A dual Ramsey theorem for finite ordered oriented graphs

In contrast to the abundance of “direct” Ramsey results for classes of finite structures (such as finite ordered graphs, finite ordered metric spaces and finite posets with a linear extension), in only a handful of cases we have a meaningful dual Ramsey result. In this paper we prove a dual Ramsey theorem for finite ordered oriented graphs. Instead of embeddings, which are crucial for “direct” Ramsey results, we consider a special class of surjective homomorphisms between finite ordered oriented graphs. Since the setting we are interested in involves both structures and morphisms, all our results are spelled out using the reinterpretation of the (dual) Ramsey property in the language of category theory.

On the satisfiability problem for fragments of two-variable logic with one transitive relation

We study the satisfiability problem for two-variable first-order logic over structures with one transitive relation. We show that the problem is decidable in 2-NExpTime for the fragment consisting of formulas where existential quantifiers are guarded by transitive atoms. As this fragment enjoys neither the finite model property nor the tree model property, to show decidability we introduce a novel model construction technique based on the infinite Ramsey theorem. We also point out why the technique is not sufficient to obtain decidability for the full two-variable logic with one transitive relation; hence, contrary to our previous claim, [FO$^2$ with one transitive relation is decidable, STACS 2013: 317-328], the status of the latter problem remains open.

An analysis of the Podelski–Rybalchenko termination theorem via bar recursion

We present an effective proof (with explicit bounds) of the Podelski and Rybalchenko Termination Theorem. The sub-recursive bounds we obtain make use of bar recursion, in the form of the product of selection functions, as this is used to interpret the Weak Ramsey Theorem for pairs. The construction can be seen as calculating a modulus of well-foundedness for a given program given moduli of well-foundedness for the disjunctively well-founded finite set of covering relations. When the input moduli are in system T , this modulus is also definable in system T by a result of Schwichtenberg on bar recursion.