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

2.4 Encoding Programs as Strings
With the abstract data types we looked at last, we studied term structures, i.e. complex mathematical
objects that were built up from constructors, variables and parameters. The motivation
for this is that we wanted to understand SML programs. And indeed we have seen that there is a
close connection between SML programs on the one side and abstract data types and procedures
on the other side. However, this analysis only holds on a very high level, SML programs are not
terms per se, but sequences of characters we type to the keyboard or load from files. We only
interpret them to be terms in the analysis of programs.
To drive our understanding of programs further, we will first have to understand more about sequences
of characters (strings) and the interpretation process that derives structured mathematical
objects (like terms) from them. Of course, not every sequence of characters will be interpretable,
so we will need a notion of (legal) well-formed sequence.
2.4.1 Formal Languages
We will now formally define the concept of strings and (building on that) formal languages.
The Mathematics of Strings
Definition 184 An alphabet A is a finite set; we call each element a ∈ A a character, and
an n-tuple of s ∈ A n a string (of length n over A).
Definition 185 Note that A 0 = {〈〉}, where 〈〉 is the (unique) 0-tuple. With the definition
above we consider 〈〉 as the string of length 0 and call it the empty string and denote it with
ɛ
Note: Sets = Strings, e.g. {1, 2, 3} = {3, 2, 1}, but 〈1, 2, 3〉 = 〈3, 2, 1〉.
Notation 186 We will often write a string 〈c1, . . . , cn〉 as ”c1 . . . cn”, for instance ”a, b, c”
for 〈a, b, c〉
Example 187 Take A = {h, 1, /} as an alphabet. Each of the symbols h, 1, and / is a
character. The vector 〈/, /, 1, h, 1〉 is a string of length 5 over A.
Definition 188 (String Length) Given a string s we denote its length with |s|.
Definition 189 The concatenation conc(s, t) of two strings s = 〈s1, ..., sn〉 ∈ A n and
t = 〈t1, ..., tm〉 ∈ A m is defined as 〈s1, ..., sn, t1, ..., tm〉 ∈ A n+m .
We will often write conc(s, t) as s + t or simply st
(e.g. conc(”t, e, x, t”, ”b, o, o, k”) = ”t, e, x, t” + ”b, o, o, k” = ”t, e, x, t, b, o, o, k”)
c○: Michael Kohlhase 116
We have multiple notations for concatenation, since it is such a basic operation, which is used
so often that we will need very short notations for it, trusting that the reader can disambiguate
based on the context.
Now that we have defined the concept of a string as a sequence of characters, we can go on to
give ourselves a way to distinguish between good strings (e.g. programs in a given programming
language) and bad strings (e.g. such with syntax errors). The way to do this by the concept of a
formal language, which we are about to define.
Formal Languages
Definition 190 Let A be an alphabet, then we define the sets A + :=
i∈N + Ai of nonempty
strings and A ∗ := A + ∪ {ɛ} of strings.
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