463 research outputs found
A logic of graph conditions extended with paths
In this paper we tackle the problem of extending the logic of nested graph conditions with paths. This means, for instance, that we may state properties about the existence of paths between some given nodes. As a main contribution, a sound and complete tableau method is defined for reasoning about this kind of properties.Peer ReviewedPostprint (published version
On Mumford's construction of degenerating abelian varieties
We prove that a 1-dimnl family of abelian varieties with an ample sheaf
defining principal polarization can be canonically compactified (after a finite
base change) to a projective family with an ample sheaf. We show that the
central fiber (P,L), which we call an SQAV, has a canonical Cartier theta
divisor. We give a combinatorial definition for SQAVs and describe their
geometrical properties, in particular compute cohomologies of L^n, n\ge0.Comment: Final version, to appear in Tohoku Math.
Computational category-theoretic rewriting
We demonstrate how category theory provides specifications that can
efficiently be implemented via imperative algorithms and apply this to the
field of graph rewriting. By examples, we show how this paradigm of software
development makes it easy to quickly write correct and performant code. We
provide a modern implementation of graph rewriting techniques at the level of
abstraction of finitely-presented C-sets and clarify the connections between
C-sets and the typed graphs supported in existing rewriting software. We
emphasize that our open-source library is extensible: by taking new categorical
constructions (such as slice categories, structured cospans, and distributed
graphs) and relating their limits and colimits to those of their underlying
categories, users inherit efficient algorithms for pushout complements and
(final) pullback complements. This allows one to perform double-, single-, and
sesqui-pushout rewriting over a broad class of data structures
Automated reasoning for attributed graph properties
Graphs are ubiquitous in computer science. Moreover, in various application fields, graphs are equipped with attributes to express additional information such as names of entities or weights of relationships. Due to the pervasiveness of attributed graphs, it is highly important to have the means to express properties on attributed graphs to strengthen modeling capabilities and to enable analysis. Firstly, we introduce a new logic of attributed graph properties, where the graph part and attribution part are neatly separated. The graph part is equivalent to first-order logic on graphs as introduced by Courcelle. It employs graph morphisms to allow the specification of complex graph patterns. The attribution part is added to this graph part by reverting to the symbolic approach to graph attribution, where attributes are represented symbolically by variables whose possible values are specified by a set of constraints making use of algebraic specifications. Secondly, we extend our refutationally complete tableau-based reasoning method as well as our symbolic model generation approach for graph properties to attributed graph properties. Due to the new logic mentioned above, neatly separating the graph and attribution parts, and the categorical constructions employed only on a more abstract level, we can leave the graph part of the algorithms seemingly unchanged. For the integration of the attribution part into the algorithms, we use an oracle, allowing for flexible adoption of different available SMT solvers in the actual implementation. Finally, our automated reasoning approach for attributed graph properties is implemented in the tool AutoGraph integrating in particular the SMT solver Z3 for the attribute part of the properties. We motivate and illustrate our work with a particular application scenario on graph database query validation.Peer ReviewedPostprint (author's final draft
Characterising Modal Formulas with Examples
We study the existence of finite characterisations for modal formulas. A
finite characterisation of a modal formula is a finite collection of
positive and negative examples that distinguishes from every other,
non-equivalent modal formula, where an example is a finite pointed Kripke
structure. This definition can be restricted to specific frame classes and to
fragments of the modal language: a modal fragment admits finite
characterisations with respect to a frame class if every formula
has a finite characterisation with respect to consting of
examples that are based on frames in . Finite characterisations are useful
for illustration, interactive specification, and debugging of formal
specifications, and their existence is a precondition for exact learnability
with membership queries. We show that the full modal language admits finite
characterisations with respect to a frame class only when the modal logic
of is locally tabular. We then study which modal fragments, freely
generated by some set of connectives, admit finite characterisations. Our main
result is that the positive modal language without the truth-constants
and admits finite characterisations w.r.t. the class of all frames. This
result is essentially optimal: finite characterizability fails when the
language is extended with the truth constant or with all but very
limited forms of negation.Comment: Expanded version of material from Raoul Koudijs's MSc thesis (2022
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