CSCI2467: Systems Programming Concepts

Transcription

1 CSCI2467: Systems Programming Concepts Slideset 2: Information as Data (CS:APP Chap. 2) Instructor: M. Toups Spring 2018

2 Course updates datalab out today! - due after Mardi gras... - do not wait until after holiday to start!

3 ints are not Integers Source: xkcd.com Z is infinitely large, computer memory is not. This is the fundamental challenge!

4 ints are not Integers and floats are not Reals Is x 2 0? - Floating point? Yes! - Int? ?? Is (x + y) + z = x + (y + z)? - Int (signed or unsigned): Yes! - Float? (1e20 + 1e20) e20 + ( 1e )??

5 Computer Arithmetic Does not generate random values - Arithmetic operations have important mathematical properties Cannot assume all usual mathematical properties - Due to finiteness of representations - int operations satisfy ring properties: Commutativity, associativity, distributivity - Floating point operations satisfy ordering properties: Monotonicity, values of signs Observation - Need to understand which abstractions apply in which contexts - Important issues for compiler writers and serious application programmers

6 Overview Course notes 1 Preview 2 Bits and Bytes Representing information as bits Bit-level manipulations 3 Integer Values Signed and Unsigned ints

7 Everything is bits Each bit is 0 or 1 By encoding/interpreting sets of bits in various ways computers determine what to do (instructions) and represent and manipulate numbers, sets, strings, etc #inclu de <st dio.h> int main() { print f("hel lo, wo rld\n" ); retur n 0;.}

8 Everything is bits Each bit is 0 or 1 By encoding/interpreting sets of bits in various ways computers determine what to do (instructions) and represent and manipulate numbers, sets, strings, etc #inclu de <st dio.h> int main() { print f("hel lo, wo rld\n" ); retur n 0;.}

9 Why bits? c 2005 Paul W Shaffer, University of Pennsylvania Course notes Preview Bits and Bytes Integer Values

10 ENIAC

11 Why bits? Course notes Preview Bits and Bytes Integer Values

12 Why bits? Electronic Implementation Easy to store with bistable elements Reliably transmitted on noisy and inaccurate wires

13 Counting in base-2 (binary) Base 2 Number Representation (not characters or strings) Represent as value value Bits add: Sum: 2467 Represent as [0011]... 2 value value Bits

14 Encoding Byte Values 1 Byte = 8 bits Binary to Decimal 0 10 to Hexadecimal to FF 16 Hexadecimal: Base 16 representation Use characters 0 to 9 and A to F Write FA 1D 37 B1 in C as: 0xFA1D37B1 0xfa1d37b1 Important to get comfortable with this notation Used in all subsequent labs Practice problems 2.1, 2.2, 2.3, 2.4 will help you build your hex-literacy hex decimal binary A B C D E F

15 Example Data Representations Size in Bytes C Data Type Typical 32-bit Typical 64-bit x86-64 char short int long float double long double /16 pointer 4 8 8

16 Course notes 1 Preview 2 Bits and Bytes Representing information as bits Bit-level manipulations 3 Integer Values

17 Boolean Algebra Algebraic representation of logic, developed by Boole in 1850s Encodes True as 1 and False as 0 Binary AND: A&B = 1 when both A = 1 and B = 1 & Binary NOT (complement): A = 1 when A = Based on Figure 2.7 in CS:APP3e Binary OR: A B = 1 when either A = 1 or B = Exclusive-Or (XOR): A B = 1 when either A = 1 or B = 1 but not both

18 Boolean Algebra extended The connection between Boolean algebra and digital logic was first proposed by Claude Shannon in a 1937 Master s thesis. Can operate on bit vectors, applying operation bitwise & ˆ & 85 = 65?? (Bitwise operations look strange when using decimal representations!)

19 Boolean Algebra and finite sets Width w bit vector represents subsets of {0,..., w-1} a j = 1 if j A { 0, 3, 5, 6 } { 0, 2, 4, 6 } Operations (on the two sets given above): & Intersection { 0, 6 } Union { 0, 2, 3, 4, 5, 6 } ˆ Symmetric difference { 2, 3, 4, 5 } Complement { 1, 3, 5, 7 }

20 Logical operators Don t confuse bitwise and logical operators! They look similar but are very different. &&,,! - View 0 as False - View anything non-zero as True - Always return 0 or 1 - Early termination! Examples:!0x41 0x00!0x00 0x01!!0x41 0x01 0x69 && 0x55 0x01 0x69 0x55 0x01 a && 5/a (will never divide by zero) p && *p (avoids null pointer access)

21 Shift operators Left Shift: x << y - Shift bitvector x left y positions (Throw away extra bits on left) - Fill with 0s on right Right Shift: x >> y - Shift bitvector x right y positions (Throw away extra bits on right) Logical shift: fill with 0s on left Arithmetic shift: Replicate most significant bit on left Undefined: Shift < 0 or word size Example 1 Argument x << Log. >> Arith. >> Example 2 Argument x << Log. >> Arith. >>

22 Some useful properties of Boolean Algebra Shared properties Property Integer ring Boolean algebra Commutativity a + b = b + a a b = b a a b = b a a & b = b & a Associativity (a + b) + c = a + (b + c) (a b) c = a (b c) (a b) c = a (b c) (a & b) & c = a & (b & c) Distributivity a (b + c) = (a b) + (a c) a & (b c) = (a & b) (a & c) Identities a + 0 = a a 0 = a a 1 = a a & 1 = a Annihilator a 0 = 0 a & 0 = 0 Cancellation ( a) = a ( a) = a Unique to Rings Inverse a + a = 0 Unique to Boolean Algebras Distributivity a (b & c) = (a b) & (a c) Complement a a = 1 a & a = 0 Idempotency a & a = a a a = a Absorption a (a & b) = a a & (a b) = a DeMorgan s laws (a & b) = a b (a b) = a & b Figure 2: Comparison of integer ring and Boolean algebra. The two mathematical structures share many properties, but there are key differences, particularly between and.

23 Bit Masks data & mask = result Unwanted bits are masked out :

24 Course notes 1 Preview 2 Bits and Bytes 3 Integer Values Signed and Unsigned ints

25 Encoding Integer values Unsigned B2U(X ) = w 1 i=0 x i 2 i Signed B2T (X ) = x w 1 2 w 1 + Change: Sign bit! B2T (X ) = x w 1 2 w 1 + Change: Sign bit! Example using short in C (2 bytes): Decimal Hex Binary A F65D This is called Two s complement Sign bit indicates sign - 0 for non-negative - 1 for negative w 2 i=0 w 2 i=0 x i 2 i x i 2 i

26 Unsigned Integers 2 3 = = = = [0001] [0101] [1011] [1111]

27 Signed Integers 2 3 = = = = = [1011] +16 [1111] +16

28 Back to the 2 s complement encoding example short int x= 2467: short int y= -2467: Weight Sum: