By Wolfgang Bibel (auth.)

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According to definition is false. t1) If F is a literal there is a model then in which F Hence F is not valid. pl) there is exactly one path through F , viz. {F} which obviously is not complementary. Since for F=0 , the theorem trivially holds, we are done in this case. (ii) If a (F) = 1 then by definition (1. ,. - (1 , L i) = T 1=1 ~ thus F is valid in this case. It T""" T""" ~ complementary by ~ default since by definition there are no paths through such a matrix, case. e. F is valid. 1) this means that for m)1, n)O • Consider the m matrices Fi = Ei i U {E2"" ,En} i=1, ••• ,m (regarding any literal Ei i as singleton set).

4 37 these illustrations have prepared the method to a point where the following understanding of this precise formalism may support this understanding rather than confusing it. 0. A matrix structured in view of the linear normal-form connection method for propositional logic is a matrix F in normal form paired with a structure S defined below~ i f the kind of deduction is clear from the context, the pair (F,S) will be simply called a structured matrix • The structure S is a triple (a,a,y) of functions wich satisfy the properties (i), (ii) and (iii).

T. t. the number of items the in each fenced area. 1) because paths in matrices are fundamental for the approach to ATP taken in this book. With the following definition we are now approaching the main result of this section. D. A ( partial path through F. path in a matrix F is any subset of a II. 3 A connection { Lro , 1 Lr 1} • 29 in a matrix F is a path in F of the form A set W of connections in a matrix F is called spanning for F if for each path p through F there is an wEW such that p contains w which is to say ~p.