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

Definition 236 We will write (−A) as A and (A∗B) as A ∗ B (and similarly for +).
Furthermore we will write variables such as x71 as x71 and elide brackets for sums and
products according to their usual precedences.
Example 237 σ(((−(x1∗x2))+(x3∗x4))) = x1 ∗ x2 + x3 ∗ x4
: Do not confuse + and ∗ (Boolean sum and product) with their arithmetic counterparts.
(as members of a formal language they have no meaning!)
c○: Michael Kohlhase 134
Now that we have defined the formal language, we turn the process of giving the strings a meaning.
We make explicit the idea of providing meaning by specifying a function that assigns objects that
we already understand to representations (strings) that do not have a priori meaning.
The first step in assigning meaning is to fix a set of objects what we will assign as meanings: the
“universe (of discourse)”. To specify the meaning mapping, we try to get away with specifying
as little as possible. In our case here, we assign meaning only to the constants and functions and
induce the meaning of complex expressions from these. As we have seen before, we also have to
assign meaning to variables (which have a different ontological status from constants); we do this
by a special meaning function: a variable assignment.
Boolean Expressions: Semantics via Models
Definition 238 A model 〈U, I〉 for Ebool is a set U of objects (called the universe) together
with an interpretation function I on A with I(Cbool) ⊆ U, I(F 1 bool ) ⊆ F(U; U), and
I(F 2 bool ) ⊆ F(U 2 ; U).
Definition 239 A function ϕ: V → U is called a variable assignment.
Definition 240 Given a model 〈U, I〉 and a variable assignment ϕ, the evaluation function
Iϕ : Ebool → U is defined recursively: Let c ∈ Cbool, a, b ∈ Ebool, and x ∈ V , then
Iϕ(c) = I(c), for c ∈ Cbool
Iϕ(x) = ϕ(x), for x ∈ V
Iϕ(a) = I(−)(Iϕ(a))
Iϕ(a + b) = I(+)(Iϕ(a), Iϕ(b)) and Iϕ(a ∗ b) = I(∗)(Iϕ(a), Iϕ(b))
U = {T, F} with 0 ↦→ F, 1 ↦→ T, + ↦→ ∨, ∗ ↦→ ∧, − ↦→ ¬.
U = Eun with 0 ↦→ /, 1 ↦→ //, + ↦→ div, ∗ ↦→ mod, − ↦→ λx.5.
U = {0, 1} with 0 ↦→ 0, 1 ↦→ 1, + ↦→ min, ∗ ↦→ max, − ↦→ λx.1 − x.
c○: Michael Kohlhase 135
Note that all three models on the bottom of the last slide are essentially different, i.e. there is
no way to build an isomorphism between them, i.e. a mapping between the universes, so that all
Boolean expressions have corresponding values.
To get a better intuition on how the meaning function works, consider the following example.
We see that the value for a large expression is calculated by calculating the values for its subexpressions
and then combining them via the function that is the interpretation of the constructor
at the head of the expression.
Evaluating Boolean Expressions
Example 241 Let ϕ := [T/x1], [F/x2], [T/x3], [F/x4], and I =
{0 ↦→ F, 1 ↦→ T, + ↦→ ∨, ∗ ↦→ ∧, − ↦→ ¬}, then
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