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

objects (unary arithmetic expressions) that we want to encode.
The formal language Eun is constructed in stages, making explicit use of the respective roles of
the characters in the alphabet. Constants and variables form the basic inventory in E 1 un, the
respective next stage is built up using the function names and the structural characters to encode
the applicative structure of the encoded terms.
Note that with this construction E i un ⊆ E i+1
un .
A formal Language for Unary Arithmetics (Examples)
Example 229 add(//////,mul(x1902,///)) ∈ Eun
Proof: we proceed according to the definition
P.1 We have ////// ∈ Cun, and x1902 ∈ V , and /// ∈ Cun by definition
P.2 Thus ////// ∈ E 1 un, and x1902 ∈ E 1 un and /// ∈ E 1 un,
P.3 Hence, ////// ∈ E 2 un and mul(x1902,///) ∈ E 2 un
P.4 Thus add(//////,mul(x1902,///)) ∈ E 3 un
P.5 And finally add(//////,mul(x1902,///)) ∈ Eun
other examples:
div(x201,add(////,x12))
sub(mul(///,div(x23,///)),///)
what does it all mean? (nothing, Eun is just a set of strings!)
c○: Michael Kohlhase 131
To show that a string is an expression s of unary arithmetics, we have to show that it is in the
formal language Eun. As Eun is the union over all the E i un, the string s must already be a member
of a set E j un for some j ∈ N. So we reason by the definintion establising set membership.
Of course, computer science has better methods for defining languages than the ones used here
(context free grammars), but the simple methods used here will already suffice to make the relevant
points for this course.
Syntax and Semantics (a first glimpse)
Definition 230 A formal language is also called a syntax, since it only concerns the “form”
of strings.
to give meaning to these strings, we need a semantics, i.e. a way to interpret these.
Idea (Tarski Semantics): A semantics is a mapping from strings to objects we already know
and understand (e.g. arithmetics).
e.g. add(//////,mul(x1902,///)) ↦→ 6 + (x1907 · 3) (but what does this mean?)
looks like we have to give a meaning to the variables as well, e.g. x1902 ↦→ 3, then
add(//////,mul(x1902,///)) ↦→ 6 + (3 · 3) = 15
c○: Michael Kohlhase 132
So formal languages do not mean anything by themselves, but a meaning has to be given to them
via a mapping. We will explore that idea in more detail in the following.
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