TY - JOUR
AB - Compositional theories are crucial when designing large and complex systems from smaller components. In this work we propose such a theory for synchronous concurrent systems. Our approach follows so-called interface theories, which use game-theoretic interpretations of composition and refinement. These are appropriate for systems with distinct inputs and outputs, and explicit conditions on inputs that must be enforced during composition. Our interfaces model systems that execute in an infinite sequence of synchronous rounds. At each round, a contract must be satisfied. The contract is simply a relation specifying the set of valid input/output pairs. Interfaces can be composed by parallel, serial or feedback composition. A refinement relation between interfaces is defined, and shown to have two main properties: (1) it is preserved by composition, and (2) it is equivalent to substitutability, namely, the ability to replace an interface by another one in any context. Shared refinement and abstraction operators, corresponding to greatest lower and least upper bounds with respect to refinement, are also defined. Input-complete interfaces, that impose no restrictions on inputs, and deterministic interfaces, that produce a unique output for any legal input, are discussed as special cases, and an interesting duality between the two classes is exposed. A number of illustrative examples are provided, as well as algorithms to compute compositions, check refinement, and so on, for finite-state interfaces.
AU - Tripakis, Stavros
AU - Lickly, Ben
AU - Henzinger, Thomas A
AU - Lee, Edward
ID - 3353
IS - 4
JF - ACM Transactions on Programming Languages and Systems (TOPLAS)
TI - A theory of synchronous relational interfaces
VL - 33
ER -
TY - CONF
AB - Byzantine Fault Tolerant (BFT) protocols aim to improve the reliability of distributed systems. They enable systems to tolerate arbitrary failures in a bounded number of nodes. BFT protocols are usually proven correct for certain safety and liveness properties. However, recent studies have shown that the performance of state-of-the-art BFT protocols decreases drastically in the presence of even a single malicious node. This motivates a formal quantitative analysis of BFT protocols to investigate their performance characteristics under different scenarios. We present HyPerf, a new hybrid methodology based on model checking and simulation techniques for evaluating the performance of BFT protocols. We build a transition system corresponding to a BFT protocol and systematically explore the set of behaviors allowed by the protocol. We associate certain timing information with different operations in the protocol, like cryptographic operations and message transmission. After an elaborate state exploration, we use the time information to evaluate the performance characteristics of the protocol using simulation techniques. We integrate our framework in Mace, a tool for building and verifying distributed systems. We evaluate the performance of PBFT using our framework. We describe two different use-cases of our methodology. For the benign operation of the protocol, we use the time information as random variables to compute the probability distribution of the execution times. In the presence of faults, we estimate the worst-case performance of the protocol for various attacks that can be employed by malicious nodes. Our results show the importance of hybrid techniques in systematically analyzing the performance of large-scale systems.
AU - Halalai, Raluca
AU - Henzinger, Thomas A
AU - Singh, Vasu
ID - 3355
TI - Quantitative evaluation of BFT protocols
ER -
TY - CONF
AB - There is recently a significant effort to add quantitative objectives to formal verification and synthesis. We introduce and investigate the extension of temporal logics with quantitative atomic assertions, aiming for a general and flexible framework for quantitative-oriented specifications. In the heart of quantitative objectives lies the accumulation of values along a computation. It is either the accumulated summation, as with the energy objectives, or the accumulated average, as with the mean-payoff objectives. We investigate the extension of temporal logics with the prefix-accumulation assertions Sum(v) ≥ c and Avg(v) ≥ c, where v is a numeric variable of the system, c is a constant rational number, and Sum(v) and Avg(v) denote the accumulated sum and average of the values of v from the beginning of the computation up to the current point of time. We also allow the path-accumulation assertions LimInfAvg(v) ≥ c and LimSupAvg(v) ≥ c, referring to the average value along an entire computation. We study the border of decidability for extensions of various temporal logics. In particular, we show that extending the fragment of CTL that has only the EX, EF, AX, and AG temporal modalities by prefix-accumulation assertions and extending LTL with path-accumulation assertions, result in temporal logics whose model-checking problem is decidable. The extended logics allow to significantly extend the currently known energy and mean-payoff objectives. Moreover, the prefix-accumulation assertions may be refined with "controlled-accumulation", allowing, for example, to specify constraints on the average waiting time between a request and a grant. On the negative side, we show that the fragment we point to is, in a sense, the maximal logic whose extension with prefix-accumulation assertions permits a decidable model-checking procedure. Extending a temporal logic that has the EG or EU modalities, and in particular CTL and LTL, makes the problem undecidable.
AU - Boker, Udi
AU - Chatterjee, Krishnendu
AU - Henzinger, Thomas A
AU - Kupferman, Orna
ID - 3356
TI - Temporal specifications with accumulative values
ER -
TY - CONF
AB - The static scheduling problem often arises as a fundamental problem in real-time systems and grid computing. We consider the problem of statically scheduling a large job expressed as a task graph on a large number of computing nodes, such as a data center. This paper solves the large-scale static scheduling problem using abstraction refinement, a technique commonly used in formal verification to efficiently solve computationally hard problems. A scheduler based on abstraction refinement first attempts to solve the scheduling problem with abstract representations of the job and the computing resources. As abstract representations are generally small, the scheduling can be done reasonably fast. If the obtained schedule does not meet specified quality conditions (like data center utilization or schedule makespan) then the scheduler refines the job and data center abstractions and, again solves the scheduling problem. We develop different schedulers based on abstraction refinement. We implemented these schedulers and used them to schedule task graphs from various computing domains on simulated data centers with realistic topologies. We compared the speed of scheduling and the quality of the produced schedules with our abstraction refinement schedulers against a baseline scheduler that does not use any abstraction. We conclude that abstraction refinement techniques give a significant speed-up compared to traditional static scheduling heuristics, at a reasonable cost in the quality of the produced schedules. We further used our static schedulers in an actual system that we deployed on Amazon EC2 and compared it against the Hadoop dynamic scheduler for large MapReduce jobs. Our experiments indicate that there is great potential for static scheduling techniques.
AU - Henzinger, Thomas A
AU - Singh, Vasu
AU - Wies, Thomas
AU - Zufferey, Damien
ID - 3358
TI - Scheduling large jobs by abstraction refinement
ER -
TY - CONF
AB - A discounted-sum automaton (NDA) is a nondeterministic finite automaton with edge weights, which values a run by the discounted sum of visited edge weights. More precisely, the weight in the i-th position of the run is divided by lambda^i, where the discount factor lambda is a fixed rational number greater than 1. Discounted summation is a common and useful measuring scheme, especially for infinite sequences, which reflects the assumption that earlier weights are more important than later weights. Determinizing automata is often essential, for example, in formal verification, where there are polynomial algorithms for comparing two deterministic NDAs, while the equivalence problem for NDAs is not known to be decidable. Unfortunately, however, discounted-sum automata are, in general, not determinizable: it is currently known that for every rational discount factor 1 < lambda < 2, there is an NDA with lambda (denoted lambda-NDA) that cannot be determinized. We provide positive news, showing that every NDA with an integral factor is determinizable. We also complete the picture by proving that the integers characterize exactly the discount factors that guarantee determinizability: we show that for every non-integral rational factor lambda, there is a nondeterminizable lambda-NDA. Finally, we prove that the class of NDAs with integral discount factors enjoys closure under the algebraic operations min, max, addition, and subtraction, which is not the case for general NDAs nor for deterministic NDAs. This shows that for integral discount factors, the class of NDAs forms an attractive specification formalism in quantitative formal verification. All our results hold equally for automata over finite words and for automata over infinite words.
AU - Boker, Udi
AU - Henzinger, Thomas A
ID - 3360
TI - Determinizing discounted-sum automata
VL - 12
ER -
TY - CONF
AB - In this paper, we investigate the computational complexity of quantitative information flow (QIF) problems. Information-theoretic quantitative relaxations of noninterference (based on Shannon entropy)have been introduced to enable more fine-grained reasoning about programs in situations where limited information flow is acceptable. The QIF bounding problem asks whether the information flow in a given program is bounded by a constant $d$. Our first result is that the QIF bounding problem is PSPACE-complete. The QIF memoryless synthesis problem asks whether it is possible to resolve nondeterministic choices in a given partial program in such a way that in the resulting deterministic program, the quantitative information flow is bounded by a given constant $d$. Our second result is that the QIF memoryless synthesis problem is also EXPTIME-complete. The QIF memoryless synthesis problem generalizes to QIF general synthesis problem which does not impose the memoryless requirement (that is, by allowing the synthesized program to have more variables then the original partial program). Our third result is that the QIF general synthesis problem is EXPTIME-hard.
AU - Cerny, Pavol
AU - Chatterjee, Krishnendu
AU - Henzinger, Thomas A
ID - 3361
TI - The complexity of quantitative information flow problems
ER -
TY - CONF
AB - State-transition systems communicating by shared variables have been the underlying model of choice for applications of model checking. Such formalisms, however, have difficulty with modeling process creation or death and communication reconfigurability. Here, we introduce “dynamic reactive modules” (DRM), a state-transition modeling formalism that supports dynamic reconfiguration and creation/death of processes. The resulting formalism supports two types of variables, data variables and reference variables. Reference variables enable changing the connectivity between processes and referring to instances of processes. We show how this new formalism supports parallel composition and refinement through trace containment. DRM provide a natural language for modeling (and ultimately reasoning about) biological systems and multiple threads communicating through shared variables.
AU - Fisher, Jasmin
AU - Henzinger, Thomas A
AU - Nickovic, Dejan
AU - Piterman, Nir
AU - Singh, Anmol
AU - Vardi, Moshe
ID - 3362
TI - Dynamic reactive modules
VL - 6901
ER -
TY - GEN
AB - We consider probabilistic automata on infinite words with acceptance defined by safety, reachability, Büchi, coBüchi, and limit-average conditions. We consider quantitative and qualitative decision problems. We present extensions and adaptations of proofs for probabilistic finite automata and present a complete characterization of the decidability and undecidability frontier of the quantitative and qualitative decision problems for probabilistic automata on infinite words.
AU - Chatterjee, Krishnendu
AU - Henzinger, Thomas A
AU - Tracol, Mathieu
ID - 3363
TI - The decidability frontier for probabilistic automata on infinite words
ER -
TY - JOUR
AB - Molecular noise, which arises from the randomness of the discrete events in the cell, significantly influences fundamental biological processes. Discrete-state continuous-time stochastic models (CTMC) can be used to describe such effects, but the calculation of the probabilities of certain events is computationally expensive. We present a comparison of two analysis approaches for CTMC. On one hand, we estimate the probabilities of interest using repeated Gillespie simulation and determine the statistical accuracy that we obtain. On the other hand, we apply a numerical reachability analysis that approximates the probability distributions of the system at several time instances. We use examples of cellular processes to demonstrate the superiority of the reachability analysis if accurate results are required.
AU - Didier, Frédéric
AU - Henzinger, Thomas A
AU - Mateescu, Maria
AU - Wolf, Verena
ID - 3364
IS - 21
JF - Theoretical Computer Science
TI - Approximation of event probabilities in noisy cellular processes
VL - 412
ER -
TY - CONF
AB - We present the tool Quasy, a quantitative synthesis tool. Quasy takes qualitative and quantitative specifications and automatically constructs a system that satisfies the qualitative specification and optimizes the quantitative specification, if such a system exists. The user can choose between a system that satisfies and optimizes the specifications (a) under all possible environment behaviors or (b) under the most-likely environment behaviors given as a probability distribution on the possible input sequences. Quasy solves these two quantitative synthesis problems by reduction to instances of 2-player games and Markov Decision Processes (MDPs) with quantitative winning objectives. Quasy can also be seen as a game solver for quantitative games. Most notable, it can solve lexicographic mean-payoff games with 2 players, MDPs with mean-payoff objectives, and ergodic MDPs with mean-payoff parity objectives.
AU - Chatterjee, Krishnendu
AU - Henzinger, Thomas A
AU - Jobstmann, Barbara
AU - Singh, Rohit
ID - 3365
TI - QUASY: quantitative synthesis tool
VL - 6605
ER -
TY - CONF
AB - We present an algorithmic method for the quantitative, performance-aware synthesis of concurrent programs. The input consists of a nondeterministic partial program and of a parametric performance model. The nondeterminism allows the programmer to omit which (if any) synchronization construct is used at a particular program location. The performance model, specified as a weighted automaton, can capture system architectures by assigning different costs to actions such as locking, context switching, and memory and cache accesses. The quantitative synthesis problem is to automatically resolve the nondeterminism of the partial program so that both correctness is guaranteed and performance is optimal. As is standard for shared memory concurrency, correctness is formalized "specification free", in particular as race freedom or deadlock freedom. For worst-case (average-case) performance, we show that the problem can be reduced to 2-player graph games (with probabilistic transitions) with quantitative objectives. While we show, using game-theoretic methods, that the synthesis problem is Nexp-complete, we present an algorithmic method and an implementation that works efficiently for concurrent programs and performance models of practical interest. We have implemented a prototype tool and used it to synthesize finite-state concurrent programs that exhibit different programming patterns, for several performance models representing different architectures.
AU - Cerny, Pavol
AU - Chatterjee, Krishnendu
AU - Henzinger, Thomas A
AU - Radhakrishna, Arjun
AU - Singh, Rohit
ED - Gopalakrishnan, Ganesh
ED - Qadeer, Shaz
ID - 3366
TI - Quantitative synthesis for concurrent programs
VL - 6806
ER -
TY - JOUR
AB - Tissue surface tension (TST) is an important mechanical property influencing cell sorting and tissue envelopment. The study by Manning et al. (1) reported on a mathematical model describing TST on the basis of the balance between adhesive and tensile properties of the constituent cells. The model predicts that, in high-adhesion cell aggregates, surface cells will be stretched to maintain the same area of cell–cell contact as interior bulk cells, resulting in an elongated and flattened cell shape. The authors (1) observed flat and elongated cells at the surface of high-adhesion zebrafish germ-layer explants, which they argue are undifferentiated stretched germ-layer progenitor cells, and they use this observation as a validation of their model.
AU - Krens, Gabriel
AU - Möllmert, Stephanie
AU - Heisenberg, Carl-Philipp J
ID - 3368
IS - 3
JF - PNAS
TI - Enveloping cell layer differentiation at the surface of zebrafish germ layer tissue explants
VL - 108
ER -
TY - JOUR
AB - Supertree methods are widely applied and give rise to new conclusions about phylogenies (e.g., Bininda-Emonds et al. 2007). Although several desiderata for supertree methods exist (Wilkinson, Thorley, et al. 2004), only few of them have been studied in greater detail, examples include shape bias (Wilkinson et al. 2005) or pareto properties (Wilkinson et al. 2007). Here I look more closely at two matrix representation methods, matrix representation with compatibility (MRC) and matrix representation with parsimony (MRP). Different null models of random data are studied and the resulting tree shapes are investigated. Thereby I consider unrooted trees and a bias in tree shape is determined by a tree balance measure. The measure for unrooted trees is a modification of a tree balance measure for rooted trees. I observe that depending on the underlying null model of random data, the methods may resolve conflict in favor of more balanced tree shapes. The analyses refer only to trees with the same taxon set, also known as the consensus setting (e.g., Wilkinson et al. 2007), but I will be able to draw conclusions on how to deal with missing data.
AU - Kupczok, Anne
ID - 3370
IS - 2
JF - Systematic Biology
TI - Consequences of different null models on the tree shape bias of supertree methods
VL - 60
ER -
TY - JOUR
AB - The Minisymposium “Cell Migration and Motility” was attended by approximately 500 visitors and covered a broad range of questions in the field using diverse model systems. Topics comprised actin dynamics, cell polarity, force transduction, signal transduction, bar- rier transmigration, and chemotactic guidance.
AU - Sixt, Michael K
AU - Parent, Carole
ID - 3371
IS - 6
JF - Molecular Biology and Evolution
TI - Cells on the move in Philadelphia
VL - 22
ER -
TY - JOUR
AB - Nowak et al.1 argue that inclusive fitness theory has been of little value in explaining the natural world, and that it has led to negligible progress in explaining the evolution of eusociality. However, we believe that their arguments are based upon a misunderstanding of evolutionary theory and a misrepresentation of the empirical literature. We will focus our comments on three general issues.
AU - Abbot, Patrick
AU - Abe, Jun
AU - Alcock, John
AU - Alizon, Samuel
AU - Alpedrinha, Joao
AU - Andersson, Malte
AU - Andre, Jean
AU - Van Baalen, Minus
AU - Balloux, Francois
AU - Balshine, Sigal
AU - Barton, Nicholas H
AU - Beukeboom, Leo
AU - Biernaskie, Jay
AU - Bilde, Trine
AU - Borgia, Gerald
AU - Breed, Michael
AU - Brown, Sam
AU - Bshary, Redouan
AU - Buckling, Angus
AU - Burley, Nancy
AU - Burton Chellew, Max
AU - Cant, Michael
AU - Chapuisat, Michel
AU - Charnov, Eric
AU - Clutton Brock, Tim
AU - Cockburn, Andrew
AU - Cole, Blaine
AU - Colegrave, Nick
AU - Cosmides, Leda
AU - Couzin, Iain
AU - Coyne, Jerry
AU - Creel, Scott
AU - Crespi, Bernard
AU - Curry, Robert
AU - Dall, Sasha
AU - Day, Troy
AU - Dickinson, Janis
AU - Dugatkin, Lee
AU - El Mouden, Claire
AU - Emlen, Stephen
AU - Evans, Jay
AU - Ferriere, Regis
AU - Field, Jeremy
AU - Foitzik, Susanne
AU - Foster, Kevin
AU - Foster, William
AU - Fox, Charles
AU - Gadau, Juergen
AU - Gandon, Sylvain
AU - Gardner, Andy
AU - Gardner, Michael
AU - Getty, Thomas
AU - Goodisman, Michael
AU - Grafen, Alan
AU - Grosberg, Rick
AU - Grozinger, Christina
AU - Gouyon, Pierre
AU - Gwynne, Darryl
AU - Harvey, Paul
AU - Hatchwell, Ben
AU - Heinze, Jürgen
AU - Helantera, Heikki
AU - Helms, Ken
AU - Hill, Kim
AU - Jiricny, Natalie
AU - Johnstone, Rufus
AU - Kacelnik, Alex
AU - Kiers, E Toby
AU - Kokko, Hanna
AU - Komdeur, Jan
AU - Korb, Judith
AU - Kronauer, Daniel
AU - Kümmerli, Rolf
AU - Lehmann, Laurent
AU - Linksvayer, Timothy
AU - Lion, Sébastien
AU - Lyon, Bruce
AU - Marshall, James
AU - Mcelreath, Richard
AU - Michalakis, Yannis
AU - Michod, Richard
AU - Mock, Douglas
AU - Monnin, Thibaud
AU - Montgomerie, Robert
AU - Moore, Allen
AU - Mueller, Ulrich
AU - Noë, Ronald
AU - Okasha, Samir
AU - Pamilo, Pekka
AU - Parker, Geoff
AU - Pedersen, Jes
AU - Pen, Ido
AU - Pfennig, David
AU - Queller, David
AU - Rankin, Daniel
AU - Reece, Sarah
AU - Reeve, Hudson
AU - Reuter, Max
AU - Roberts, Gilbert
AU - Robson, Simon
AU - Roze, Denis
AU - Rousset, Francois
AU - Rueppell, Olav
AU - Sachs, Joel
AU - Santorelli, Lorenzo
AU - Schmid Hempel, Paul
AU - Schwarz, Michael
AU - Scott Phillips, Tom
AU - Shellmann Sherman, Janet
AU - Sherman, Paul
AU - Shuker, David
AU - Smith, Jeff
AU - Spagna, Joseph
AU - Strassmann, Beverly
AU - Suarez, Andrew
AU - Sundström, Liselotte
AU - Taborsky, Michael
AU - Taylor, Peter
AU - Thompson, Graham
AU - Tooby, John
AU - Tsutsui, Neil
AU - Tsuji, Kazuki
AU - Turillazzi, Stefano
AU - Úbeda, Francisco
AU - Vargo, Edward
AU - Voelkl, Bernard
AU - Wenseleers, Tom
AU - West, Stuart
AU - West Eberhard, Mary
AU - Westneat, David
AU - Wiernasz, Diane
AU - Wild, Geoff
AU - Wrangham, Richard
AU - Young, Andrew
AU - Zeh, David
AU - Zeh, Jeanne
AU - Zink, Andrew
ID - 3372
IS - 7339
JF - Nature
TI - Inclusive fitness theory and eusociality
VL - 471
ER -
TY - JOUR
AB - The use of optical traps to measure or apply forces on the molecular level requires a precise knowledge of the trapping force field. Close to the trap center, this field is typically approximated as linear in the displacement of the trapped microsphere. However, applications demanding high forces at low laser intensities can probe the light-microsphere interaction beyond the linear regime. Here, we measured the full nonlinear force and displacement response of an optical trap in two dimensions using a dual-beam optical trap setup with back-focal-plane photodetection. We observed a substantial stiffening of the trap beyond the linear regime that depends on microsphere size, in agreement with Mie theory calculations. Surprisingly, we found that the linear detection range for forces exceeds the one for displacement by far. Our approach allows for a complete calibration of an optical trap.
AU - Jahnel, Marcus
AU - Behrndt, Martin
AU - Jannasch, Anita
AU - Schaeffer, Erik
AU - Grill, Stephan
ID - 3373
IS - 7
JF - Optics Letters
TI - Measuring the complete force field of an optical trap
VL - 36
ER -
TY - JOUR
AB - Genetic regulatory networks enable cells to respond to changes in internal and external conditions by dynamically coordinating their gene expression profiles. Our ability to make quantitative measurements in these biochemical circuits has deepened our understanding of what kinds of computations genetic regulatory networks can perform, and with what reliability. These advances have motivated researchers to look for connections between the architecture and function of genetic regulatory networks. Transmitting information between a network's inputs and outputs has been proposed as one such possible measure of function, relevant in certain biological contexts. Here we summarize recent developments in the application of information theory to gene regulatory networks. We first review basic concepts in information theory necessary for understanding recent work. We then discuss the functional complexity of gene regulation, which arises from the molecular nature of the regulatory interactions. We end by reviewing some experiments that support the view that genetic networks responsible for early development of multicellular organisms might be maximizing transmitted 'positional information'.
AU - Tkacik, Gasper
AU - Walczak, Aleksandra
ID - 3374
IS - 15
JF - Journal of Physics: Condensed Matter
TI - Information transmission in genetic regulatory networks a review
VL - 23
ER -
TY - JOUR
AB - By exploiting an analogy between population genetics and statistical mechanics, we study the evolution of a polygenic trait under stabilizing selection, mutation and genetic drift. This requires us to track only four macroscopic variables, instead of the distribution of all the allele frequencies that influence the trait. These macroscopic variables are the expectations of: the trait mean and its square, the genetic variance, and of a measure of heterozygosity, and are derived from a generating function that is in turn derived by maximizing an entropy measure. These four macroscopics are enough to accurately describe the dynamics of the trait mean and of its genetic variance (and in principle of any other quantity). Unlike previous approaches that were based on an infinite series of moments or cumulants, which had to be truncated arbitrarily, our calculations provide a well-defined approximation procedure. We apply the framework to abrupt and gradual changes in the optimum, as well as to changes in the strength of stabilizing selection. Our approximations are surprisingly accurate, even for systems with as few as five loci. We find that when the effects of drift are included, the expected genetic variance is hardly altered by directional selection, even though it fluctuates in any particular instance. We also find hysteresis, showing that even after averaging over the microscopic variables, the macroscopic trajectories retain a memory of the underlying genetic states.
AU - de Vladar, Harold
AU - Barton, Nicholas H
ID - 3375
IS - 58
JF - Journal of the Royal Society Interface
TI - The statistical mechanics of a polygenic character under stabilizing selection mutation and drift
VL - 8
ER -
TY - JOUR
AB - Regulatory conflicts occur when two signals that individually trigger opposite cellular responses are present simultaneously. Here, we investigate regulatory conflicts in the bacterial response to antibiotic combinations. We use an Escherichia coli promoter-GFP library to study the transcriptional response of many promoters to either additive or antagonistic drug pairs at fine two-dimensional (2D) resolution of drug concentration. Surprisingly, we find that this data set can be characterized as a linear sum of only two principal components. Component one, accounting for over 70% of the response, represents the response to growth inhibition by the drugs. Component two describes how regulatory conflicts are resolved. For the additive drug pair, conflicts are resolved by linearly interpolating the single drug responses, while for the antagonistic drug pair, the growth-limiting drug dominates the response. Importantly, for a given drug pair, the same conflict resolution strategy applies to almost all genes. These results provide a recipe for predicting gene expression responses to antibiotic combinations.
AU - Bollenbach, Mark Tobias
AU - Kishony, Roy
ID - 3376
IS - 4
JF - Molecular Cell
TI - Resolution of gene regulatory conflicts caused by combinations of antibiotics
VL - 42
ER -
TY - JOUR
AB - By definition, transverse intersections are stable under in- finitesimal perturbations. Using persistent homology, we ex- tend this notion to sizeable perturbations. Specifically, we assign to each homology class of the intersection its robust- ness, the magnitude of a perturbation necessary to kill it, and prove that robustness is stable. Among the applications of this result is a stable notion of robustness for fixed points of continuous mappings and a statement of stability for con- tours of smooth mappings.
AU - Edelsbrunner, Herbert
AU - Morozov, Dmitriy
AU - Patel, Amit
ID - 3377
IS - 3
JF - Foundations of Computational Mathematics
TI - Quantifying transversality by measuring the robustness of intersections
VL - 11
ER -