Binary Decisions

1
Binary Decisions
To be (1) ornot to be (0) that is
the question

• Binary representations
• Algorithms
• Functions

2
Dichotomy

• A dichotomy is a situation in which there are
  two possible states of being vertebrate or
  invertebrate male or female right or wrong
  good or bad on or off 1 or 0, etc.
• A binary decision is a choice made between the
  two states of a dichotomy.
• The question for decision is often presented as
  a proposition which should be determined as being
  either true or false for example, (32) gt v26
  is false.
• Binary decisions can be made by computers
  current will flow if there is charge and a path
  and the result can be stored.
• If 1 represents true and 0 false, the outcome to
  the proposition (32) gt v26 is 0.

3
Base 2 numeral system

• In a base 2 numeral system every digit is 2
  times as large as the digit immediately to its
  right. For example, the place values of the
  bits 1 0 0 1 1 0 1are 26
  25 24 23 22 21 20
• The same pattern is followed for fractional
  numbers (to the right of a binary point) 1 0
  0 1 1 0 1 . 0 1 1 26 25
  24 23 22 21 20 2-1 2-2 2-3

4
Successive binary decisions can give precise
values e.g. for v2

• Fact if x2 is bigger than y2 then x is bigger
  than y.
• Suppose s is the square root of 2, then s2
  2.

Is s bigger than 1 ? yes because 12 lt 2. Is s
bigger than 11/2 ? no because 1.52 gt 2. Is s
bigger than 10/21/4 ? yes because 1.252 lt
2. Is s bigger than 10/21/41/8? yes because
1.3752 lt 2. Is s bigger than 10/21/41/81/16?
no because 1.43752 gt 2. Answers so far 1 0 1
1 0 Is s bigger than 1 0/2 1/4 1/8 0/16
1/32 ? etc.
5
Computation of binary expansion for v2

• The sequence of decisions (1 for yes, 0 for no)
  is 1 0 1 1 0 1 0 1 0 0 0 0 0 1 0
  0 1 . . .It stores the place values which show
  that s is greater than the sum of the indicated
  powers of ½
• 20 2-2 2-3 2-5 2-7
  2-13 2-16
• 1.0110101000001001 is called the binary
  expansion of v2 Note the use of the binary
  point.
• The decimal equivalent is the sum of 20, 2-2,
  2-3, 2-5, 2-7, 2-13, 2-16 or 1.41419983 (cf.
  1.41421356)

6
Binary representation of any number
It is possible to find the binary representation
of any number by asking questions about its size.

• For example, 23.71gt 25 (32) or higher power No
  0
• gt 24 (16) Yes 1
• gt 2423 (24) No 0
• gt 2422 (20) Yes 1
• gt 242221 (22) Yes 1
• gt24222120 (23) Yes 1
• Integer part 10111.

• gt232-1 (23.5) Yes 1
• gt232-12-2 (23.75) No 0
• gt232-1 2-3 (23.625) Yes 1
• gt232-1 2-3 2-4 (23.6875) 1
• gt23 0.71875 0
• gt23 0.703125 1
• gt23 0.7109375 0
• . . .

23.71 has binary representation 10111.1011010
7
Storing numbers in a computer

• Obviously the bits of the binary representation
  of a number , e.g. 10111.1011010 will be
  stored for the number in a computer, but what do
  we do about the binary point, or the minus sign
  for a negative number, and what do we do about
  binary expansions that never terminate?
• We will return to these questions later, but
  first we examine the process of recovering the
  decimal representation from the binary form and
  what might be needed to program a computer to do
  the job for us.

8
Binary representations

• The place values in a binary representation go up
  in powers of 2 to the left of the binary point
  and down by powers of 2 to the right, just as
  with decimals which grow by a factor of ten for a
  shift of one position to the left.

9
Binary to decimal conversion

• What we need to do is
• write down the list of required powers
• multiply the powers by the bits
• add up to get the result.
• (The sum of corresponding products is sometimes
  called the dot product)Notice how convenient it
  is that bits are numbers and not arbitrary
  symbols like F and T .

10
An algorithm

• The steps for converting a binary numeral to a
  decimal are
• write down the list of required powers
• add up the products of the powers with the bits
• They constitute an algorithm (or procedure) for
  the conversion. The algorithm makes sense to a
  human being and we could all manage to do it with
  a bit of practice.
• But could we expect to be able to give
  instructions like this to a computer?

11
Be clear about objectives

• Whenever we want to give something to a computer
  to do we must be clear about exactly what it is
  that we want the computer to do. This applies
  particularly to writing programs for a computer.
• Our goal here is to start with something like
  101100101.100111001 on the understanding that
  this is a binary representation for a number and
  then have the computer do stuff and eventually
  return the decimal equivalent (in this case,
  357.611328125).

12
Computer conventions

• We cannot expect the computer to recognize a
  number in binary form from the context as human
  beings might. Computer languages are designed to
  recognize numbers in decimal representation (for
  our convenience).
• In most computer languages something like
  10111.101 will be interpreted as a decimal,
  here ten thousand, one hundred and eleven and one
  hundred-and-one thousandths with the point being
  recognized as a decimal point.
• If we want the computer to do something with
  10111.101 as a binary representation, then it
  will have to be given as character data.

13
Character data

• Character data in most computer languages is
  distinguished from numeric data by enclosing it
  in single quotes.
• For example, 5.25 is text. The ASCII code
  for the characters 5.25 is 0011010100101110001
  1001000110101
• 5.25 (without quotes) will be interpreted as a
  number. Its binary representation, 101.01, will
  form the basis of how it is stored in a computer
  rather than some arbitrary code.

14
Commands understood by a computer

1. write down the list of required powers
2. add up the products of the powers with the bits

• Ultimately the computer can only let electricity
  do its thing. Computer languages are designed to
  let people send commands without having to worry
  about the wiring and the electric charges (the
  bits of data).
• It turns out that computers can do arithmetic
  and so we expect that they can do dot products,
  but we cant command the computer to look at a
  number and see where the binary point is not
  the way a human being would.

15
Locating the binary point

• The list of powers (of 2) required for the
  binary to decimal conversion is determined by the
  number of bits to the left and right of the
  binary point. We need a way to locate this point
  on the basis of binary decisions rather than
  eyesight.
• A computer can respond to a question like is
  there charge there or not (current will flow
  from a charge if its allowed to, i.e. if the
  question is asked). And we can get a response
  to a question like, do the bits in a particular
  8-bit word match the bits in, say, 00101110 (the
  ASCII code for . )

16
Comparing character data
The ASCII code for the characters 5.25 written
out in 8-bit (or one byte) words is 00110101
00101110 00110010 00110101

• A decision that we could reasonably expect a
  computer language to provide an answer for is
  whether the byte (8 bits) for . is found in
  the bytes for 5.25

17
The syntax of a language

• The question Is the byte for '.' found in the
  bytes for '5.25' is written in the English
  language with the accepted syntax for English.
  There are many understandable ways of asking the
  same question in English.
• In a computer language the options are restricted
  by technology. Syntax must be precise, perhaps
  something like
• '.' is_in '5.25'

18
Functions

• If '.' is_in '5.25' is an interpretable
  statement in a computer language then is_in is a
  recognized word in that language with a very
  specific meaning, namely that characters in the
  text data '5.25' should be scanned to see if the
  code for '.' is in there and return 1 if it is,
  0 if it is not.
• is_in is an action word, a verb, or in
  mathematical jargon, the name of a function.
• '.' is the left data input (or left argument)
  and
• '5.25' is the right data input (or right
  argument)
• When the language is asked to execute the
  statement '.' is_in '5.25' we say it is a
  function call and the syntax of the call is
• (left argument) (function name) (right argument)

19
Function syntax

• In-fix syntax the function name appears between
  data to the left and right of the name e.g. 3
  4, 2 5, 6 8, 12 2, a dot b
• Pre-fix syntax the function name appears before
  the data, sometimes the data is required to be
  enclosed in parentheses, e.g. sin x , square(12)
• Post-fix syntax the function name comes after
  the data e.g. 6 ! , 12 squared , g'

20
What is a function?

• 122, square(12) and 12 squared all return
  144 and, in fact, x2, square(x) and x
  squared will all return the same value no matter
  what value is input, or given, to x. There are
  three different names, perhaps three different
  ways of doing the job i.e. three different
  algorithms, but all three return the same output
  given the same input. So, they are all
  implementations of the same function.
• A function is the relationship between input and
  output.
• Implementations of a function will have their
  own names together with their own syntax and
  algorithms. (Usually, we dont have multiple
  names for the same function in one environment,
  but different languages may have different names
  and algorithms for the same function.)

21
Using a computer language

• Working with a particular computer language
  allows us to translate the algorithms discussed
  into that language. This provides a check on
  whether the algorithm will actually work or not.
• The textbook uses pseudo-code and accepts
  instructions like
• write down the list of required powers
• add up the products of the powers with the bits
• which verge on the computer illiterate.
• We will also write pseudo-code but computer
  literate code that can be re-written in J or
  Matlab in the special vocabulary and syntax for
  each language.
• It is not a part of this course to use any
  particular computer language but the instructor
  will support your use of J or Matlab. J can
  be downloaded for free from jsoftware.com
  Matlab you have to purchase.

22
Is the byte for '.' found in the bytes for
'5.25' ?
Notice the differences between the languages. J
uses only symbols Matlab has some symbols, and
many words for doing different tasks. Both are
interpreters, and unlike spreadsheets, there is
only one execution (or command ) line which is
invoked by pressing the Enter key.
23
Primitive functions

• e. gt i. lt''1 are
  examples of primitive functions in J.
• any find
• are examples of primitive functions in Matlab.

Primitive functions are identified by reserved
symbols or words in a language and perform
specific tasks.
The functions named in J and in Matlab
have in-fix syntax with data on either side of
the function name. The functions called lt''1
in J and find in Matlab have a pre-fix syntax
with the data appearing only to the right of
the functions name or symbol.
24
Finding the range of powers
Notice that a function to count the number of
characters in a string ( in J length in
Matlab) was needed. More functions would be
needed to list the integers between 8 and -9 to
take 2 to each of these powers, to convert the
character string to a numeric list of 1s and 0s
and to take the dot product.
25
The algorithm implemented

• It turned out that our algorithm
• write down the list of required powers
• take the dot product of the powers with the bits
• can, in fact, be implemented. It is the
  algorithm for a function. If it is not provided
  as a primitive function, it might be possible to
  create a user-defined function.
• If we call the function b2d and give it a
  pre-fix syntax, a development environment (like J
  or Matlab) will allow us to save the function and
  its algorithm for later use, for example,
  executing the function call b2d
  '101100101.100111001'should return 357.6113281
  25

26
Whats the big idea?

1. The computer not only stores data, it also
   executes operations like comparisons and
   arithmetic.Action words, or verbs, in computer
   languages are the names of functions which return
   specified results, provided they are called with
   the correct syntax.
2. Character and numeric data cannot be mixed.
   Character storage can use an arbitrary code
   number storage is based on the binary
   representation of the numbers. Binary
   representations can be found from a sequence of
   binary decisions.
3. By working with a computer language we can be
   confident that the procedures we contemplate are
   indeed computable, but we have to learn a bit
   about the language itself its vocabulary and
   syntax.
4. Software packages try to provide the user with
   the best set of tools (functions) for a
   particular set of tasks. A development
   environment tries to provide the most flexible
   set of tools for producing software packages.