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

Idea: Unicode supports multiple encodings (but not character sets) for efficiency
Definition 226 (Unicode Transformation Format) UTF-8, 8-bit, variable-width
encoding, which maximizes compatibility with ASCII.
UTF-16, 16-bit, variable-width encoding (popular in Asia)
UTF-32, a 32-bit, fixed-width encoding (for safety)
Definition 227 The UTF-8 encoding follows the following encoding scheme
Unicode Byte1 Byte2 Byte3 Byte4
U+000000 − U+00007F 0xxxxxxx
U+000080 − U+0007F F 110xxxxx 10xxxxxx
U+000800 − U+00F F F F 1110xxxx 10xxxxxx 10xxxxxx
U+010000 − U+10F F F F 11110xxx 10xxxxxx 10xxxxxx 10xxxxxx
Example 228 $ = U+0024 is encoded as 00100100 (1 byte)
¢ = U+00A2 is encoded as 11000010,10100010 (two bytes)
e = U+20AC is encoded as 11100010,10000010,10101100 (three bytes)
c○: Michael Kohlhase 129
Note how the fixed bit prefixes in the encoding are engineered to determine which of the four cases
apply, so that UTF-8 encoded documents can be safely decoded..
2.4.4 Formal Languages and Meaning
After we have studied the elementary theory of codes for strings, we will come to string representations
of structured objects like terms. For these we will need more refined methods.
As we have started out the course with unary natural numbers and added the arithmetical
operations to the mix later, we will use unary arithmetics as our running example and study
object.
A formal Language for Unary Arithmetics
Idea: Start with something very simple: Unary Arithmetics
(i.e. N with addition, multiplication, subtraction, and integer division)
Eun is based on the alphabet Σun := Cun ∪ V ∪ F 2 un ∪ B, where
Cun := {/} ∗ is a set of constant names,
V := {x} × {1, . . . , 9} × {0, . . . , 9} ∗ is a set of variable names,
F 2 un := {add, sub, mul, div, mod} is a set of (binary) function names, and
B := {(, )} ∪ {,} is a set of structural characters. ( “,”,”(“,”)” characters!)
define strings in stages: Eun :=
i∈N Ei un, where
E 1 un := Cun ∪ V
E i+1
un := {a, add(a,b), sub(a,b), mul(a,b), div(a,b), mod(a,b) | a, b ∈ E i un}
We call a string in Eun an expression of unary arithmetics.
c○: Michael Kohlhase 130
The first thing we notice is that the alphabet is not just a flat any more, we have characters
with different roles in the alphabet. These roles have to do with the symbols used in the complex
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