On the apparent failure of the topological theory of phase transitions

The topological theory of phase transitions has its strong point in two theorems proving that, for a wide class of physical systems, phase transitions necessarily stem from topological changes of some submanifolds of configuration space. It has been recently argued that the $2D$ lattice $\phi^4$-model provides a counterexample that falsifies this theory. It is here shown that this is not the case: the phase transition of this model stems from an asymptotic ($N\to\infty$) change of topology of the energy level sets, in spite of the absence of critical points of the potential in correspondence of the transition energy.Comment: 5 pages, 4 figure

Code obfuscation against abstraction refinement attacks

Code protection technologies require anti reverse engineering transformations to obfuscate programs in such a way that tools and methods for program analysis become ineffective. We introduce the concept of model deformation inducing an effective code obfuscation against attacks performed by abstract model checking. This means complicating the model in such a way a high number of spurious traces are generated in any formal verification of the property to disclose about the system under attack.We transform the program model in order to make the removal of spurious counterexamples by abstraction refinement maximally inefficient. Because our approach is intended to defeat the fundamental abstraction refinement strategy, we are independent from the specific attack carried out by abstract model checking. A measure of the quality of the obfuscation obtained by model deformation is given together with a corresponding best obfuscation strategy for abstract model checking based on partition refinement

On the origin of Phase Transitions in the absence of Symmetry-Breaking

In this paper we investigate the Hamiltonian dynamics of a lattice gauge model in three spatial dimension. Our model Hamiltonian is defined on the basis of a continuum version of a duality transformation of a three dimensional Ising model. The system so obtained undergoes a thermodynamic phase transition in the absence of symmetry-breaking. Besides the well known use of quantities like the Wilson loop we show how else the phase transition in such a kind of models can be detected. It is found that the first order phase transition undergone by this model is characterised according to an Ehrenfest-like classification of phase transitions applied to the configurational entropy. On the basis of the topological theory of phase transitions, it is discussed why the seemingly divergent behaviour of the third derivative of configurational entropy can be considered as the "shadow" of some suitable topological transition of certain submanifolds of configuration space.Comment: 31 pages, 9 figure

Geometrical aspects in the analysis of microcanonical phase-transitions

In the present work, we discuss how the functional form of thermodynamic observables can be deduced from the geometric properties of subsets of phase space. The geometric quantities taken into account are mainly extrinsic curvatures of the energy level sets of the Hamiltonian of a system under investigation. In particular, it turns out that peculiar behaviours of thermodynamic observables at a phase transition point are rooted in more fundamental changes of the geometry of the energy level sets in phase space. More specifically, we discuss how microcanonical and geometrical descriptions of phase-transitions are shaped in the special case of $\phi^4$ models with either nearest-neighbours and mean-field interactions

Persistent Homology analysis of Phase Transitions

Persistent homology analysis, a recently developed computational method in algebraic topology, is applied to the study of the phase transitions undergone by the so-called XY-mean field model and by the phi^4 lattice model, respectively. For both models the relationship between phase transitions and the topological properties of certain submanifolds of configuration space are exactly known. It turns out that these a-priori known facts are clearly retrieved by persistent homology analysis of dynamically sampled submanifolds of configuration space.Comment: 10 pages; 10 figure

Generalized contexts for reaction systems: definition and study of dynamic causalities

Reaction systems are a qualitative formalism for the modelling of systems of biochemical reactions. In their original formulation, a reaction system executes in an environment (or context) that can supply it with new objects at each evolution step. The context drives the behaviour of a reaction system: it can provide different inputs to the system that can lead to different behaviours. In order to more faithfully deal with open systems, in this paper we propose a more powerful notion of context having not only the capability to provide objects, but also to absorb (or remove) objects at each evolution step. For such reaction systems with generalized context we investigate properties of dynamic causality by revising the previously proposed concept of formula based predictor. A formula based predictor is a Boolean formula characterising all contexts that lead to the production of a certain object after a given number of steps. In this paper, we revise the theory of formula based predictors in order to deal with reaction systems executed in a context of the new kind. As applications, we show an example of interaction between biochemical pathways and a reaction system modelling cell metabolism and respiration

Optimizing transformations for contrastive learning in a differentiable framework

Current contrastive learning methods use random transformations sampled from a large list of transformations, with fixed hyperparameters, to learn invariance from an unannotated database. Following previous works that introduce a small amount of supervision, we propose a framework to find optimal transformations for contrastive learning using a differentiable transformation network. Our method increases performances at low annotated data regime both in supervision accuracy and in convergence speed. In contrast to previous work, no generative model is needed for transformation optimization. Transformed images keep relevant information to solve the supervised task, here classification. Experiments were performed on 34000 2D slices of brain Magnetic Resonance Images and 11200 chest X-ray images. On both datasets, with 10% of labeled data, our model achieves better performances than a fully supervised model with 100% labels.Comment: Accepted at MILLanD workshop (MICCAI