Succinct Population Protocols for Presburger Arithmetic

International audienceIn [5], Angluin et al. proved that population protocols compute exactly the predicates definable in Presburger arithmetic (PA), the first-order theory of addition. As part of this result, they presented a procedure that translates any formula $ϕ$ of quantifier-free PA with remainder predicates (which has the same expressive power as full PA) into a population protocol with $2 O(poly(|ϕ|))$ states that computes $ϕ$. More precisely, the number of states of the protocol is exponential in both the bit length of the largest coefficient in the formula, and the number of nodes of its syntax tree. In this paper, we prove that every formula $ϕ$ of quantifier-free PA with remainder predicates is computable by a leaderless population protocol with $O(poly(|ϕ|))$ states. Our proof is based on several new constructions, which may be of independent interest. Given a formula $ϕ$ of quantifier-free PA with remainder predicates, a first construction produces a succinct protocol (with $O(|ϕ| 3)$ leaders) that computes ϕ; this completes the work initiated in [8], where we constructed such protocols for a fragment of PA. For large enough inputs, we can get rid of these leaders. If the input is not large enough, then it is small, and we design another construction producing a succinct protocol with one leader that computes $ϕ$. Our last construction gets rid of this leader for small inputs

Automata Tutor v3

Computer science class enrollments have rapidly risen in the past decade. With current class sizes, standard approaches to grading and providing personalized feedback are no longer possible and new techniques become both feasible and necessary. In this paper, we present the third version of Automata Tutor, a tool for helping teachers and students in large courses on automata and formal languages. The second version of Automata Tutor supported automatic grading and feedback for finite-automata constructions and has already been used by thousands of users in dozens of countries. This new version of Automata Tutor supports automated grading and feedback generation for a greatly extended variety of new problems, including problems that ask students to create regular expressions, context-free grammars, pushdown automata and Turing machines corresponding to a given description, and problems about converting between equivalent models - e.g., from regular expressions to nondeterministic finite automata. Moreover, for several problems, this new version also enables teachers and students to automatically generate new problem instances. We also present the results of a survey run on a class of 950 students, which shows very positive results about the usability and usefulness of the tool

Fast and Succinct Population Protocols for Presburger Arithmetic

In their 2006 seminal paper in Distributed Computing, Angluin et al. present a construction that, given any Presburger predicate as input, outputs a leaderless population protocol that decides the predicate. The protocol for a predicate of size $m$ (when expressed as a Boolean combination of threshold and remainder predicates with coefficients in binary) runs in $\mathcal{O}(m \cdot n^2 \log n)$ expected number of interactions, which is almost optimal in $n$. However, the number of states of the protocol is exponential in $m$. Blondin et al. described in STACS 2020 another construction that produces protocols with a polynomial number of states, but exponential expected number of interactions. We present a construction that produces protocols with $\mathcal{O}(m)$ states that run in expected $\mathcal{O}(m^{7} \cdot n^2)$ interactions, optimal in $n$, for all inputs of size $\Omega(m)$. For this we introduce population computers, a carefully crafted generalization of population protocols easier to program, and show that our computers for Presburger predicates can be translated into fast and succinct population protocols.Comment: 52 pages, 4 figures, to be published in SAND 202

Interface-mediated interactions: Entropic forces of curved membranes

Particles embedded in a fluctuating interface experience forces and torques mediated by the deformations and by the thermal fluctuations of the medium. Considering a system of two cylinders bound to a fluid membrane we show that the entropic contribution enhances the curvature-mediated repulsion between the two cylinders. This is contrary to the usual attractive Casimir force in the absence of curvature-mediated interactions. For a large distance between the cylinders, we retrieve the renormalization of the surface tension of a flat membrane due to thermal fluctuations.Comment: 11 pages, 5 figures; final version, as appeared in Phys. Rev.

Conical defects in growing sheets

A growing or shrinking disc will adopt a conical shape, its intrinsic geometry characterized by a surplus angle $se$ at the apex. If growth is slow, the cone will find its equilibrium. Whereas this is trivial if $se <= 0$, the disc can fold into one of a discrete infinite number of states if $se$ is positive. We construct these states in the regime where bending dominates, determine their energies and how stress is distributed in them. For each state a critical value of $se$ is identified beyond which the cone touches itself. Before this occurs, all states are stable; the ground state has two-fold symmetry.Comment: 4 pages, 4 figures, LaTeX, RevTeX style. New version corresponds to the one published in PR

Main phase transition in lipid bilayers: phase coexistence and line tension in a soft, solvent-free, coarse-grained model

We devise a soft, solvent-free, coarse-grained model for lipid bilayer membranes. The non-bonded interactions take the form of a weighted-density functional which allows us to describe the thermodynamics of self-assembly and packing effects of the coarse-grained beads in terms of a density expansion of the equation of state and the weighting functions that regularize the microscopic bead densities, respectively. Identifying the length and energy scales via the bilayer thickness and the thermal energy scale, kT, the model qualitatively reproduces key characteristics (e.g., bending rigidity, area per lipid molecules, and compressibility) of lipid membranes. We employ this model to study the main phase transition between the liquid and the gel phase of the bilayer membrane. We accurately locate the phase coexistence using free energy calculations and also obtain estimates for the bare and the thermodynamic line tension.Comment: 21 pages, 12 figures. Submitted to J. Chem. Phy

Contact lines for fluid surface adhesion

When a fluid surface adheres to a substrate, the location of the contact line adjusts in order to minimize the overall energy. This adhesion balance implies boundary conditions which depend on the characteristic surface deformation energies. We develop a general geometrical framework within which these conditions can be systematically derived. We treat both adhesion to a rigid substrate as well as adhesion between two fluid surfaces, and illustrate our general results for several important Hamiltonians involving both curvature and curvature gradients. Some of these have previously been studied using very different techniques, others are to our knowledge new. What becomes clear in our approach is that, except for capillary phenomena, these boundary conditions are not the manifestation of a local force balance, even if the concept of surface stress is properly generalized. Hamiltonians containing higher order surface derivatives are not just sensitive to boundary translations but also notice changes in slope or even curvature. Both the necessity and the functional form of the corresponding additional contributions follow readily from our treatment.Comment: 8 pages, 2 figures, LaTeX, RevTeX styl