Symmetric Determinantal Representation of Formulas and Weakly Skew Circuits

We deploy algebraic complexity theoretic techniques for constructing symmetric determinantal representations of for00504925mulas and weakly skew circuits. Our representations produce matrices of much smaller dimensions than those given in the convex geometry literature when applied to polynomials having a concise representation (as a sum of monomials, or more generally as an arithmetic formula or a weakly skew circuit). These representations are valid in any field of characteristic different from 2. In characteristic 2 we are led to an almost complete solution to a question of B\"urgisser on the VNP-completeness of the partial permanent. In particular, we show that the partial permanent cannot be VNP-complete in a finite field of characteristic 2 unless the polynomial hierarchy collapses.Comment: To appear in the AMS Contemporary Mathematics volume on Randomization, Relaxation, and Complexity in Polynomial Equation Solving, edited by Gurvits, Pebay, Rojas and Thompso

Of McKay Correspondence, Non-linear Sigma-model and Conformal Field Theory

The ubiquitous ADE classification has induced many proposals of often mysterious correspondences both in mathematics and physics. The mathematics side includes quiver theory and the McKay Correspondence which relates finite group representation theory to Lie algebras as well as crepant resolutions of Gorenstein singularities. On the physics side, we have the graph-theoretic classification of the modular invariants of WZW models, as well as the relation between the string theory nonlinear $\sigma$-models and Landau-Ginzburg orbifolds. We here propose a unification scheme which naturally incorporates all these correspondences of the ADE type in two complex dimensions. An intricate web of inter-relations is constructed, providing a possible guideline to establish new directions of research or alternate pathways to the standing problems in higher dimensions.Comment: 35 pages, 4 figures; minor corrections, comments on toric geometry and references adde

Domain and range for angelic and demonic compositions

We give finite axiomatizations for the varieties generated by representable domain--range algebras when the semigroup operation is interpreted as angelic or demonic composition, respectively

Modelling Software Evolution using Algebraic Graph Rewriting

We show how evolution requests can be formalized using algebraic graph rewriting. In particular, we present a way to convert the UML class diagrams to colored graphs. Since changes in software may effect the relation between the methods of classes, our colored graph representation also employs the relations in UML interaction diagrams. Then, we provide a set of algebraic graph rewrite rules that formalizes the changes that may be caused by an evolution request, using the pushout construction in the category of marked colored graphs

Modular Theory, Non-Commutative Geometry and Quantum Gravity

This paper contains the first written exposition of some ideas (announced in a previous survey) on an approach to quantum gravity based on Tomita-Takesaki modular theory and A. Connes non-commutative geometry aiming at the reconstruction of spectral geometries from an operational formalism of states and categories of observables in a covariant theory. Care has been taken to provide a coverage of the relevant background on modular theory, its applications in non-commutative geometry and physics and to the detailed discussion of the main foundational issues raised by the proposal.Comment: Special Issue "Noncommutative Spaces and Fields

Ten virtues of structured graphs

This paper extends the invited talk by the first author about the virtues of structured graphs. The motivation behind the talk and this paper relies on our experience on the development of ADR, a formal approach for the design of styleconformant, reconfigurable software systems. ADR is based on hierarchical graphs with interfaces and it has been conceived in the attempt of reconciling software architectures and process calculi by means of graphical methods. We have tried to write an ADR agnostic paper where we raise some drawbacks of flat, unstructured graphs for the design and analysis of software systems and we argue that hierarchical, structured graphs can alleviate such drawbacks

Graph models for reachability analysis of concurrent programs

Reachability analysis is an attractive technique for analysis of concurrent programs because it is simple and relatively straightforward to automate, and can be used in conjunction with model-checking procedures to check for application-specific as well as general properties. Several techniques have been proposed differing mainly on the model used; some of these propose the use of flowgraph based models, some others of Petri nets.This paper addresses the question: What essential difference does it make, if any, what sort of finite-state model we extract from program texts for purposes of reachability analysis? How do they differ in expressive power, decision power, or accuracy? Since each is intended to model synchronization structure while abstracting away other features, one would expect them to be roughly equivalent.We confirm that there is no essential semantic difference between the most well known models proposed in the literature by providing algorithms for translation among these models. This implies that the choice of model rests on other factors, including convenience and efficiency.Since combinatorial explosion is the primary impediment to application of reachability analysis, a particular concern in choosing a model is facilitating divide-and-conquer analysis of large programs. Recently, much interest in finite-state verification systems has centered on algebraic theories of concurrency. Yeh and Young have exploited algebraic structure to decompose reachability analysis based on a flowgraph model. The semantic equivalence of graph and Petri net based models suggests that one ought to be able to apply a similar strategy for decomposing Petri nets. We show this is indeed possible through application of category theory