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

Example 191 If A = {a, b, c}, then A ∗ = {ɛ, a, b, c, aa, ab, ac, ba, . . ., aaa, . . .}.
Definition 192 A set L ⊆ A ∗ is called a formal language in A.
Definition 193 We use c [n] for the string that consists of n times c.
Example 194 # [5] = 〈#, #, #, #, #〉
Example 195 The set M = {ba [n] | n ∈ N} of strings that start with character b followed
by an arbitrary numbers of a’s is a formal language in A = {a, b}.
Definition 196 The concatenation conc(L1, L2) of two languages L1 and L2 over the same
alphabet is defined as conc(L1, L2) := {s1s2 | s1 ∈ L1 ∧ s2 ∈ L2}.
c○: Michael Kohlhase 117
There is a common misconception that a formal language is something that is difficult to understand
as a concept. This is not true, the only thing a formal language does is separate the “good”
from the bad strings. Thus we simply model a formal language as a set of stings: the “good”
strings are members, and the “bad” ones are not.
Of course this definition only shifts complexity to the way we construct specific formal languages
(where it actually belongs), and we have learned two (simple) ways of constructing them by
repetition of characters, and by concatenation of existing languages.
Substrings and Prefixes of Strings
Definition 197 Let A be an alphabet, then we say that a string s ∈ A ∗ is a substring of a
string t ∈ A ∗ (written s ⊆ t), iff there are strings v, w ∈ A ∗ , such that t = vsw.
Example 198 conc(/, 1, h) is a substring of conc(/, /, 1, h, 1), whereas conc(/, 1, 1) is not.
Definition 199 A string p is a called a prefix of s (write p s), iff there is a string t, such
that s = conc(p, t). p is a proper prefix of s (write p ⊳ s), iff t = ɛ.
Example 200 text is a prefix of textbook = conc(text, book).
Note: A string is never a proper prefix of itself.
c○: Michael Kohlhase 118
We will now define an ordering relation for formal languages. The nice thing is that we can induce
an ordering on strings from an ordering on characters, so we only have to specify that (which is
simple for finite alphabets).
Lexical Order
Definition 201 Let A be an alphabet and