Partition Refinement Research Articles

Efficient Algorithm for Proportional Lumpability and Its Application to Selfish Mining in Public Blockchains

This paper explores the concept of proportional lumpability as an extension of the original definition of lumpability, addressing the challenges posed by the state space explosion problem in computing performance indices for large stochastic models. Lumpability traditionally relies on state aggregation techniques and is applicable to Markov chains demonstrating structural regularity. Proportional lumpability extends this idea, proposing that the transition rates of a Markov chain can be modified by certain factors, resulting in a lumpable new Markov chain. This concept facilitates the derivation of precise performance indices for the original process. This paper establishes the well-defined nature of the problem of computing the coarsest proportional lumpability that refines a given initial partition, ensuring a unique solution exists. Additionally, a polynomial time algorithm is introduced to solve this problem, offering valuable insights into both the concept of proportional lumpability and the broader realm of partition refinement techniques. The effectiveness of proportional lumpability is demonstrated through a case study that consists of designing a model to investigate selfish mining behaviors on public blockchains. This research contributes to a better understanding of efficient approaches for handling large stochastic models and highlights the practical applicability of proportional lumpability in deriving exact performance indices.

Algorithmic Minimization of Uncertain Continuous-Time Markov Chains

The assumption of perfect knowledge of rate parameters in continuous-time Markov chains (CTMCs) is undermined when confronted with reality, where they may be uncertain due to lack of information or because of measurement noise. Here we consider uncertain CTMCs (UCTMCs), where rates are assumed to vary non-deterministically with time from bounded continuous intervals. An uncertain CTMC can be therefore seen as a specific type of Markov decision process for which the analysis is computationally difficult. To tackle this, we develop a theory of minimization which generalizes the notion of lumpability for CTMCs. Our first result is a quantitative and logical characterization of minimization. Specifically, we show that the reduced UCTMC model has a macro-state for each block of a partition of the state space, which preserves value functions and logical formulae whenever rewards are equal within each block. The second result is an efficient minimization algorithm for UCTMCs by means of partition refinement. As application, we show that reductions in a number of CTMCbenchmark models are robust with respect to uncertainties in original rates.

Semi-attention Partition for Occluded Person Re-identification

This paper proposes a Semi-Attention Partition (SAP) method to learn well-aligned part features for occluded person re-identification (re-ID). Currently, the mainstream methods employ either external semantic partition or attention-based partition, and the latter manner is usually better than the former one. Under this background, this paper explores a potential that the weak semantic partition can be a good teacher for the strong attention-based partition. In other words, the attention-based student can substantially surpass its noisy semantic-based teacher, contradicting the common sense that the student usually achieves inferior (or comparable) accuracy. A key to this effect is: the proposed SAP encourages the attention-based partition of the (transformer) student to be partially consistent with the semantic-based teacher partition through knowledge distillation, yielding the so-called semi-attention. Such partial consistency allows the student to have both consistency and reasonable conflict with the noisy teacher. More specifically, on the one hand, the attention is guided by the semantic partition from the teacher. On the other hand, the attention mechanism itself still has some degree of freedom to comply with the inherent similarity between different patches, thus gaining resistance against noisy supervision. Moreover, we integrate a battery of well-engineered designs into SAP to reinforce their cooperation (e.g., multiple forms of teacher-student consistency), as well as to promote reasonable conflict (e.g., mutual absorbing partition refinement and a supervision signal dropout strategy). Experimental results confirm that the transformer student achieves substantial improvement after this semi-attention learning scheme, and produces new state-of-the-art accuracy on several standard re-ID benchmarks.

Lowerbounds for Bisimulation by Partition Refinement

We provide time lower bounds for sequential and parallel algorithms deciding bisimulation on labeled transition systems that use partition refinement. For sequential algorithms this is $\Omega((m \mkern1mu {+} \mkern1mu n ) \mkern-1mu \log \mkern-1mu n)$ and for parallel algorithms this is $\Omega(n)$, where $n$ is the number of states and $m$ is the number of transitions. The lowerbounds are obtained by analysing families of deterministic transition systems, ultimately with two actions in the sequential case, and one action for parallel algorithms. For deterministic transition systems with one action, bisimilarity can be decided sequentially with fundamentally different techniques than partition refinement. In particular, Paige, Tarjan, and Bonic give a linear algorithm for this specific situation. We show, exploiting the concept of an oracle, that this approach is not of help to develop a faster generic algorithm for deciding bisimilarity. For parallel algorithms there is a similar situation where these techniques may be applied, too.

Coalgebras for Bisimulation of Weighted Automata over Semirings

Weighted automata are a generalization of nondeterministic automata that associate a weight drawn from a semiring $K$ with every transition and every state. Their behaviours can be formalized either as weighted language equivalence or weighted bisimulation. In this paper we explore the properties of weighted automata in the framework of coalgebras over (i) the category $\mathsf{SMod}$ of semimodules over a semiring $K$ and $K$-linear maps, and (ii) the category $\mathsf{Set}$ of sets and maps. We show that the behavioural equivalences defined by the corresponding final coalgebras in these two cases characterize weighted language equivalence and weighted bisimulation, respectively. These results extend earlier work by Bonchi et al. using the category $\mathsf{Vect}$ of vector spaces and linear maps as the underlying model for weighted automata with weights drawn from a field $K$. The key step in our work is generalizing the notions of linear relation and linear bisimulation of Boreale from vector spaces to semimodules using the concept of the kernel of a $K$-linear map in the sense of universal algebra. We also provide an abstract procedure for forward partition refinement for computing weighted language equivalence. Since for weighted automata defined over semirings the problem is undecidable in general, it is guaranteed to halt only in special cases. We provide sufficient conditions for the termination of our procedure. Although the results are similar to those of Bonchi et al., many of our proofs are new, especially those about the coalgebra in $\mathsf{SMod}$ characterizing weighted language equivalence.

Adaptive piecewise Poly-Sinc methods for function approximation

We propose a new method of adaptive piecewise function approximation based on Sinc points. The adaptive method is a Lagrange interpolation scheme which utilizes Poly-Sinc interpolation to reach a preset level of accuracy for the approximation. We prove the exponential convergence of our algorithm. We derive an a priori error estimate for our adaptive method and show its exponential convergence. We use a statistical approach for partition refinement. The adaptive greedy piecewise Poly-Sinc algorithm is validated on continuous, singular, and piecewise functions. We demonstrate that Runge's phenomenon can be handled within a given accuracy goal.

Quasilinear-time Computation of Generic Modal Witnesses for Behavioural Inequivalence

We provide a generic algorithm for constructing formulae that distinguish behaviourally inequivalent states in systems of various transition types such as nondeterministic, probabilistic or weighted; genericity over the transition type is achieved by working with coalgebras for a set functor in the paradigm of universal coalgebra. For every behavioural equivalence class in a given system, we construct a formula which holds precisely at the states in that class. The algorithm instantiates to deterministic finite automata, transition systems, labelled Markov chains, and systems of many other types. The ambient logic is a modal logic featuring modalities that are generically extracted from the functor; these modalities can be systematically translated into custom sets of modalities in a postprocessing step. The new algorithm builds on an existing coalgebraic partition refinement algorithm. It runs in time O((m+n) log n) on systems with n states and m transitions, and the same asymptotic bound applies to the dag size of the formulae it constructs. This improves the bounds on run time and formula size compared to previous algorithms even for previously known specific instances, viz. transition systems and Markov chains; in particular, the best previous bound for transition systems was O(mn).

Skeleton Abstraction for Universal Temporal Properties

Uniform coloured Petri nets can be abstracted to their skeleton, the place/transition net that simply turns the coloured tokens into black tokens. A coloured net and its skeleton are related by a net morphism [1, 2]. For the application of the skeleton as an abstraction method in the model checking process, we need to establish a simulation relation [3] between the state spaces of the two nets. Then, universal temporal properties (properties of the ACTL* logic) are preserved. The abstraction relation induced by a net morphism is not necessarily a simulation relation, due to a subtle issue related to deadlocks [4]. We discuss several situations where the abstraction relation induced by a net morphism is as well a simulation relation, thus preserving ACTL* properties. We further propose a partition refinement algorithm for folding a place/transition net into a coloured net. This way, skeleton abstraction becomes available for models given as place/transition nets. Experiments demonstrate the capabilities of the proposed technology. Using skeleton abstraction, we are capable of solving problems that have not been solved before in the Model Checking Contest [5].

Adaptive Piecewise Poly-Sinc Methods for Ordinary Differential Equations

We propose a new method of adaptive piecewise approximation based on Sinc points for ordinary differential equations. The adaptive method is a piecewise collocation method which utilizes Poly-Sinc interpolation to reach a preset level of accuracy for the approximation. Our work extends the adaptive piecewise Poly-Sinc method to function approximation, for which we derived an a priori error estimate for our adaptive method and showed its exponential convergence in the number of iterations. In this work, we show the exponential convergence in the number of iterations of the a priori error estimate obtained from the piecewise collocation method, provided that a good estimate of the exact solution of the ordinary differential equation at the Sinc points exists. We use a statistical approach for partition refinement. The adaptive greedy piecewise Poly-Sinc algorithm is validated on regular and stiff ordinary differential equations.

Morphisms and Minimisation of Weighted Automata

This paper studies the algorithms for the minimisation of weighted automata. It starts with the definition of morphisms — which generalises and unifies the notion of bisimulation to the whole class of weighted automata — and the unicity of a minimal quotient for every automaton, obtained by partition refinement. From a general scheme for the refinement of partitions, two strategies are considered for the computation of the minimal quotient: the Domain Split and the Predecesor Class Split algorithms. They correspond respectivly to the classical Moore and Hopcroft algorithms for the computation of the minimal quotient of deterministic Boolean automata. We show that these two strategies yield algorithms with the same quadratic complexity and we study the cases when the second one can be improved in order to achieve a complexity similar to the one of Hopcroft algorithm.

Abstraction-based synthesis for stochastic systems with omega-regular objectives

This paper studies the synthesis of controllers for discrete-time, continuous state stochastic systems subject to omega-regular specifications using finite-state abstractions. Omega-regular properties allow specifying complex behaviors and encompass, for example, linear temporal logic. First, we present a synthesis algorithm for minimizing or maximizing the probability that a discrete-time switched stochastic system with a finite number of modes satisfies an omega-regular property. Our approach relies on a finite-state abstraction of the underlying dynamics in the form of a Bounded-parameter Markov Decision Process arising from a finite partition of the system’s domain. Such Markovian abstractions allow for a range of probabilities of transition between states for each selected action representing a mode of the original system. Our method is built upon an analysis of the Cartesian product between the abstraction and a Deterministic Rabin Automaton encoding the specification of interest or its complement. Specifically, we show that synthesis can be decomposed into a qualitative problem, where the so-called greatest permanent winning components of the product automaton are created, and a quantitative problem, which requires maximizing the probability of reaching this component in the worst-case instantiation of the transition intervals. Additionally, we propose a quantitative metric for measuring the quality of the designed controller with respect to the continuous abstracted states and devise a specification-guided domain partition refinement heuristic with the objective of reaching a user-defined optimality target. Next, we present a method for computing control policies for stochastic systems with a continuous set of available inputs. In this case, the system is assumed to be affine in input and disturbance, and we derive a technique for solving the qualitative and quantitative problems in the resulting finite-state abstractions of such systems. For this, we introduce a new type of abstractions called Controlled Interval-valued Markov Chains. Specifically, we show that the greatest permanent winning component of such abstractions are found by appropriately partitioning the continuous input space in order to generate a bounded-parameter Markov decision process that accounts for all possible qualitative transitions between the finite set of states. Then, the problem of maximizing the probability of reaching these components is cast as a (possibly non-convex) optimization problem over the continuous set of available inputs. A metric of quality for the synthesized controller and a partition refinement scheme are described for this framework as well. Finally, we present a detailed case study.

On a conservative partition refinement (CPR) method for a class of two-stage stochastic programming problems

On a conservative partition refinement (CPR) method for a class of two-stage stochastic programming problems

Certain linear and weakly linear systems of matrix equations over semirings. Applications in a state reduction of weighted automata

In this paper we study particular linear and weakly linear systems of matrix equations over semirings, with the aim of describing and computing functions as solutions to those systems. Our special attention is devoted to solutions whose ranks are as small as possible. We prove the existence of solutions with the smallest ranks, give their characterizations, and present a method for their computations. Regarding weakly linear systems, the method is based on the well known partition refinement algorithm by Kanellakis and Smolka, adapted to work with weighted labeled transition systems. Moreover, we give a state reduction procedure of weighted automata based on a decomposition of solutions to particular linear and weakly linear systems.

Multilevel Graph Partitioning for Three-Dimensional Discrete Fracture Network Flow Simulations

We present a topology-based method for mesh-partitioning in three-dimensional discrete fracture network (DFN) simulations that takes advantage of the intrinsic multi-level nature of a DFN. DFN models are used to simulate flow and transport through low-permeability fractured media in the subsurface by explicitly representing fractures as discrete entities. The governing equations for flow and transport are numerically integrated on computational meshes generated on the interconnected fracture networks. Modern high-fidelity DFN simulations require high-performance computing on multiple processors where performance and scalability depends partially on obtaining a high-quality partition of the mesh to balance work-loads and minimize communication across all processors. The discrete structure of a DFN naturally lends itself to various graph representations, which can be thought of as coarse-scale representations of the computational mesh. Using this concept, we develop two applications of the multilevel graph partitioning algorithm to partition the mesh of a DFN. In the first, we project a partition of the graph based on the DFN topology onto the mesh of the DFN and in the second, this DFN-based projection is used as the initial condition for further partitioning refinement of the mesh. We compare the performance of these methods with standard multi-level graph partitioning using graph-based metrics (cut, imbalance, partitioning time), computational-based metrics (FLOPS, iterations, solver time), and total run time. The DFN-based and the mesh-based partitioning methods are comparable in terms of the graph-based metrics, but the time required to obtain the partition is several orders of magnitude faster using the DFN-based partitions. The computation-based metrics show comparable performance between both methods so, in combination, the DFN-based partitions are several orders of magnitude faster than the mesh-based partition. Moreover, the method which uses the DFN-partition solution as the initial condition of the mesh partition provided cut and imbalance values that were close to the mesh-based partition but in a fraction of the time. In turn, this hybrid method outperformed both of the other methods in terms of the total run time.

From generic partition refinement to weighted tree automata minimization

Partition refinement is a method for minimizing automata and transition systems of various types. Recently,we have developed a partition refinement algorithm that is generic in the transition type of the given systemand matches the run time of the best known algorithms for many concrete types of systems, e.g. deterministicautomata as well as ordinary, weighted, and probabilistic (labelled) transition systems. Genericity is achieved bymodelling transition types as functors on sets, and systems as coalgebras. In the present work, we refine the runtime analysis of our algorithm to cover additional instances, notably weighted automata and, more generally,weighted tree automata. For weights in a cancellative monoid we match, and for non-cancellative monoids suchas (the additive monoid of) the tropical semiring even substantially improve, the asymptotic run time of the bestknown algorithms. We have implemented our algorithm in a generic tool that is easily instantiated to concretesystem types by implementing a simple refinement interface. Moreover, the algorithm and the tool are modular,and partition refiners for new types of systems are obtained easily by composing pre-implemented basic functors.Experiments show that even for complex system types, the tool is able to handle systems with millions oftransitions.

Finite Characterization of the Coarsest Balanced Coloring of a Network

Balanced colorings of networks correspond to flow-invariant synchrony spaces. It is known that the coarsest balanced coloring is equivalent to nodes having isomorphic infinite input trees, but this condition is not algorithmic. We provide an algorithmic characterization: two nodes have the same color for the coarsest balanced coloring if and only if their [Formula: see text]th input trees are isomorphic, where [Formula: see text] is the number of nodes. Here [Formula: see text] is the best possible. The proof is analogous to that of Leighton’s theorem in graph theory, using the universal cover of the network and the notion of a symbolic adjacency matrix to set up a partition refinement algorithm whose output is the coarsest balanced coloring. The running time of the algorithm is cubic in [Formula: see text].

Characteristic invariants in Hennessy\u2013Milner logic

In this paper, we prove that Hennessy–Milner Logic (HML), despite its structural limitations, is sufficiently expressive to specify an initial property varphi _0 and a characteristic invariant upchi _{_I} for an arbitrary finite-state process P such that varphi _0 wedge mathbf{AG }(upchi _{_I}) is a characteristic formula for P. This means that a process Q, even if infinite state, is bisimulation equivalent to P iff Q models varphi _0 wedge mathbf{AG }(upchi _{_I}). It follows, in particular, that it is sufficient to check an HML formula for each state of a finite-state process to verify that it is bisimulation equivalent to P. In addition, more complex systems such as context-free processes can be checked for bisimulation equivalence with P using corresponding model checking algorithms. Our characteristic invariant is based on so called class-distinguishing formulas that identify bisimulation equivalence classes in P and which are expressed in HML. We extend Kanellakis and Smolka’s partition refinement algorithm for bisimulation checking in order to generate concise class-distinguishing formulas for finite-state processes.

Finding compact proofs for infinite-data parameterised Boolean equation systems

Parameterised Boolean Equation Systems (PBESs) can be used to represent many different kinds of decision problems. Most notably, model checking and equivalence problems can be encoded in a PBES. Traditional techniques to solve PBESs, such as instantiation techniques, cannot deal with PBESs with an infinite data domain. We propose an approach that can solve PBESs with infinite data by computing the bisimulation quotient of the underlying graph structure. Furthermore, we show how this technique can be improved by repeatedly searching for finite proofs. We also apply knowledge of intermediate solutions in an early termination heuristic. Unlike existing approaches, our technique is not restricted to subfragments of PBESs. Compared to similar procedures that operate on behavioural models, our technique is also more general: it is not restricted to model checking with finite action sets. Experimental results show that our ideas work well in practice and support a wider range of models and properties than state-of-the-art techniques. • Parameterised Boolean equation systems (PBES) can encode model checking problems. • A normal form for PBESs facilitates symbolic approaches such as partition refinement. • Symbolic partition refinement for PBESs enables infinite-state model checking. • The setting of PBESs allows several optimisations to the standard algorithm. • Symbolic partition refinement for PBESs is more scalable than existing techniques.

Aggregation-based minimization of finite state automata

We present a minimization algorithm for non-deterministic finite state automata that finds and merges bisimulation-equivalent states. The bisimulation relation is computed through partition aggregation, in contrast to existing algorithms that use partition refinement. The algorithm simultaneously generalises and simplifies an earlier one by Watson and Daciuk for deterministic devices. We show the algorithm to be correct and run in time O left( n^2 r^2 left| varSigma right| right) , where n is the number of states of the input automaton M, r is the maximal out-degree in the transition graph for any combination of state and input symbol, and left| varSigma right| is the size of the input alphabet. The algorithm has a higher time complexity than derivatives of Hopcroft’s partition-refinement algorithm, but represents a promising new solution approach that preserves language equivalence throughout the computation process. Furthermore, since the algorithm essentially computes the maximal model of a logical formula derived from M, optimisation techniques from the field of model checking become applicable.

Cost-Aware Partitioning for Efficient Large Graph Processing in Geo-Distributed Datacenters

Graph processing is an emerging computation model for a wide range of applications and graph partitioning is important for optimizing the cost and performance of graph processing jobs. Recently, many graph applications store their data on geo-distributed datacenters (DCs) to provide services worldwide with low latency. This raises new challenges to existing graph partitioning methods, due to the multi-level heterogeneities in network bandwidth and communication prices in geo-distributed DCs. In this article, we propose an efficient graph partitioning method named Geo-Cut, which takes both the cost and performance objectives into consideration for large graph processing in geo-distributed DCs. Geo-Cut adopts two optimization stages. First, we propose a cost-aware streaming heuristic and utilize the one-pass streaming graph partitioning method to quickly assign edges to different DCs while minimizing inter-DC data communication cost. Second, we propose two partition refinement heuristics which identify the performance bottlenecks of geo-distributed graph processing and refine the partitioning result obtained in the first stage to reduce the inter-DC data transfer time while satisfying the budget constraint. Geo-Cut can be also applied to partition dynamic graphs thanks to its lightweight runtime overhead. We evaluate the effectiveness and efficiency of Geo-Cut using real-world graphs with both real geo-distributed DCs and simulations. Evaluation results show that Geo-Cut can reduce the inter-DC data transfer time by up to 79 percent (42 percent as the median) and reduce the monetary cost by up to 75 percent (26 percent as the median) compared to state-of-the-art graph partitioning methods with a low overhead.