Wednesday, January 28, 2018

Size: px

Start display at page:

Download "Wednesday, January 28, 2018"

Curtis Shepherd
5 years ago
Views:

1 Wednesday, January 28, 2018 Topics for today History of Computing (brief) Encoding data in binary Unsigned integers Signed integers Arithmetic operations and status bits Number conversion: binary to/from decimal Other number systems: Binary-coded decimal, Gray codes Brief coverage of history of computing Programmable Devices Computing Devices (1837) Babbage's Analytical Engine (designed but not built 1 ) was the first programmable calculator. Prior to that date there had been: programmable devices (e.g. music boxes, weaving looms) calculators (e.g., abacuses and simple machines devised by Pascal and by Leibnitz) but nothing that combined programming with calculation. Babbage s machine had components that we can map to elements of modern computers, for example, store (memory) and mill (processor). His collaborator, Ada Countess of Lovelace (Lord Byron s daughter), summed up the capabilities of the machine: It can do whatever we know how to order it to perform Developments included the punched card (Hollerith) which helped speed up the tabulation of the 1890 census. Later widely used as input medium for computers. (early 1940s) Computers were programmed externally by inserting wires into plugboards. See, for example, 1 The Difference Engine by William Gibson and Bruce Sterling is an alternative-history novel in which Babbage engines play a large part. William Gibson is also the coiner of the term cyberspace. Comp 162 Notes Page 1 of 15 January 24, 2018

2 (1945) John von Neumann (and others) published a design for a stored program computer. The idea of storing the instructions for processing the data along with the data itself in the same memory may not have originated with von Neumann but his name is now associated with the concept. Having the program stored in internal memory leads to (a) (b) (c) the possibility of a program modifying its own instructions - see examples of this later in the course need for a "program counter" - some mechanism to remember where the next instruction is counter is misleading, pointer would be better. bottleneck in memory access when CPU is both getting instructions and reading/writing data from the same memory. Harvard architecture A computer system that uses the Harvard architecture stores instructions and data in separate memories. (Named after an IBM system at Harvard in the 1940s that used tape for instructions and electro-mechanical counters for data.) This separation allows a CPU to write the results of one instruction to data memory while reading the next instruction from instruction memory. The memories may have different components: the instruction memory might be read-only (ROM) and the data memory might be read-write (RAM). Since 1945 the technology used to implement the von Neumann model has changed from valves to transistors to integrated circuits to LSI (large-scale integration) to VLSI (very large scale integration) but the basic mode of operation is the same. What does the future hold? Probably increased throughput (operations per second) achieved through parallelism. Multicore processors are becoming ubiquitous more than one CPU packaged on the same chip perhaps sharing some part of the memory. It is common for CPUs to use pipelining overlapping the processing of instructions. So rather than complete the fetch-decode-execute phases of one instruction before starting the next, the phases are overlapped (think car assembly line). The world's most powerful systems (see ) typically have millions of cores. Here are the top 5 systems in the latest list (November 2017). The Japanese system is a new entry in the top 5 2. In recent years, the top of the list has changed relatively slowly. The Wikipedia Top500 article has more detail and background. 2 The Japanese and Swiss systems are the only top 10 systems to appear also in the top 10 of the Green 500. Comp 162 Notes Page 2 of 15 January 24, 2018

3 System Location Cores Max Petaflops Power Consumption (Kwatts) TaihuLight China 10,649, Tianhe-2 China 3,120, Piz Daint Switzerland 361, Gyoukou Japan 19,860, Titan Oak Ridge 560, The June 2008 list was the first in which the top system is rated at more than 1 Petaflop (10 15 floating point operations per second). A challenge is to design software that can take advantage of the hardware. Non Von Neumann systems Systems that do not follow the von Neumann model have been investigated Dataflow Computer In a dataflow computer, tokens flow along a graph. Many tokens can be active at once. See Quantum Computing Rather than using bits (binary digits), which are either 1 or 0, a quantum computer uses qubits (quantum bits) which can be in both states at the same time. A quantum computer can perform the equivalent of many conventional operations simultaneously and may be able to solve problems that cannot be solved in reasonable time on a conventional computer. This technology is still in its infancy. Applications in cryptography are likely to be significant. Comp 162 Notes Page 3 of 15 January 24, 2018

4 Chapter 3 Given our stored program model, we need to find ways to represent in memory both data items and program instructions. We can classify items to be represented as follows Data Numeric Non-numeric Integer Real Booleans, Unsigned, Fixed point, Characters, Signed Floating point Strings Instructions Machine-specific Instructions tend to be specific to particular processors so we will look at them later. Data items will include integers, characters, real numbers and Booleans (true/false values). Because it is easier to build devices that are stable in each of 2 states rather than stable in each of 10 states, we use binary devices rather than decimal devices to store information. Hence there is a need to devise mappings of our data types onto binary. Typically we will have a fixed number of binary digits (bits) in which to store a data item. We start with integers then look at characters. We postpone discussion of floating point (real) numbers until later in the course because Pep/9 does not have instructions for operating on floating point numbers. Unsigned integers (section 3.1) We use normal base N ideas. Base 2 numbers (1 upwards) are 1, 10, 11, 100, 101, 110, 111, 1000, and so on. If we have a storage device with N binary digits (bits) then we can store 2 N different numbers. Typically we choose to store 0 through 2 N -1. So if we have 8 bits we can store if we have 12 bits we can store if we have 16 bits we can store and so on. When we add two unsigned numbers, a carry out of the most significant stage of the addition is an indication of an error. It means that the true result of the addition is outside the range that we can represent. For example, if N is 5, our range of representable numbers is so an attempt to add 19 and 15 is not going to work. Comp 162 Notes Page 4 of 15 January 24, 2018

1 0 0 1 1 (19) + 0 1 1 1 1 (15) --------- (1) 0 0 0 1 0 (2!!) The conversion ideas between decimal and binary (see number conversion below) also work for other bases.

5 (19) (15) (1) (2!!) The conversion ideas between decimal and binary (see number conversion below) also work for other bases. Swiss train control Or, presumably, a multiple thereof. Signed integers (section 3.2) There are a variety of schemes for mapping numbers that might be positive or negative. The schemes mostly differ in the way that they represent negative numbers. There is a trade-off between simplicity of the system and complexity of the circuitry that would be needed to perform addition and subtraction. For example, a very simple system would use the first bit to hold the sign of a number and the remaining bits to hold its magnitude (see Sign and magnitude below) but a corresponding adder circuit would be relatively complex. The most commonly used scheme is termed "two's complement" in which -1 is represented by is represented by and so on. This seems like a good choice because when we add 1 ( ) to -1 ( ) we get zero ( ); similarly for 2 and -2 and so on.. Here is a complete two s complement table for 4-bit storage Comp 162 Notes Page 5 of 15 January 24, 2018

6 Bit pattern Represents Note that the representation of a negative number begins with 1 and the representation of a positive number begins with a 0 so we can still treat the leftmost bit as a sign bit. However, also note that just changing that bit will not negate a number. To find the representation of -K in the two's complement scheme: Method 1 Method 2 Find the representation of +K (regular base 2 number) change all the bits add 1 take the representation of +K working right to left, leave unchanged all the bits up to and including the first "1" change the remaining bits. You can check that these two methods both give the same result (for example, in our 5-bit scheme, -5 is 11011). Two s complement is a popular system because the circuitry to perform arithmetic operations on two s complement numbers is relatively simple. No special tests are needed on the incoming numbers to see if they are positive or negative. Consider our 5-bit example again and the addition of 5 and -1. Comp 162 Notes Page 6 of 15 January 24, 2018

7 (5) (-1) (1) (4) The usual protocol for adding base 2 numbers gives the correct result. Recall that when we added unsigned numbers and got a carry out, this indicated overflow. The same is not true when we add signed numbers the carry is not an indication of error (more on this later). The range of numbers that we can store in an N-bit object using the two s complement scheme is -2 N-1 through 2 N-1-1 So for our 5-bit storage this translates to -16 to +15. An 8-bit byte can store -128 to +127 A 16-bit register can store to A 32-bit register can store to Comp 162 Notes Page 7 of 15 January 24, 2018

8 There are schemes other than two s complement for representing signed integers. We do not look at them in detail; the following table of 4-bit patterns is just for information, it shows how the bit patterns are used in each scheme. Two s complement One s complement Sign and magnitude Excess Magnitude Features: One's complement has fast negation (just change all the bits) but two representations for zero and also an end-around-carry operation needed during addition. E.g (3) (-1) ---- (1) 0001 add the carry to get the final result 0010 (2) "Sign and magnitude" and "excess magnitude" are both difficult to design addition circuitry for but each may have other advantages - see floating point later in the semester. Subtraction. Doing subtraction by hand in binary is awkward. But there is no need for the CPU to have a separate subtract circuit in a two's complement machine because A - B = A + (-B) = A + B' + 1 (where B' represents the inverse or one's complement of B) Comp 162 Notes Page 8 of 15 January 24, 2018

9 Y2.038K On some Unix TM systems, the time of day is held as a 32-bit signed integer representing the number of seconds since 12:00AM GMT on January 1 st The largest value that can be held is which is (about 68 years worth of seconds). Adding 1 to this number will give us a large negative number and apparently set the clock back to December This will happen at 3:14:08 on January 19, Had unsigned integers been used instead, the overflow problem would occur in The Wikipedia article has an illustration of the rollover. Arithmetic operations and status bits Q: What happens if we try to perform an operation on signed integers where the true result is outside the range we can represent? A: We get overflow. Q: How do we detect overflow? A: It is not quite as simple as looking for a carry out of the most significant stage as we did with unsigned numbers. In the case of addition, we can only get overflow if we are adding two numbers with the same sign (if we add one positive and one negative number we cannot get overflow). Overflow is indicated by the sign bit of the result being different to what we would expect, i.e., if we add two positive numbers and get a negative result or add two negative numbers and get a positive result. In contrast to operations on unsigned numbers, getting a carry out of the mostsignificant stage is not an error. The Arithmetic and Logic Unit inside a typical computer gives us 4 indicators (status bits) when we do an arithmetic or logical operation. Each indicator is 1 or 0 (true or false). Input 1 Input 2 N Operation ALU Z Select V C Result Comp 162 Notes Page 9 of 15 January 24, 2018

10 N - the result was negative Z - the result was zero C - there was a carry out from the most significant stage V - there was overflow (=error) We will look at some examples next time. Number conversions for integers and fractions (1) binary to decimal (a) if number is positive, add the weights of digits thus = = 26 decimal same applies if number is a fraction. Weights of digits after the binary point are 0.5, 0.25, and so on = (b) if number is signed and negative, convert the negation (assume 2 s complement) Negating gives = = 17 So original number was -17. same applies if number is a fraction. (2) decimal to binary Negating gives = So original number was (a) if number is positive, repeatedly divide by 2 and keep track of the remainders. E.g. 167 / 2 = 83 remainder 1 83 / 2 = 41 remainder 1 41 / 2 = 20 remainder 1 20 / 2 = 10 remainder 0 10 / 2 = 5 remainder 0 5 / 2 = 2 remainder 1 2 / 2 = 1 remainder 0 1 / 2 = 0 remainder 1 Thus 167(decimal) is (binary) (read up the column of remainders) Comp 162 Notes Page 10 of 15 January 24, 2018

11 if the number is negative, make it positive, convert it then negate the binary. E.g., -167 with a 9-bit representation -167 => 167 => => => (b) we can convert a decimal fraction to binary but note that few decimal fractions have an exact binary representation. To convert, multiply (just the) fraction part by 2. Example * 2 = * 2 = * 2 = * 2 = 0.75 * 2 = 1.5 * 2 = 1.0 * 2 = (decimal) is (binary) (read down the column of digits before the decimal point). This is 19/64. If the fraction is negative, make it positive, convert it then negate the binary. E.g => * 2 = * 2 = * 2 = * 2 = 1.5 * 2 = 1.0 * 2 = 0.0 Fraction is Negative gives Other number representations: Binary-coded decimal The decimal digits 0 through 9 could be encoded using 4 bits 0000 to Thus each requires just a nybble (half a byte). One way to code a decimal number is then to map each decimal digit onto a nybble. So, for example, 4385 would be represented Advantages: input/output would be quite efficient. Disadvantages: representation of negative numbers, arithmetic operations would be complex. Comp 162 Notes Page 11 of 15 January 24, 2018

12 There are many variations on coding decimal numbers in this way Other number representations: Binary-reflected (Gray) code A disadvantage of a numbering system such as two s complement is that incrementing a counter often requires changes in multiple bit positions. For example, 7 => => If the counter is sampled during such transitions, a spurious result may be obtained because the bit positions may not change simultaneously. Similarly, an N-bit object representing the angle of a rotating device may also give spurious readouts if the angle is represented using 2 s complement. A Gray code (after Frank Gray) is an ordering of the integers in which only one bit position changes between successive numbers. It is also cyclic in that incrementing the last number gives us the first one. The table below shows how an N-bit Gray code can be constructed by reflecting the (N-1)-bit code then prepending 0 to the first half and 1 to the second. N=1 Reflected N=1 N=2 Reflected N=2 N=3 Reflected N=3 N= Comp 162 Notes Page 12 of 15 January 24, 2018

13 Reading The topics covered today are, for the most part, in Chapter 3. Next we will look at status bit examples and more topics from Chapter 3: logical operations such as AND, OR and EXOR shift operations which are also classified as logical operators hexadecimal (base 16 notation) notation. how characters are commonly represented (section 3.4). We will also look at Huffman codes. These are variable-length codes. The topic is not covered in the book but there will be links on the course page to sites describing Huffman codes. Comp 162 Notes Page 13 of 15 January 24, 2018

14 Review Questions 1. A 5-bit unsigned binary counter is counting down to zero. Unfortunately, it doesn t stop at zero. What does the display show next? 2. In signed binary, what is the largest 3-bit, 5-bit, N-bit number expressed in binary? 3. Suppose the Unix designers had chosen to use a 48-bit signed integer instead of a 32-bit to count seconds. How you could you calculate when the time rollover would occur? 4. In a 5 bit Gray code, what number follows 11010? 5. If Swiss locomotives and wagons all have 4 axles and each is 20 meters long, how long is the shortest ghost train? 6. Your computer uses the 2 s complement system. How would you describe the binary representation of Put the following 6-bit two s complement numbers in the appropriate boxes below Odd Positive Negative Even Comp 162 Notes Page 14 of 15 January 24, 2018

15 Review answers , 01111, 0 followed by N-1 1s 3. Problem arises 2 47 seconds after base time, i.e seconds or / (60*60*24) days or some time around the year 4,473, Reflect the 4-bit code to get the 5-bit code So the answer is Problem arises when train has 256 axles which translates to 64 locos/wagons. Total length would be 1280 meters. 6. All bits are 1 7. Positive Negative Odd Even Comp 162 Notes Page 15 of 15 January 24, 2018

Monday, January 27, 2014

Monday, January 27, 2014 Topics for today History of Computing (brief) Encoding data in binary Unsigned integers Signed integers Arithmetic operations and status bits Number conversion: binary to/from