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

Definition 355 (Resolution Proof) We call a resolution refutation P : CNF (A) ⊢R T
a resolution sproof for A ∈ wff o(Vo).
c○: Michael Kohlhase 210
Note: Note that the C-terms in the definition of the resolution calculus are necessary, since
we assumed that the assumptions of the inference rule must match full formulae. The C-terms
are used with the convention that they are optional. So that we can also simplify (A ∨ B) T to
A T ∨ B T .
The background behind this notation is that A and T ∨ A are equivalent for any A. That
allows us to interpret the C-terms in the assumptions as T and thus leave them out.
The resolution calculus as we have formulated it here is quite frugal; we have left out rules for the
connectives ∨, ⇒, and ⇔, relying on the fact that formulae containing these connectives can be
translated into ones without before CNF transformation. The advantage of having a calculus with
few inference rules is that we can prove meta-properties like soundness and completeness with
less effort (these proofs usually require one case per inference rule). On the other hand, adding
specialized inference rules makes proofs shorter and more readable.
Fortunately, there is a way to have your cake and eat it. Derived inference rules have the property
that they are formally redundant, since they do not change the expressive power of the calculus.
Therefore we can leave them out when proving meta-properties, but include them when actually
using the calculus.
Derived Rules of Inference
A1 . . . An
Definition 356 Let C be a calculus, a rule of inference
is called a derived
C
inference rule in C, iff there is a C-proof of A1, . . . , An ⊢ C.
Example 357
Others:
C ∨ (A ⇒ B) T
C ∨ A F ∨ B T
C ∨ (A ⇒ B) T
C ∨ (¬A ∨ B) T
C ∨ ¬A T ∨ B T
C ∨ A F ∨ B T
↦→
C ∨ (A ⇒ B) F
C ∨ A T ; C ∨ B F
C ∨ (A ⇒ B) T
C ∨ A F ∨ B T
C ∨ A ∧ B T
C ∨ A T ; C ∨ B T
c○: Michael Kohlhase 211
C ∨ A ∧ B F
C ∨ A F ∨ B F
With these derived rules, theorem proving becomes quite efficient. To get a better understanding
of the calculus, we look at an example: we prove an axiom of the Hilbert Calculus we have studied
above.
Example: Proving Axiom S
Example 358 Clause Normal Form transformation
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