Eulerian digraphs and toric Calabi-Yau varieties

We investigate the structure of a simple class of affine toric Calabi-Yau varieties that are defined from quiver representations based on finite eulerian directed graphs (digraphs). The vanishing first Chern class of these varieties just follows from the characterisation of eulerian digraphs as being connected with all vertices balanced. Some structure theory is used to show how any eulerian digraph can be generated by iterating combinations of just a few canonical graph-theoretic moves. We describe the effect of each of these moves on the lattice polytopes which encode the toric Calabi-Yau varieties and illustrate the construction in several examples. We comment on physical applications of the construction in the context of moduli spaces for superconformal gauged linear sigma models.Comment: 27 pages, 8 figure

Partial Quantifier Elimination By Certificate Clauses

We study partial quantifier elimination (PQE) for propositional CNF formulas. In contrast to full quantifier elimination, in PQE, one can limit the set of clauses taken out of the scope of quantifiers to a small subset of target clauses. The appeal of PQE is twofold. First, PQE can be dramatically simpler than full quantifier elimination. Second, it provides a language for performing incremental computations. Many verification problems (e.g. equivalence checking and model checking) are inherently incremental and so can be solved in terms of PQE. Our approach is based on deriving clauses depending only on unquantified variables that make the target clauses $\mathit{redundant}$. Proving redundancy of a target clause is done by construction of a ``certificate'' clause implying the former. We describe a PQE algorithm called $\mathit{START}$ that employs the approach above. We apply $\mathit{START}$ to generating properties of a design implementation that are not implied by specification. The existence of an $\mathit{unwanted}$ property means that this implementation is buggy. Our experiments with HWMCC-13 benchmarks suggest that $\mathit{START}$ can be used for generating properties of real-life designs

Minimization for Generalized Boolean Formulas

The minimization problem for propositional formulas is an important optimization problem in the second level of the polynomial hierarchy. In general, the problem is Sigma-2-complete under Turing reductions, but restricted versions are tractable. We study the complexity of minimization for formulas in two established frameworks for restricted propositional logic: The Post framework allowing arbitrarily nested formulas over a set of Boolean connectors, and the constraint setting, allowing generalizations of CNF formulas. In the Post case, we obtain a dichotomy result: Minimization is solvable in polynomial time or coNP-hard. This result also applies to Boolean circuits. For CNF formulas, we obtain new minimization algorithms for a large class of formulas, and give strong evidence that we have covered all polynomial-time cases

On the Structure of Covers of Sofic Shifts

A canonical cover generalizing the left Fischer cover to arbitrary sofic shifts is introduced and used to prove that the left Krieger cover and the past set cover of a sofic shift can be divided into natural layers. These results are used to find the range of a flow-invariant and to investigate the ideal structure of the universal C*-algebra associated to a sofic shift space.Comment: To appear in Documenta Mathematica. Section 2 has been shortened. Three sections concerning the layered structure of the left Krieger cover and the past set cover have been merged and rewritten. Non-essential examples have been omitted. 21 pages, 8 figure

Provenance Circuits for Trees and Treelike Instances (Extended Version)

Query evaluation in monadic second-order logic (MSO) is tractable on trees and treelike instances, even though it is hard for arbitrary instances. This tractability result has been extended to several tasks related to query evaluation, such as counting query results [3] or performing query evaluation on probabilistic trees [10]. These are two examples of the more general problem of computing augmented query output, that is referred to as provenance. This article presents a provenance framework for trees and treelike instances, by describing a linear-time construction of a circuit provenance representation for MSO queries. We show how this provenance can be connected to the usual definitions of semiring provenance on relational instances [20], even though we compute it in an unusual way, using tree automata; we do so via intrinsic definitions of provenance for general semirings, independent of the operational details of query evaluation. We show applications of this provenance to capture existing counting and probabilistic results on trees and treelike instances, and give novel consequences for probability evaluation.Comment: 48 pages. Presented at ICALP'1

A Limit Theorem for Stochastically Decaying Partitions at the Edge

In this paper, we study the asymptotic behavior of the first, second, and so on rows of stochastically decaying partitions. We establish that, with appropriate scaling in time and length, the sequence of rows converges to the Airy$_2$ line ensemble. This result was first established, in a more general setting, by Borodin and Olshanski, who relied on the determinantal structure of the Poissonized correlation functions. Our argument is based on a different, combinatorial approach, developed by Okounkov. This approach may be useful in other problems in which no determinantal structure is available, and also highlights the similarity between random partitions and random matrices.Comment: 39 pages, 8 figure