Model-checking for successor-invariant first-order formulas on graph classes of bounded expansion

A successor-invariant first-order formula is a formula that has access to an auxiliary successor relation on a structure's universe, but the model relation is independent of the particular interpretation of this relation. It is well known that successor-invariant formulas are more expressive on finite structures than plain first-order formulas without a successor relation. This naturally raises the question whether this increase in expressive power comes at an extra cost to solve the model-checking problem, that is, the problem to decide whether a given structure together with some (and hence every) successor relation is a model of a given formula. It was shown earlier that adding successor-invariance to first-order logic essentially comes at no extra cost for the model-checking problem on classes of finite structures whose underlying Gaifman graph is planar [1], excludes a fixed minor [2] or a fixed topological minor [3], [4]. In this work we show that the model-checking problem for successor-invariant formulas is fixed-parameter tractable on any class of finite structures whose underlying Gaifman graphs form a class of bounded expansion. Our result generalises all earlier results and comes close to the best tractability results on nowhere dense classes of graphs currently known for plain first-order logic.

We establish that the decidability of the first order theory of a class of finite structures C is a simple and useful condition for guaranteeing that the expressive power of FO+LFP properly extends that of FO on C, unifying separation results for various classes of structures that have been studied. We then apply this result to show that it encompasses certain constructive pebble game techniques which are widely used to establish separations between FO and FO+LFP, and demonstrate that these same techniques cannot succeed in performing separations from any complexity class that contains DLOGTIME.

Big Ramsey degrees in ultraproducts of finite structures

A class of structures satisfies the extension preservation theorem if, on this class, every first order sentence is preserved under extension iff it is equivalent to an existential sentence. We consider different acyclicity notions for hypergraphs (\gamma, \beta and \alpha-acyclicity and also acyclicity on hypergraph quotients) and estimate their influence on the validity of the extension preservation theorem on classes of finite structures. More precisely, we prove that \gamma-acyclic classes satisfy the extension preservation theorem, whereas \beta-acyclic classes do not. We also extend the validity of the extension preservation theorem for a generalization of \gamma-acyclicity that we call \gamma-acyclic k-quotient. To achieve this, we make a reduction from finite structures to their k-quotients and we use combinatorial arguments on \gamma-acyclic hypergraphs.

We give examples of classes of rigid structures which are of unbounded rigidity but least fixed point (Partial fixed point) logic can express all Boolean PTIME (PSPACE) queries on these classes. This shows that definability of linear order in FO+LEP although sufficient for it to capture Boolean PTIME queries, is not necessary even on the classes of rigid structures. The situation however appears very different for nonzero-ary queries. Next, we turn to the study of fixed point logics on arbitrary classes of structures. We completely characterize the recursively enumerable classes of finite structures on which PFP captures all PSPACE queries of arbitrary arities. We also state in some alternative forms several natural necessary and some sufficient conditions for PFP to capture PSPACE queries on classes of finite structures. The conditions similar to the ones proposed above work for LFP and PTIME also in some special cases but to prove the same necessary conditions in general for LFP to capture PTIME seems harder and remains open.

The concept of bounded expansion provides a robust way to capture sparse graph classes with interesting algorithmic properties. Most notably, every problem definable in first-order logic can be solved in linear time on bounded expansion graph classes. First-order interpretations and transductions of sparse graph classes lead to more general, dense graph classes that seem to inherit many of the nice algorithmic properties of their sparse counterparts.In this work we introduce lacon- and shrub-decompositions and use them to characterize transductions of bounded expansion graph classes and other graph classes. If one can efficiently compute sparse shrub- or lacon-decompositions of transductions of bounded expansion classes then one can solve every problem definable in first-order logic in linear time on these classes.

On the Complexity of RSRL

FPT algorithms for domination in sparse graphs and beyond

We propose a new, general and easy method for proving nonexistence of asymptotic probabilities of monadic second-order sentences in classes of finite structures where first-order extension axioms hold almost surely.KeywordsRandom GraphTuring MachineFree GraphSubgraph IsomorphicProof MethodThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

We study the complexity of the model checking problem, for fixed models A, over certain fragments $\mathcal{L}$ of first-order logic, obtained by restricting which of the quantifiers and boolean connectives we permit. These are sometimes known as the expression complexities of $\mathcal{L}$. We obtain various full and partial complexity classification theorems for these logics $\mathcal{L}$ as each ranges over models A, in the spirit of the dichotomy conjecture for the Constraint Satisfaction Problem --- which itself may be seen as the model checking problem for existential conjunctive positive first-order logic.

Capturing complexity classes by fragments of second-order logic

We study the expressive power in the finite of the logic Fixed-Point+Counting, the extension of first-order logic which is obtained through adding both the fixed-point constructor and the ability to count.To this end an isomorphism preserving (‘generic’) model of computation is introduced whose PTime restriction exactly corresponds to this level of expressive power, while its PSpace restriction corresponds to While+Counting. From this model we obtain a normal form which shows a rather clear separation of the relational vs. the arithmetical side of the algorithms involved.In parallel, we study the relations of Fixed-Point+Counting with the infinitary logics and the corresponding pebble games.The main result, however, involves the concept of an arithmetical invariant. By this we mean a functor taking every finite relational structure to an expansion of (an initial segment of) the standard arithmetical structure. In particular its values are linearly ordered structures. We establish the existence of a family of arithmetical invariants with the following properties:• The invariants themselves can be evaluated in polynomial time.• A class of finite relational structures is definable in Fixed-Point+Counting if and only if membership can be decided in polynomial time on the basis of the values of one of the invariants.• The invariant r classifies all finite relational structures exactly up to equivalence with respect to the logic We also give a characterization of Fixed-Point+Counting in terms of sequences of formulae in the : It corresponds exactly to the polynomial time computable families (φn)n ∈ ω in these logics.Towards a positive assessment of the expressive power of Fixed-Point+Counting, it is shown that the natural extension of fixed-point logic by Lindström quantifiers, which capture all the PTime computable properties of cardinalities of definable predicates, is strictly weaker than what we get here. This implies in particular that every extension of fixed-point logic by means of monadic Lindström quantifiers, which stays within PTime, must be strictly contained in Fixed-Point+Counting.

For first-order logic model checking on monotone graph classes the borderline between tractable and intractable is well charted: it is tractable on all nowhere dense classes of graphs, and this is essentially the limit. In contrast to this, there are few results concerning the tractability of model checking on general, i.e. not necessarily monotone, graph classes.

Rainbow independent sets on dense graph classes

Liveness properties are traditionally proven using a ranking function that maps system states to some well-founded set. Carrying out such proofs in first-order logic enables automation by SMT solvers. However, reasoning about many natural ranking functions is beyond reach of existing solvers. To address this, we introduce the notion of implicit rankings — first-order formulas that soundly approximate the reduction of some ranking function without defining it explicitly. We provide recursive constructors of implicit rankings that can be instantiated and composed to induce a rich family of implicit rankings. Our constructors use quantifiers to approximate reasoning about useful primitives such as cardinalities of sets and unbounded sums that are not directly expressible in first-order logic. We demonstrate the effectiveness of our implicit rankings by verifying liveness properties of several intricate examples, including Dijkstra’s k-state, 4-state and 3-state self-stabilizing protocols.