Computing Monadic Fixed-Points in Linear-Time on Doubly-Linked Data Structures

Detlef Seese has shown that first-order queries on bounded-degree graphs can be computed in linear-time. We extend this result by using connected doubly-linked data structures (modeled in logic over a singulary vocabulary -one which permits only monadic predicates and functions). The first result is that first-order sentences can then be evaluated by an automaton which works directly in place on these singulary models, without changing their size or shape, and using no external resources whatsoever. In particular, this evaluation algorithm satisfies the finite-visit property: the number of times each datum is read from or written to is a uniformly limited constant. The second result analyzes the complexity of monadic fixed-points in the same vocabulary and shows that they too are in linear-time (though we use a RAM model for this).