller to callee
    • How values are returned from callee to caller
    • How control flows from caller to callee, and then back again
    • What goes on the stack
    • Who is allowed to use which registers, and for what purposes
    • Which registers must be preserved across calls, and which can be trashed
  • In MIPS, part of this is the s register contract
    • If you want to use some s register sx, you:
      • push sx at the beginning of the function that wants to use it
      • pop sx at the end of the function that wants to use it
    • By following this protocol, it’s as if every function gets its own s registers
      • But everyone has to follow the protocol, or the guarantee is gone!
    • If you look back at your labs 3 and 4, and in some functions you used s0 but didn’t push s0 at the beginning and pop s0 at the end, but it still worked…
      • you just got lucky.
      • you were actually messing up the caller’s copy of s0, but the caller never used s0 for anything, so you never noticed it.
• Bitwise AND and its uses
  • Two main uses:
    1. masking: isolating the lowest n bits of a number by ANDing with 2ⁿ - 1
    2. doing fast modulo by 2ⁿ by ANDing with 2ⁿ - 1
  • Notice both of those are the same operation, just different interpretations
  • Do not confuse bitwise AND (&, works on ints) with logical AND (&&, works on booleans, is lazy)!
• Bit shifting
  • Shifting left by n places is like multiplying by 2ⁿ
    • Shifting left writes 0s on the right side of the number and then erases bits on the left side, which means it has a truncation “built in”
    • Truncation can give you weird results if you lose meaningful bits!
  • Shifting right by n places is like dividing by 2ⁿ
    • Shifting right erases bits on the right side of the number, which forces you to add bits on the left side, which means it has an extension “built in”
    • Because of that, there are two flavors of right-shift:
      • Logical (unsigned) right shift >>> puts 0s to the left of the number
      • Arithmetic (signed) right shift >> puts copies of the sign bit to the left of the number
• Bitsets
  • Simplification of bitfields where each field is 1 bit (0 or 1)
  • Bits are numbered starting with bit 0 on the right side and increasing to the left
    • (this is because bit numbers are the powers of 2 that they represent)
  • To turn on bit n: sets |= (1 << n)
  • To turn off bit n: sets &= ~(1 << n) - note the ~ in there
  • To test if bit n is 1: if((sets & (1 << n)) != 0) - do NOT use ~ in there!
• Bitfields
  • Given the specification for a bitfield, you can determine these for each field:
    • Position: the low bit number (the one on the right)
      • this indicates how far to shift left/right for encoding/decoding that field
    • Size: high bit + 1 - low bit
      • this is how many bits the field is
    • Mask: 2^size - 1 (where size is calculated in the previous point)
      • another way of thinking of it is writing size 1 bits in binary
      • and then turn that into hex
      • e.g. if size = 6, in binary that’s 11 1111 (6 1s in a row)
      • turn that to hex, it’s 0x3F
  • Then, to decode (get a field OUT of an encoded bitfield):
    • shift value right by position and AND with mask
    • so, field = (encoded >> FIELD_POSITION) & FIELD_MASK
    • e.g. with a position of 7 and a mask of 0x3F, field = (encoded >> 7) & 0x3F
  • Finally, to encode (put fields together into an encoded bitfield):
    • shift each field left by position, and or them all together
    • e.g. with 3 fields it might look like encoded = (A << 9) | (B << 7) | (C << 0)
• Floats
  • IEEE 754 standard is the only way floating-point numbers are encoded and manipulated on modern computers
  • Based on binary scientific notation, e.g. +1.10101 x 2⁶
  • Represented in sign-magnitude, not 2’s complement
  • Three parts of a number: sign, fraction, and exponent
    • Sign is the MSB and follows same rule as ints (0 = positive, 1 = negative)
    • Fraction is just the bits after the binary point, left-aligned
      • e.g. if significand is 1.001, then fraction is 00100000.... (many 0s after it)
    • Exponent: if you have 2ⁿ, n is encoded as an unsigned number n + k, where k is the bias constant
      • The bias constant is given to you, e.g. for single-precision floats, k = 127.
      • So for a float, an exponent of +6 is encoded as 127 + 6 = 133, as an unsigned integer.
• Fixed-point numbers
  • A fixed-point number is an integer where we decide to interpret the place values differently, and we consider some places to be fractional.
  • We decide in advance how many fractional places there are.
    • e.g. if we have a 32-bit value, we might say we have 10 fractional places, giving us a “22.10” fixed-point number - 22 bits of whole number, then the binary point, then 10 bits of fraction.
    • Note that to make some bits fractional, you always have to sacrifice an equal number of whole number bits
    • That means that you can have either precision (tiny fractions) or range (big numbers), but never both
  • Since fixed-point numbers are just integers, then they are just as fast as integers (compared to floats, which can be 2-4x slower on average)
    • Fixed-point numbers can also be used in situations when floats are unavailable (e.g. small/cheap CPUs inside appliances; or when writing code for the operating system)
  • However there are some special considerations:
    • After multiplying, you have to shift the product right to get the correct answer
    • Before dividing, you have to shift the dividend (first number) left to get the correct answer
    • You need to write your own functions to print them out properly
• What should you use: floats, BigDecimal, or fixed-point numbers?

    if(you need to represent fractions like 1/10, 1/100, 1/1000) {
        use BigDecimal;
    } else if(floats are not available || floats are too slow) {
        use fixedPointNumbers;
    } else {
        use floats;
    }
  

  • NEVER EVER EVER USE FLOATS TO REPRESENT CURRENCY/MONEY/FINANCIAL TRANSACTIONS. This is because floats use binary (base-2) fractions, and 1/10, 1/100, 1/1000 etc. are infinitely repeating fractions in binary.
    • it is not a matter of “not having enough precision” or “the numbers get rounded off.” it’s that they are infinite in size and computers are incapable of representing infinitely sized values, it’s just a mathematical impossibility
• HOW TO DETECT OVERFLOW
  • AN OVERFLOW OCCURRED IF:

  	Addition	Subtraction
  Unsigned	MSB carry out is 1	MSB carry out is 0 (i.e. there is no carry out)
  Signed	same sign inputs, different sign output	same as addition, but after negating second input

  • For signed addition: you get an overflow only if you add two numbers of the same sign and get the opposite sign out (e.g. add two positives, get a negative)
    • it’s totally possible to add two numbers of the same sign and not have overflow
    • also if the inputs are opposite signs, then overflow is impossible.
  • Remember that detecting overflow is only the first step.
    • Once it has been detected, you can respond to it in 3 ways: store, ignore, fall on the floor (crash)
      • in MIPS, add/sub crash on signed overflow, and addu/subu ignore all overflow.
      • not all architectures are this limited.
• Responding to overflow
  • After detecting an overflow occurred (see above), there are three possible ways in which the addition or subtraction instructions can respond:
    1. Store the extra bit into a special 1-bit carry register
       • This can be checked after the addition or subtraction to see what happened
       • Or it can be used as an input to a subsequent addition or subtraction to perform arbitrary-precision arithmetic, which lets you add or subtract numbers of any number of bits
    2. Ignore that an overflow occurred, and use the result truncated back to n bits
       • This sucks and is the most popular way to respond because it’s Easy
    3. Fall on the floor (crash the program) instantly
       • This lets the programmer/user know right away that something went wrong
       • But in some high-reliability environments (e.g. aerospace) this might be a Bad Idea
       • It really depends