A Logic for Graph with the Edges as the Domain

In this paper we introduce a logic for graph, where the domain is the edges, not the vertices. Such logic behaves differently from the usual graph logic, where the domain is the vertices, because reachability can be expressed in monadic NP in this logic, while it cannot in the standard graph logic. We establish that in this logic to express a path of length 2k or less, quantifier rank k is necessary and sufficient. The proof is non standard in the sense that we use pebble automata over infinite alphabet, rather than the standard Ehrenfeucht-Fräıssé game. Notation. Let u and v be two vertices. and e be an edge from u to v. We say that the head of e is the vertex v, while the tail of e is the vertex u. 1 On graphs with the edges as the domain In this section we are going to apply our results in the previous sections in finite model theory. We will need a different kind of vocabulary for logic for graphs. Let τ = {S, T, HeadHead, HeadTail, TailTail} be a vocabulary on directed graph structures whose domains are the edges, not the vertices. The symbols S, T are both unary predicates, whereas HeadHead, HeadTail, TailTail are binary predicates. The intentions of each predicate are as follows. 1 T ec hn io n C om pu te r Sc ie nc e D ep ar tm en t T eh ni ca l R ep or t C S20 09 -0 4 20 09 • S(e) means that the head of the edge e is a special source vertex. • T (e) means that the tail of the edge e is a special target vertex. • TailTail(e1, e2) means that the tails of e1 and e2 are the same. • HeadHead(e1, e2) means that the heads of e1 and e2 are the same. • HeadTail(e1, e2) means that the head of e1 is the tail of e2. A A A AK e1 ¢ ¢ ¢ ¢̧ e2 TailTail(e1, e2) ¢ ¢ ¢ ¢̧ e1 A A A AK e2 HeadHead(e1, e2) ¢ ¢ ¢ ¢̧ e1 A A A AU e2