General Computer Science 320201 GenCS I & II Lecture ... - Kwarc

Now, checking whether a code is a prefix code can be a tedious undertaking: the naive algorithm
for this needs to check all pairs of codewords. Therefore we will look at a couple of properties of
character codes that will ensure a prefix code and thus decodeability.
Sufficient Conditions for Prefix Codes
Theorem 215 If c is a code with |c(a)| = k for all a ∈ A for some k ∈ N, then c is prefix
code.
Proof: by contradiction.
P.1 If c is not at prefix code, then there are a, b ∈ A with c(a) ⊳ c(b).
P.2 clearly |c(a)| < |c(b)|, which contradicts our assumption.
Theorem 216 Let c: A → B + be a code and ∗ ∈ B be a character, then there is a prefix
code c ∗ : A → (B ∪ {∗}) + , such that c(a) ⊳ c ∗ (a), for all a ∈ A.
Proof: Let c ∗ (a) := c(a) + ”∗” for all a ∈ A.
P.1 Obviously, c(a) ⊳ c ∗ (a).
P.2 If c ∗ is not a prefix code, then there are a, b ∈ A with c ∗ (a) ⊳ c ∗ (b).
P.3 So, c ∗ (b) contains the character ∗ not only at the end but also somewhere in the middle.
P.4 This contradicts our construction c ∗ (b) = c(b) + ”∗”, where c(b) ∈ B +
c○: Michael Kohlhase 124
2.4.3 Character Codes in the Real World
We will now turn to a class of codes that are extremely important in information technology:
character encodings. The idea here is that for IT systems we need to encode characters from
our alphabets as bit strings (sequences of binary digits 0 and 1) for representation in computers.
Indeed the Morse code we have seen above can be seen as a very simple example of a character
encoding that is geared towards the manual transmission of natural languages over telegraph lines.
For the encoding of written texts we need more extensive codes that can e.g. distinguish upper
and lowercase letters.
The ASCII code we will introduce here is one of the first standardized and widely used character
encodings for a complete alphabet. It is still widely used today. The code tries to strike a balance
between a being able to encode a large set of characters and the representational capabiligies
in the time of punch cards (cardboard cards that represented sequences of binary numbers by
rectangular arrays of dots). 6 EdNote:6
The ASCII Character Code
Definition 217 The American Standard Code for Information Interchange (ASCII) code
assigns characters to numbers 0-127
Code ···0 ···1 ···2 ···3 ···4 ···5 ···6 ···7 ···8 ···9 ···A ···B ···C ···D ···E ···F
0··· NUL SOH STX ETX EOT ENQ ACK BEL BS HT LF VT FF CR SO SI
1··· DLE DC1 DC2 DC3 DC4 NAK SYN ETB CAN EM SUB ESC FS GS RS US
2··· ! ” # $ % & ′
( ) ∗ + , − . /
3··· 0 1 2 3 4 5 6 7 8 9 : ; < = > ?
4··· @ A B C D E F G H I J K L M N O
5··· P Q R S T U V W X Y Z [ \ ] ˆ
6··· ‘ a b c d e f g h i j k l m n o
7··· p q r s t u v w x y z { | } ∼ DEL
6 EdNote: is the 7-bit grouping really motivated by the cognitive limit?
67