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

2.7 Machine-Oriented Calculi
Now we have studied the Hilbert-style calculus in some detail, let us look at two calculi that work
via a totally different principle. Instead of deducing new formulae from axioms (and hypotheses)
and hoping to arrive at the desired theorem, we try to deduce a contradiction from the negation
of the theorem. Indeed, a formula A is valid, iff ¬A is unsatisfiable, so if we derive a contradiction
from ¬A, then we have proven A. The advantage of such “test-calculi” (also called negative
calculi) is easy to see. Instead of finding a proof that ends in A, we have to find any of a broad
class of contradictions. This makes the calculi that we will discuss now easier to control and
therefore more suited for mechanization.
2.7.1 Calculi for Automated Theorem Proving: Analytical Tableaux
Analytical Tableaux
Before we can start, we will need to recap some nomenclature on formulae.
Recap: Atoms and Literals
Definition 330 We call a formula atomic, or an atom, iff it does not contain connectives.
We call a formula complex, iff it is not atomic.
Definition 331 We call a pair A α a labeled formula, if α ∈ {T, F}. A labeled atom is
called literal.
Definition 332 Let Φ be a set of formulae, then we use Φ α := {A α | A ∈ Φ}.
c○: Michael Kohlhase 199
The idea about literals is that they are atoms (the simplest formulae) that carry around their
intended truth value.
Now we will also review some propositional identities that will be useful later on. Some of
them we have already seen, and some are new. All of them can be proven by simple truth table
arguments.
Test Calculi: Tableaux and Model Generation
Idea: instead of showing ∅ ⊢ T h, show ¬T h ⊢ trouble (use ⊥ for trouble)
Example 333 Tableau Calculi try to construct models.
Tableau Refutation (Validity) Model generation (Satisfiability)
|=P ∧ Q ⇒ Q ∧ P |=P ∧ (Q ∨ ¬R) ∧ ¬Q
P ∧ Q ⇒ Q ∧ P F
P ∧ QT Q ∧ P F
P T
P ∧ (Q ∨ ¬R) ∧ ¬QT P ∧ (Q ∨ ¬R) T
¬QT QF P T
P F
⊥
Q T
Q F
⊥
Q ∨ ¬R T
Q T
⊥
¬R T
R F
No Model Herbrand Model {P T , Q F , R F }
ϕ := {P ↦→ T, Q ↦→ F, R ↦→ F}
Algorithm: Fully expand all possible tableaux, (no rule can be applied)
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