## Section: New Results

### Distributed complexity of classical logics

Participants : Stéphane Grumbach, Zhilin Wu.

We have obtained results on the distributed complexity of first-order (FO), Fixpoint (FP) and monadic second order (MSO) logic on various classes of graphs. In [11] , we show that first-order properties can be frugally evaluated, that is, with only a bounded number of messages, of size logarithmic in the number of nodes, sent over each link, over bounded degree networks as well as planar networks. Moreover, we show that the result carries over for the extension of first-order logic with unary counting. These results relate the locality of the logic, in the sense of Gaifman [28] , to the complexity of the distributed computation in terms of the number of messages handled on each node, which can be shown to be constant, a property weaker but which resembles the locality of distributed computations [32] .

In [10] , we considered fixpoint logic, and showed that it can be evaluated with a polynomial number of messages of logarithmic size. We then showed that the (global) logical formulas can be translated into rule programs describing the local behavior of the nodes of the distributed system, which compute equivalent results. We also introduced local fragments of the logic which have a nice expressive power and admit tighter upper-bounds with bounded number of messages of bounded size.

In [15] , we considered monadic second order logic, and showed that MSO can be evaluated in distributed linear time with only a constant number of messages sent over each link for planar networks with bounded diameter, as well as for networks with bounded degree and bounded tree-length. The distributed algorithms rely on the translation of MSO sentences into finite automata over trees, and on nontrivial transformations of linear time sequential algorithms for the tree decomposition of bounded tree-width graphs.