Monday, January 27, 2014

Transcription

1 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 decimal Tutoring Here is a link to the Computer Science tutoring schedule (BEL2362) and the tutor s expertise In addition, there is tutoring at the STEM Center in El Dorado Hall for students in any gateway (first or second-year) STEM course: The STEM Center offers free drop-in tutoring for all gateway STEM courses.. Also check out the Learning Resource Center in the Broome Library. 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) 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 11 January 27, 2014

2 but nothing that combined programming with calculation. Babbage s machine had components that we can map to elements of modern computers: 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, (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 getting both data and instructions 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 may allow 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. Comp 162 Notes Page 2 of 11 January 27, 2014

3 What does the future hold? Probably increased throughput (operations per second) achieved through parallelism. Multicore processors are becoming common 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 thousands of CPUs. Here are the top 5 in the latest list (November 2013). System Location Cores Max Petaflops Power Consumption (Kwatts) Tianhe-2 China Titan Oak Ridge Sequoia Lawrence Livermore K Computer Japan Mira Argonne Nat. Labs 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 many states at the same time. A quantum computer can perform the equivalent of conventional operations at the same time. This technology is still in its infancy. Comp 162 Notes Page 3 of 11 January 27, 2014

4 Chapter 3 Given our stored program model, we need to find ways to represent both data items and instructions in memory. We can classify items to be represented as follows Data Numeric Integer Real Unsigned, Signed Fixed point, Floating point Non-numeric Booleans, Characters, 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/8 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 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 11 January 27, 2014

5 (19) (15) (1) (2!!) The conversion ideas between decimal and binary (see number conversion below) also work for other bases. 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 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 Bit pattern Represents Comp 162 Notes Page 5 of 11 January 27, 2014

6 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, note that just changing that bit will not negate a number. In general, to find the representation of -K in the two's complement scheme: Find the representation of +K (regular base 2 number) change all the bits add 1 An alternative equivalent technique is: take the representation of +K working right to left, leave nchanged 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 (5) (-1) (1) (4) The same basic add-base-2 logic gives the correct result. With addition of signed numbers the carry is not an indication of error (more on this later) and we get the correct result. 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 6 of 11 January 27, 2014

7 There are schemes other than two s complement for representing signed integers. We do not look at them in detail; the following table 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 7 of 11 January 27, 2014

8 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 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 Operation Select N Z V C Result Comp 162 Notes Page 8 of 11 January 27, 2014

9 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. Comp 162 Notes Page 9 of 11 January 27, 2014

10 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) if the number is negative, make it positive, convert it then negate the binary. E.g., -167 with a 9-bit representation -167 => 167 => => => Comp 162 Notes Page 10 of 11 January 27, 2014

11 (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. Reading 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 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; there will be links on the course page to sites describing Huffman codes. Comp 162 Notes Page 11 of 11 January 27, 2014