CSE 111, Fall 2000

Great Ideas in Computer Science

Lecture Notes #3

(3. Insight I, continued)

b) How to convert between decimal & binary

        e.g., how to convert decimal 87
                (= letter W, in ASCII)
                to 8-bit binary (01010111)

i.e., how to code a decimal numeral with
only 2 objects

Note: An 8-bit binary numeral has these places:

___	___	___	___	___	___	___	___
128s place	64s place	32s place	16s place	8s place	4s place	2s place	1s place

        Note:    Max decimal numeral that can be
                    represented in 8 bits is:
                    11111111 = 128 + 64 + ... + 2 + 1 = 255

Link to the Algorithm for Converting a Decimal D to an 8-Bit Binary B.

e.g., to convert 87 (decimal) to 8-bit binary:

1	2	3
Decimal Numeral	How many of these?	Bit
D = 87	128	0 = Q	87 / 128 = 0, R=87
R = 87	N = 64	1 = Q	87 / 64 = 1, R = 23
R = 23	N = 32	0 = Q	23 / 32 = 0, R = 23
R = 23	N = 16	1 = Q	23 / 16 = 1, R = 7
R = 7	N = 8	0 = Q	7 / 8 = 0, R = 7
R = 7	N = 4	1 = Q	7 / 4 = 1, R = 3
R = 3	N = 2	1 = Q	3 / 2 = 1, R = 1
R = 1	N = 1	1 = Q	1 / 1 = 1, R = 0 (so, stop)

So, 87 (decimal) = 01010111 (binary),
reading column 3 from the top down.

e.g., How to convert 8-bit binary 00101001
to decimal 41? Easy!

128s	64s	32s	16s	8s	4s	2s	1s
0	0	1	0	1	0	0	1
		32	+	8	+		1
		= 41 (!)

Link to the Binary Magic Trick.

c) How to add binary numbers:

Easy!
(Just remember that 1+1 = 10 (= decimal 2))

	87		01010111
+	41	+	00101001
	128		10000000

            Multiplication is similar, even easier!:
            In decimal, 2 * 2 = 4
            (remember: In computer science "*" = multiplication)
            In binary, 10 * 10 = 100
            (which is NOT the same as decimal 10 * 10 = 100 (!) )

4. Insight II

(Turing's insight about only needing 5 actions)

    - Every algorithm can be expressed in a language
       for a computer, called a Turing Machine (TM),
       that consists of :
            *    a tape (i.e., an unlimited supply of paper)
            *    & read/write heads
         & whose only nouns are "0" and "1"
        & whose only verbs (actions) are:
            *    MOVELEFT
            *    MOVERIGHT
            *    PRINT-0
            *    PRINT-1
            *    ERASE

- We'll return to this later in the course,
but for more information, see:

            Schagrin, Morton L.; Dipert, Randall R.;
            & Rapaport, William J. (1985),
            Logic: A Computer Approach
            (New York: McGraw-Hill)
            in Lockwood Library, BC138 .S32 1985

- We'll use Pascal, which is a much more
expressive language, but no more powerful

    -    Church-Turing Thesis:
            Everything that can be computed
            can be computed by a TM

            i.e., for each computable problem P,
                  there is a TM that can
                  compute P's solution
                    (n problems, n TMs)

    -    Turing's Theorem:
            There is a universal TM that can compute
            anything that can be computed.

            i.e., there is a universal TM, call it "U",
                  such that for each computable problem,
                  U can compute its solution
                    (1 TM, n problems)

- Karel the Robot is a TM (of a special kind)

5. Insight III (Boehm & Jacopini's Theorem)

- The 3 (or 4) (or 5) ways of creating more
complex actions out of the 5 basic ones are:

* sequence (do this; do that)

        *    selection (or choice):
               if such and such is the case (or is true),
                   then do this
                   else do that

        *    repetition (or looping):
               while such and such is the case (or is true),
                   do this

(* stop)

(* define new actions by naming them
(procedures, or subroutines) )

- These are the 3 basic features of Pascal
(& Karel) that we'll learn.

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