Padding and the Expressive Power of Existential Second-Order Logics

Padding techniques are well-known from Computational Complexity Theory. Here, an analogous concept is considered in the context of existential second-order logics. Informally, a graph H is a padded version of a graph G, if H consists of an isomorphic copy of G and some isolated vertices. A set A of graphs is called weakly expressible by a formula ' in the presence of padding, if ' is able to distinguish between (suuciently) padded versions of graphs from A and padded versions of graphs that are not in A. From results of Lynch Lyn82, Lyn92] it can be easily concluded that (essentially) every NP-set of graphs is weakly expressible by an existential monadic second-order (Monn 1 1) formula with polynomial padding and built-in addition. In particular, NP 6 = coNP if and only if there is a coNP-set of graphs that is not weakly expressible by a Monn 1 1-formula in the presence of addition, even if polynomial padding is allowed. In some sense, this implies that Monn 1 1 is well suited to investigate the NP vs. coNP question. In this paper, it is shown, that in the above statements, addition can be replaced by two unary functions, by built-in relations of degree O(n), for every > 0, and by built-in relations with (1 +)n edges, respectively; on the other hand, Mon 1 1 with built-in relations of degree n o(1) or with n + n o(1) edges is weak, in the sense that not every P-set of graphs is weakly expressible with polynomial padding in this logic; and Monn 1 1 with a built-in linear order or built-in coloured trees is very weak, in the sense that they are weak and padding does not help at all. Corresponding results are shown for several sublogics of binary 1 1 .