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

Even though the definition of the lexical ordering is relatively involved, we know it very well, it is
the ordering we know from the telephone books.
The next task for understanding programs as mathematical objects is to understand the process
of using strings to encode objects. The simplest encodings or “codes” are mappings from strings
to strings. We will now study their properties.
2.4.2 Elementary Codes
The most characterizing property for a code is that if we encode something with this code, then
we want to be able to decode it again: We model a code as a function (every character should
have a unique encoding), which has a partial inverse (so we can decode). We have seen above,
that this is is the case, iff the function is injective; so we take this as the defining characteristic of
a code.
Character Codes
Definition 204 Let A and B be alphabets, then we call an injective function c: A → B +
a character code. A string c(w) ∈ {c(a) | a ∈ A} := B + is called a codeword.
Definition 205 A code is a called binary iff B = {0, 1}.
Example 206 Let A = {a, b, c} and B = {0, 1}, then c: A → B + with c(a) = 0011,
c(b) = 1101, c(c) = 0110 c is a binary character code and the strings 0011, 1101, and 0110
are the codewords of c.
Definition 207 The extension of a code (on characters) c: A → B + to a function c ′ : A ∗ →
B ∗ is defined as c ′ (〈a1, . . . , an〉 = 〈c(a1), . . . , c(an)〉).
Example 208 The extension c ′ of c from the above example on the string ”b, b, a, b, c”
Morse Code
c ′ (”b, b, a, b, c”) =
1101,
1101
, 0011
, 1101
, 0110
c(b) c(b) c(a) c(b) c(c)
c○: Michael Kohlhase 120
In the early days of telecommunication the “Morse Code” was used to transmit texts, using
long and short pulses of electricity.
Definition 209 (Morse Code) The following table gives the Morse code for the text
characters:
A .- B -... C -.-. D -.. E .
F ..-. G –. H .... I .. J .—
K -.- L .-.. M – N -. O —
P .–. Q –.- R .-. S ... T -
U ..- V ...- W .– X -..- Y -.–
Z –..
1 .—- 2 ..— 3 ...– 4 ....- 5 .....
6 -.... 7 –... 8 —.. 9 —-. 0 —–
Furthermore, the Morse code uses .−.−.− for full stop (sentence termination), −−..−− for
comma, and ..−−.. for question mark.
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