Padding and the expressive power of existential second-order logics

Abstract

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 ϕ in the presence of padding, if ϕ is able to distinguish between (sufficiently) 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 (Monσ ¹₁ ) sentence with polynomial padding and built-in addition. In particular, NP ≠ coNP if and only if there is a coNP-set of graphs that is not weakly expressible by a Monσ ¹₁ -formula in the presence of addition, even if polynomial padding is allowed. In some sense, this implies that Monσ ¹₁ 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 ^ε), for every ε > 0, and by built-in relations with at most (1 + ε)n edges, respectively;

• - on the other hand, Monσ ¹₁ with built-in relations of degree n° ^(l) or with n + n° ^(l) edges is weak, in the sense that not every P-set of graphs is weakly expressible with polynomial padding in this logic;

• - Monσ ¹₁ 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 increase their expressive power at all.

Corresponding results are shown for several sublogics of binary σ ¹₁ .