Layered and object-based game semantics

Large-scale software verification relies critically on the use of compositional languages, semantic models, specifications, and verification techniques. Recent work on certified abstraction layers synthesizes game semantics, the refinement calculus, and algebraic effects to enable the composition of heterogeneous components into larger certified systems. However, in existing models of certified abstraction layers, compositionality is restricted by the lack of encapsulation of state. In this paper, we present a novel game model for certified abstraction layers where the semantics of layer interfaces and implementations are defined solely based on their observable behaviors. Our key idea is to leverage Reddy's pioneer work on modeling the semantics of imperative languages not as functions on global states but as objects with their observable behaviors. We show that a layer interface can be modeled as an object type (i.e., a layer signature) plus an object strategy. A layer implementation is then essentially a regular map, in the sense of Reddy, from an object with the underlay signature to that with the overlay signature. A layer implementation is certified when its composition with the underlay object strategy implements the overlay object strategy. We also describe an extension that allows for non-determinism in layer interfaces. After formulating layer implementations as regular maps between object spaces, we move to concurrency and design a notion of concurrent object space, where sequential traces may be identified modulo permutation of independent operations. We show how to express protected shared object concurrency, and a ticket lock implementation, in a simple model based on regular maps between concurrent object spaces.

BLOWUPS AND FIBERS OF MORPHISMS

Our object of study is a rational map defined by homogeneous forms $g_{1},\ldots ,g_{n}$, of the same degree $d$, in the homogeneous coordinate ring $R=k[x_{1},\ldots ,x_{s}]$ of $\mathbb{P}_{k}^{s-1}$. Our goal is to relate properties of $\unicode[STIX]{x1D6F9}$, of the homogeneous coordinate ring $A=k[g_{1},\ldots ,g_{n}]$ of the variety parameterized by $\unicode[STIX]{x1D6F9}$, and of the Rees algebra ${\mathcal{R}}(I)$, the bihomogeneous coordinate ring of the graph of $\unicode[STIX]{x1D6F9}$. For a regular map $\unicode[STIX]{x1D6F9}$, for instance, we prove that ${\mathcal{R}}(I)$ satisfies Serre’s condition $R_{i}$, for some $i>0$, if and only if $A$ satisfies $R_{i-1}$ and $\unicode[STIX]{x1D6F9}$ is birational onto its image. Thus, in particular, $\unicode[STIX]{x1D6F9}$ is birational onto its image if and only if ${\mathcal{R}}(I)$ satisfies $R_{1}$. Either condition has implications for the shape of the core, namely, $\text{core}(I)$ is the multiplier ideal of $I^{s}$ and $\text{core}(I)=(x_{1},\ldots ,x_{s})^{sd-s+1}.$ Conversely, for $s=2$, either equality for the core implies birationality. In addition, by means of the generalized rows of the syzygy matrix of $g_{1},\ldots ,g_{n}$, we give an explicit method to reduce the nonbirational case to the birational one when $s=2$.

Bridge Graphs and Deodhar Parametrizations for Positroid Varieties

International audience A <i>parametrization</i> of a positroid variety $\Pi$ of dimension $d$ is a regular map $(\mathbb{C}^{\times})^{d} \rightarrow \Pi$ which is birational onto a dense subset of $\Pi$. There are several remarkable combinatorial constructions which yield parametrizations of positroid varieties. We investigate the relationship between two families of such parametrizations, and prove they are essentially the same. Our first family is defined in terms of Postnikov’s <i>boundary measurement map</i>, and the domain of each parametrization is the space of edge weights of a planar network. We focus on a special class of planar networks called <i>bridge graphs</i>, which have applications to particle physics. Our second family arises from Marsh and Rietsch’s parametrizations of Deodhar components of the flag variety, which are indexed by certain subexpressions of reduced words. Projecting to the Grassmannian gives a family of parametrizations for each positroid variety. We show that each Deodhar parametrization for a positroid variety corresponds to a bridge graph, while each parametrization from a bridge graph agrees with some projected Deodhar parametrization. Soit $\Pi$ une variété positroïde. Nous appellerons <i>paramétrisation</i> toute application régulière $(\mathbb{C}^{\times})^{d} \rightarrow \Pi$ qui est un isomorphisme birégulier sur un sous-ensemble dense de $\Pi$. On sait que plusieurs constructions combinatoires donnent des paramétrisations intéressantes. Le but du présent article est d’investiguer deux familles de telles paramétrisations et de montrer, essentiellement, qu’elles coïncident. La première famille trouve son origine dans la <i>fonction de mesure des bords</i> de Postnikov. Le domaine de chaque paramétrisation est en ce cas-ci l’ensemble de poids des arêtes d’un réseau planaire pondéré. Nous nous concentrons sur une classe particulière de réseaux planaires, les <i>graphes de ponts</i>, ayant des applications à la physique subatomique. La deuxième famille provient des paramétrisations de Marsh et de Rietsch des composantes de Deodhar (indexées par certaines sous-expressions de mots réduits de permutations) de la variété de drapeaux. On obtient alors des paramétrisations de cellules de positroïdes en appliquant la projection à la grassmannienne. Nous montrons que chaque paramétrisation de Deodhar correspond à un graphe de ponts; d’autre part, chaque paramétrisation provenant d’un graphe de ponts s’accorde avec quelque paramétrisation de Deodhar.

COMPLEX CYCLES AS OBSTRUCTIONS ON REAL ALGEBRAIC VARIETIES

LetYbe a compact nonsingular real algebraic variety of positive dimension. Then one can find a compact connected nonsingular real algebraic varietyX, which admits a continuous map intoYthat is not homotopic to any regular map. It is hard to determine the minimum dimension of such a varietyX. In this paper, new upper bounds for dimXare obtained. The main role in the constructions is played by complex algebraic cycles onY.

A NEW CHARACTERIZATION OF WIMAN SURFACES

Let ℳ be a regular map of genus g>1 and X be the underlying Riemann surface. A reflection of ℳ fixes some simple closed curves on X, which we call mirrors. Each mirror passes through at least two of the geometric points (vertices, face-centers and edge-centers) of ℳ. In this paper we study the surfaces which contain mirrors passing through just two geometric points, and show that only Wiman surfaces have this property.

THE SEPARATRIX VALUES OF A PLANAR HOMOCLINIC LOOP

It is well known that the stability of a homoclinic loop for planar vector fields is closely related to the cyclicity of this homoclinic loop. For a planar homoclinic loop consisting of a hyperbolic saddle, the loop values are crucial to the stability. The loop values are divided into two classes: saddle values and separatrix values. The saddle values are related to Dulac map near the saddle, and the separatrix values are related to the regular map near the homoclinic loop. The alternation of these quantities determines the stability of the homoclinic loop. So, it is important to investigate the separatrix values in both theory and for practical applications. For a given planar vector field, we can try to calculate the saddle values by means of dual Liapunov constants or by finding elementary invariants developed by Liu and Li [1990]. The first separatrix value was obtained by Dulac. The second separatrix value was given by Han and Zhu [2007] and by Hu and Feng [2001] independently. The third separatrix value was obtained by Luo and Li [2005] by means of Tkachev's method. In this paper, we shall establish the formulae for the third and fourth separatrix values. As applications, we will give an example with the homoclinic bifurcation of order 9 and prove that the cyclicity of homoclinic loop together with double homoclinic loops is 57.

On holomorphic maps with only fold singularities

Let f : N ≡ P be a holomorphic map between n-dimensional complex manifolds which has only fold singularities. Such a map is called a holomorphic fold map. In the complex 2-jet space J2(n,n;C), let Ω10 denote the space consisting of all 2-jets of regular map germs and fold map germs. In this paper we prove that Ω10 is homotopy equivalent to SU(n + 1). By using this result we prove that if the tangent bundles TN and TP are equipped with SU(n)-structures in addition, then a holomorphic fold map f canonically determines the homotopy class of an SU(n + 1)-bundle map of TN ⊕ θN to TP⊕ θP, where θN and θP are the trivial line bundles.