Tableau-Based Theorem Proving

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30 Reiner HahnleNow consider the clause set T n = fp i _:p i j i 2f1;:::;ngg. S n [ T n has ashort proof (displayed for n = 3 in Figure 10). The point is that the tautologiesin T n can be used to enumerate all interpretations over fp 1 ;:::;p n g. Then eachclause in S n is contradicted by exactly one interpretation. The resulting tableauhas n2 n +2 n+1 , 1 2O(jS n [ T n j) nodes.truep 1:p 1p 2:p 2p 3:p 3 p 3 :p 3 :p 1 :p 2 :p 3 :p 1 :p 2 p 3p 3p 2:p 3p 3:p 2:p 3p 1 p 2 p 3Figure 10. Clause tableau proof for S3 [ T3.The eect of the clauses T n can also be achieved by adding a new tableau extensionrulep :p(4.7)called atomic cut rule. It is closely related the well-known cut rule part of sequentcalculi [21].4.7.1. Tableaux and Sequent CalculiA propositional clausal sequent is an expression of the form , ) , where , isaVtuple of clauses and is a tuple of atoms. , ) isvalid i the formulaG2, G ! W D2D is valid. Hence, a clause set S is unsatisable if the sequentS ) is valid. The propositional clausal sequent calculus consists of only three rules:,;L 1 ; , 0 ) ,;L n ; , 0 ) ,;L 1 __L n ; , 0 _ -left) ,; , 0 (4.8)) P; ,; :P; , 0 ) :-left ,;P;, 0 ) ;P; 0 axiomIt is straightforward to show that a sequent isvalid i it has a proof tree in whichall leaves are marked with a . Moreover, there is a strong duality between clausetableaux and clausal sequent calculi, summarized in Table 3. 77 This duality extends to the non-clausal and rst-order case, see [58].
Tableau-Based Theorem Proving 31Clausal SequentsAxiom rule_-left ruleAtoms/unit clauses left of )Atoms right of)sequent proofs interpreted as logicalconjunction of their endsequentssequents interpreted as logicaldisjunction of their elementsValidity proofDuality between sequent and tableau calculi.Clause TableauxClosed branchTableau extensionpositive branch literalsnegative branch literalstableaux interpreted as logicaldisjunction of their branchesbranches interpreted as logicalconjunction of their literalsUnsatisability proofTable 3While literal occurrences can be shared among several tableau branches, theyare duplicated in sequents. In the light of this and the duality between sequent andtableau proofs, rule (4.7) is exactly the same as the usual cut rule of sequent calculi,where the cut formula is restricted to atomic formulas:,; , 0 ) ;; 0 ,;;, 0 ) ; 0,; , 0 ) ; 0 (4.9)4.7.2. Tableaux with LemmasThe main problem of the atomic cule (4.7) is that its application is completelyunrestricted. A rst restriction is obtained by allowing its application only if thefollowing extension rule constitutes a strong connection step:.L iL iL 1 L i L nThis way no more branches are generated if one had applied the extension stepimmediately. On the other hand, in n,1 of the new branches the literal L i is present.Applying n,1 atomic cuts before an extension step and writing the resulting proof
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Tableau-Based Theorem Proving

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