By Wolfgang Bibel (auth.)

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**Sample text**

P3) simply asks for a path through this submatrix E. This time we have selected P12 = {Kl,Ll,M2} • Hence, , , p = {Li} u {Kl,L 1 ,M2} U {Mil ~ {K 1 ,L 1 ,M 2 ,L 1 ,Ml} • In our illustrative "gate" interpretation we might say that any gate in the sense before wi thin matrices in normal form may be split into an entrance and an exit gate with a whole matrix in between. In that sense even the whole matrix itself may be regarded as such a generalized gate which has already been realized in the previous picture.

The question whether such a statement E is true or false in fact lies at the heart of ATP. Typically, it is posed together with information about the truth of other statements E), ••. ,E n , n)O. 1) where E is ,Sc,t and El, •.. 2 four conjuncts in its if-part, which are assumed to be true. 1), their four atomic statements Nc, Sc, t , Rc, t , Oc are to be regarded as elements in If pO. we would know explicitiy for these whether they are true or false then the question on E could be solved easily, not only in this particular example but also in general as we will see in a moment.

But in order to span out the whole matrix, each needs the support of at least one connection (like of a pillar). In which was obtained from the previous example by adding the further clause 1 M, the three connections in fact are spanning~ hence this matrix is complementary. 9). 1). Since any path containing a connection itself is a complementary matrix, we may talk of complementary paths in this case. For the same reason in the particular case of a connection its two literals are also called complementary literals.