Automated Theorem Proving by Wolfgang Bibel (auth.)

Machine Theory

By Wolfgang Bibel (auth.)

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L. that ~ Any formula ~ F F uniquely determines a matrix such Note that is represented by F • The proof is obvious (c2) has to be applied if yield F, v( II(Fi, ••• ,F n ». 3). M+,(LIIK) represented by {lKri,lLr2,Mr3} the matrix to o formula. all are derived 19 ILl before from the first formula of this list. Incidentally, the last one is its contraposition which indicates the reason why in the connection rule to be introduced in section 4 there is no action corresponding to contraposition as mentioned already in section (I.

O,F) = T by (e1). • ,m, by (e2). This yields Tl = T by (e3), hence T~(O,F) = T by (e~). Since in both directions the chain of reasoning holds for any model, this establishes (I). The proof of (II) is an immediate consequence of the following equation. a path through Fi , and vice versa. 4 arity. It remains to be seen how this may be done in an efficient algorithmic way. 4. 4) provides the basis for a powerful proof method which naturally may be called the connection method • Roughly speaking it consists in selecting connections in the given matrix F, one after the other, until the set of selected connections becomes spanning for F.

For any matrix F in normal form, ~F iff F is comple- paths through F. mentary. Proof. e. 3) the re- 43 II. 5 quired initial step may not be applied. 3) ({0},So) is a proof for F • Thus the theorem holds in these two special cases. Therefore we may now assume that F contains literals. Only-if-case. Consider the following statement. (S) I f (F,Sn) ~ (F,So) with no initial step and i f each path in D~n is complementary then F is complementary, n)O • The since t- t- only-if-case F is means that an immediate there is a consequence deduction of (F,So) (F,Sn) ~ (F,So) , for which D~n = 0 • Thus left to prove (S) which will be done by induction on n.

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