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

Proof:
P.1 It is easy to see tahat applying rules to worked off formulae will only add formulae that
are already present in its branch.
P.2 Let µ(T ) be the number of connectives in a labeled formulae in T that are not worked
off.
P.3 Then each rule application to a labeled formula in T that is not worked off reduces µ(T )
by at least one. (inspect the rules)
P.4 at some point the tableau only contains worked off formulae and literals.
P.5 since there are only finitely many literals in T , so we can only apply the tableau cut rule
a finite number of times.
c○: Michael Kohlhase 208
The Tableau calculus basically computes the disjunctive normal form: every branch is a disjunct
that is a conjunct of literals. The method relies on the fact that a DNF is unsatisfiable, iff each
monomial is, i.e. iff each branch contains a contradiction in form of a pair of complementary
literals.
2.7.2 Resolution for Propositional Logic
The next calculus is a test calculus based on the conjunctive normal form. In contrast to the
tableau method, it does not compute the normal form as it goes along, but has a pre-processing
step that does this and a single inference rule that maintains the normal form. The goal of this
calculus is to derive the empty clause (the empty disjunction), which is unsatisfiable.
Another Test Calculus: Resolution
Definition 350 A clause is a disjunction of literals. We will use for the empty disjunction
(no disjuncts) and call it the empty clause.
Definition 351 (Resolution Calculus) The resolution calculus operates a clause sets
via a single inference rule:
P T ∨ A P F ∨ B
A ∨ B
This rule allows to add the clause below the line to a clause set which contains the two clauses
above.
Definition 352 (Resolution Refutation) Let S be a clause set, and D : S ⊢R T a R
derivation then we call D resolution refutation, iff ∈ T .
c○: Michael Kohlhase 209
A calculus for CNF Transformation
Definition 353 (Transformation into Conjunctive Normal Form) The CNF
transformation calculus CN F consists of the following four inference rules on clause sets.
C ∨ (A ∨ B) T
C ∨ A T ∨ B T
C ∨ (A ∨ B) F
C ∨ A F ; C ∨ B F
C ∨ ¬A T
C ∨ A F
C ∨ ¬A F
C ∨ A T
Definition 354 We write CNF (A) for the set of all clauses derivable from A F via the rules
above.
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