Expanding the expressive power of Monadic Second-Order logic on restricted graph classes

We combine integer linear programming and recent advances in Monadic Second-Order model checking to obtain two new algorithmic meta-theorems for graphs of bounded vertex-cover. The first shows that cardMSO1, an extension of the well-known Monadic Second-Order logic by the addition of cardinality constraints, can be solved in FPT time parameterized by vertex cover. The second meta-theorem shows that the MSO partitioning problems introduced by Rao can also be solved in FPT time with the same parameter. The significance of our contribution stems from the fact that these formalisms can describe problems which are W[1]-hard and even NP-hard on graphs of bounded tree-width. Additionally, our algorithms have only an elementary dependence on the parameter and formula. We also show that both results are easily extended from vertex cover to neighborhood diversity.Comment: Accepted for IWOCA 201

Order Invariance on Decomposable Structures

Order-invariant formulas access an ordering on a structure's universe, but the model relation is independent of the used ordering. Order invariance is frequently used for logic-based approaches in computer science. Order-invariant formulas capture unordered problems of complexity classes and they model the independence of the answer to a database query from low-level aspects of databases. We study the expressive power of order-invariant monadic second-order (MSO) and first-order (FO) logic on restricted classes of structures that admit certain forms of tree decompositions (not necessarily of bounded width). While order-invariant MSO is more expressive than MSO and, even, CMSO (MSO with modulo-counting predicates), we show that order-invariant MSO and CMSO are equally expressive on graphs of bounded tree width and on planar graphs. This extends an earlier result for trees due to Courcelle. Moreover, we show that all properties definable in order-invariant FO are also definable in MSO on these classes. These results are applications of a theorem that shows how to lift up definability results for order-invariant logics from the bags of a graph's tree decomposition to the graph itself.Comment: Accepted for LICS 201

Spatial Existential Positive Logics for Hyperedge Replacement Grammars

We study a (first-order) spatial logic based on graphs of conjunctive queries for expressing (hyper-)graph languages. In this logic, each primitive positive (resp. existential positive) formula plays a role of an expression of a graph (resp. a finite language of graphs) modulo graph isomorphism. First, this paper presents a sound- and complete axiomatization for the equational theory of primitive/existential positive formulas under this spatial semantics. Second, we show Kleene theorems between this logic and hyperedge replacement grammars (HRGs), namely that over graphs, the class of existential positive first-order (resp. least fixpoint, transitive closure) formulas has the same expressive power as that of non-recursive (resp. all, linear) HRGs

A complete reified temporal logic and its applications

Temporal representation and reasoning plays a fundamental and increasingly important role in some areas of Computer Science and Artificial Intelligence. A natural approach to represent and reason about time-dependent knowledge is to associate them with instantaneous time points and/or durative time intervals. In particular, there are various ways to use logic formalisms for temporal knowledge representation and reasoning. Based on the chosen logic frameworks, temporal theories can be classified into modal logic approaches (including prepositional modal logic approaches and hybrid logic approaches) and predicate logic approaches (including temporal argument methods and temporal reification methods). Generally speaking, the predicate logic approaches are more expressive than the modal logic approaches and among predicate logic approaches, temporal reification methods are even more expressive for representing and reasoning about general temporal knowledge. However, the current reified temporal logics are so complicate that each of them either do not have a clear definition of its syntax and semantics or do not have a sound and complete axiomatization. In this thesis, a new complete reified temporal logic (CRTL) is introduced which has a clear syntax, semantics, and a complete axiomatic system by inheriting from the initial first order language. This is the main improvement made to the reification approaches for temporal representation and reasoning. It is a true reified logic since some meta-predicates are formally defined that allow one to predicate and quantify over prepositional terms, and therefore provides the expressive power to represent and reason about both temporal and non-temporal relationships between prepositional terms. For a special case, the temporal model of the simplified CRTL system (SCRTL) is defined as scenarios and graphically represented in terms of a directed, partially weighted or attributed, simple graph. Therefore, the problem of matching temporal scenarios is transformed into conventional graph matching. For the scenario graph matching problem, the traditional eigen-decomposition graph matching algorithm and the symmetric polynomial transform graph matching algorithm are critically examined and improved as two new algorithms named meta-basis graph matching algorithm and sort based graph matching algorithm respectively, where the meta-basis graph matching algorithm works better for 0-1 matrices while the sort based graph matching algorithm is more suitable for continuous real matrices. Another important contribution is the node similarity graph matching framework proposed in this thesis, based on which the node similarity graph matching algorithms can be defined, analyzed and extended uniformly. We prove that that all these node similarity graph matching algorithms fail to work for matching circles

Global and local graph modifiers

5th Workshop on Methods for Modalities 2007, Cachan, France, 29-30 novembre 2007International audienceWe dene two modal logics that allow to reason about modications of graphs. Both have a universal modal operator. The rst one only involves global modications (of some state label, or of some edge label) everywhere in the graph. The second one also allows for modications that are local to states. The global version generalizes logics of public announcements and public assignments, as well as a logic of preference modication introduced by van Benthem et Liu. By means of reduction axioms we show that it is just as expressive as the underlying logic without global modiers. We then show that adding local modiers dramatically increases the power of the logic: the logic of global and local modiers is undecidable. We nally study its relation with hybrid logic with binder