# Workshop on Logic, Graph Transformations and Discrete Structures,

### Open problem sheet

Open problems suggested by Bruno Courcelle during the workshop

For definitions, see the slides of the talk or http://www.labri.fr/~courcell/ActSci.html

1. Conjecture (Seese) : If C is a class of finite graphs for which the satisfiability problem for monadic second-order logic (MS logic) is decidable, then this class is the image of a set T of finite trees under a monadic second-order transduction, equivalently C has bounded clique-width.

Reference: D. Seese, The structure of the models of decidable monadic theories of graphs, Ann. Pure Appl. Logic 53 (1991) 169-195.

Observation: For proving Seese's Conjecture, one need to find a monadic second-order compatible graph transformation S and a function f such that, for every graph G, if G has clique-width more than f(k), then S(G) is a k x k square grid.

2. A stronger conjecture: If C and T are as above, then T is also the image of C under a monadic second-order transduction. Hence a decision algorithm for the satisfiability problem for MS logic in graphs of C can be derived from one for T and vice-versa.

3. Reductions between specific cases

Let C and D be two classes of graphs.

We write C => D, if one can prove the conjecture for all subsets of D assuming it is proved for all subsets of C.

Do we have "finite graphs" => "countable graphs" (the conjecture is actually stated by D. Seese for finite and infinite graphs) ?

Do we have "finite graphs" => "finite relational structures with relations of arity more than 2" ?

4. Which structure transformations are monadic second-order compatible (MS compatible)?

The following transformations are known to be MS compatible : MS transductions, the Shelah-Stupp-Muchnik tree expansion.

Compositions of MS compatible transformations are MS compatible. (Unfolding is a composition of an MS -transduction and a tree-expansion.)

Question: Can one find MS-compatible transformations that cannot be obtained as finite compositions of transformations of the above two basic forms.

5. For a Noetherian and confluent term rewriting system, the normal form mapping goes from finite terms to finite terms .

When is it an MS-transduction ? When is it MS-compatible?

Pascal Weil     October 13, 2002