Path Logics with Synchronization

Over trees and partial orders, chain logic and path logic are systems of monadic second-order logic in which second-order quantification is applied to paths and to chains (i.e., subsets of paths), respectively; accordingly we speak of the path theory and the chain theory of a structure. We present some known and some new results on decidability of the path theory and chain theory of structures that are enhanced by features of synchronization between paths. We start with the infinite two-dimensional grid for which the finite-path theory is shown to be undecidable. Then we consider the infinite binary tree expanded by the binary ”equal level predicate” E. We recall the (known) decidability of the chain theory of a regular tree with the predicate E and observe that this does not extend to algebraic trees. Finally, we study refined models in which the time axis (represented by the sequence of tree levels) or the tree levels themselves are supplied with additional structure.