The two-variable fragment with counting and equivalence

We consider the two-variable fragment of first-order logic with counting, subject to the stipulation that a sin-gle distinguished binary predicate be interpreted as an equivalence. We show that the satisfiability and finite satisfiability problems for this logic are both NEXPTIME-complete. We further show that the corresponding problems for two-variable first-order logic with counting and two equivalences are both undecidable. Copyright line will be provided by the publisher

Capturing Logarithmic Space and Polynomial Time on Chordal Claw-Free Graphs

We show that the class of chordal claw-free graphs admits LREC$_=$-definable canonization. LREC$_=$ is a logic that extends first-order logic with counting by an operator that allows it to formalize a limited form of recursion. This operator can be evaluated in logarithmic space. It follows that there exists a logarithmic-space canonization algorithm, and therefore a logarithmic-space isomorphism test, for the class of chordal claw-free graphs. As a further consequence, LREC$_=$ captures logarithmic space on this graph class. Since LREC$_=$ is contained in fixed-point logic with counting, we also obtain that fixed-point logic with counting captures polynomial time on the class of chordal claw-free graphs.Comment: 34 pages, 13 figure

On Generalized Records and Spatial Conjunction in Role Logic

We have previously introduced role logic as a notation for describing properties of relational structures in shape analysis, databases and knowledge bases. A natural fragment of role logic corresponds to two-variable logic with counting and is therefore decidable. We show how to use role logic to describe open and closed records, as well the dual of records, inverse records. We observe that the spatial conjunction operation of separation logic naturally models record concatenation. Moreover, we show how to eliminate the spatial conjunction of formulas of quantifier depth one in first-order logic with counting. As a result, allowing spatial conjunction of formulas of quantifier depth one preserves the decidability of two-variable logic with counting. This result applies to two-variable role logic fragment as well. The resulting logic smoothly integrates type system and predicate calculus notation and can be viewed as a natural generalization of the notation for constraints arising in role analysis and similar shape analysis approaches.Comment: 30 pages. A version appears in SAS 200

The Weisfeiler-Leman Dimension of Planar Graphs is at most 3

We prove that the Weisfeiler-Leman (WL) dimension of the class of all finite planar graphs is at most 3. In particular, every finite planar graph is definable in first-order logic with counting using at most 4 variables. The previously best known upper bounds for the dimension and number of variables were 14 and 15, respectively. First we show that, for dimension 3 and higher, the WL-algorithm correctly tests isomorphism of graphs in a minor-closed class whenever it determines the orbits of the automorphism group of any arc-colored 3-connected graph belonging to this class. Then we prove that, apart from several exceptional graphs (which have WL-dimension at most 2), the individualization of two correctly chosen vertices of a colored 3-connected planar graph followed by the 1-dimensional WL-algorithm produces the discrete vertex partition. This implies that the 3-dimensional WL-algorithm determines the orbits of a colored 3-connected planar graph. As a byproduct of the proof, we get a classification of the 3-connected planar graphs with fixing number 3.Comment: 34 pages, 3 figures, extended version of LICS 2017 pape