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

2.5 Boolean Algebra
We will now look a formal language from a different perspective. We will interpret the language
of “Boolean expressions” as formulae of a very simple “logic”: A logic is a mathematical construct
to study the association of meaning to strings and reasoning processes, i.e. to study how humans 4
derive new information and knowledge from existing one.
2.5.1 Boolean Expressions and their Meaning
In the following we will consider the Boolean Expressions as the language of “Propositional Logic”,
in many ways the simplest of logics. This means we cannot really express very much of interest,
but we can study many things that are common to all logics.
Let us try again (Boolean Expressions)
Definition 231 (Alphabet) Ebool is based on the alphabet A :=
Cbool ∪ V ∪ F 1 bool ∪ F 2 bool ∪ B, where Cbool = {0, 1}, F 1 bool = {−} and F 2 bool = {+, ∗}.
(V and B as in Eun)
Definition 232 (Formal Language) Ebool :=
i∈N Ei bool , where E1 bool := Cbool ∪ V and
E i+1
bool := {a, (−a), (a+b), (a∗b) | a, b ∈ Ei bool }.
Definition 233 Let a ∈ Ebool. The minimal i, such that a ∈ E i bool
is called the depth of a.
e1 := ((−x1)+x3) (depth 3)
e2 := ((−(x1∗x2))+(x3∗x4)) (depth 4)
e3 := ((x1+x2)+((−((−x1)∗x2))+(x3∗x4))) (depth 6)
c○: Michael Kohlhase 133
Boolean Expressions as Structured Objects.
Idea: As strings in in Ebool are built up via the “union-principle”, we can think of them as
constructor terms with variables
Definition 234 The abstract data type
via the translation
B := 〈{B}, {[1: B], [0: B], [−: B → B], [+: B × B → B], [∗: B × B → B]}〉
Definition 235 σ : Ebool → TB(B; V) defined by
σ(1) := 1 σ(0) := 0
σ((−A)) := (−σ(A))
σ((A∗B)) := (σ(A)∗σ(B)) σ((A+B)) := (σ(A)+σ(B))
We will use this intuition for our treatment of Boolean expressions and treak the strings and
constructor terms synonymouslhy. (σ is a (hidden) isomorphism)
4 until very recently, humans were thought to be the only systems that could come up with complex argumentations.
In the last 50 years this has changed: not only do we attribute more reasoning capabilities to animals, but
also, we have developed computer systems that are increasingly capable of reasoning.
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