Transition Invariants Research Articles

Neural Closure Certificates

Notions of transition invariants and closure certificates have seen recent use in the formal verification of controlled dynamical systems against \omega-regular properties. Unfortunately, existing approaches face limitations in two directions. First, they require a closed-form mathematical expression representing the model of the system. Such an expression may be difficult to find, too complex to be of any use, or unavailable due to security or privacy constraints. Second, finding such invariants typically rely on optimization techniques such as sum-of-squares (SOS) or satisfiability modulo theory (SMT) solvers. This restricts the classes of systems that need to be formally verified. To address these drawbacks, we introduce a notion of neural closure certificates. We present a data-driven algorithm that trains a neural network to represent a closure certificate. Our approach is formally correct under some mild assumptions, i.e., one is able to formally show that the unknown system satisfies the \omega-regular property of interest if a neural closure certificate can be computed. Finally, we demonstrate the efficacy of our approach with relevant case studies.

Boundedness and liveness enforcement for labeled Petri nets using transition priority

This paper deals with the supervisory control problem of discrete event systems modeled by labeled Petri nets. The system is originally unbounded. First, the solvability of the problem is confirmed. A necessary condition is given and proven for the existence of a feasible priority-based controller based on the notions of liveness and transition invariants. Next, a cyclic behavior graph is constructed, which shows the reachable markings that guarantee the maximum liveness of the system within a given bound vector. Finally, an on-line control strategy is proposed to enforce boundedness and liveness to the given system by appending priority relations to transitions. The dynamic priority relation changes flexibly according to the current state of the system and enforces the system evolving in a bounded and live manner. In addition, numerical examples are studied to verify the validity of the proposed approach that remains the structure of the plant net and is efficient for on-line control.

Holistic View on the Structure of Immune Response: Petri Net Model

The simulation of immune response is a challenging task because quantitative data are scarce. Quantitative theoretical models either focus on specific cell-cell interactions or have to make assumptions about parameters. The broad variation of, e.g., the dimensions and abundance between lymph nodes as well as between individual patients hampers conclusive quantitative modeling. No theoretical model has been established representing a consensus on the set of major cellular processes involved in the immune response. In this paper, we apply the Petri net formalism to construct a semi-quantitative mathematical model of the lymph nodes. The model covers the major cellular processes of immune response and fulfills the formal requirements of Petri net models. The intention is to develop a model taking into account the viewpoints of experienced pathologists and computer scientists in the field of systems biology. In order to verify formal requirements, we discuss invariant properties and apply the asynchronous firing rule of a place/transition net. Twenty-five transition invariants cover the model, and each is assigned to a functional mode of the immune response. In simulations, the Petri net model describes the dynamic modes of the immune response, its adaption to antigens, and its loss of memory.

On anti-occurrence of subsets of transitions in Petri net-based models of complex biological systems

Background and Objective:In the last two decades there can be observed a rapid development of systems biology. The basis of systems methods is a formal model of an analyzed system. It can be created in a language of some branch of mathematics and recently Petri net-based biological models seem to be especially promising since they have a great expressive power. One of the methods of analysis of such models is based on transition invariants. They correspond to some subprocesses which do not change a state of the modeled biological system. During such analysis, a need arose to study the subsets of transitions, what leads to interesting combinatorial problems — which have been considered in theory and practice. Methods & Results:Two problems of anti-occurrence were considered. These problems concern a set of transitions which is not a subset of any of t-invariant supports or is not a subset of t-invariant supports from some collection of such supports. They are defined in a formal way, their computational complexity is analyzed and an exact algorithm is provided for one of them. Conclusions:A comprehensive analysis of complex biological phenomena is challenging. Finding elementary processes that do not affect subprocesses belonging to the entire studied biological system may be necessary for a complete understanding of such a model and it is possible thanks to the proposed algorithm.

Lifted structural invariant analysis of Petri net product lines

Petri nets are commonly used to represent concurrent systems. However, they lack support for modelling and analysing system families, like variants of controllers, different variations of a process model, or the possible configurations of a flexible assembly line.To facilitate modelling potentially large collections of similar systems, in this paper, we enrich Petri nets with variability mechanisms based on product line engineering. Moreover, we present methods for the efficient analysis of the place and transition invariants in all defined versions of a Petri net. Efficiency is achieved by analysing the system family as a whole, instead of analysing each possible net variant separately. For this purpose, we lift the notion of incidence matrix to the product line level, and rely on constraint solving techniques. We present tool support and evaluate the benefits of our techniques on synthetic and realistic examples, achieving in some cases speed-ups of two orders of magnitude with respect to analysing each net variant separately.

Disentangling a complex response in cell reprogramming and probing the Waddington landscape by automatic construction of Petri nets

We analyzed the developmental switch to sporulation of a multinucleate Physarum polycephalum plasmodial cell, a complex response to phytochrome photoreceptor activation. Automatic construction of Petri nets representing finite state machines assembled from trajectories of differential gene expression in single cells revealed alternative, genotype-dependent interconnected developmental routes and identified reversible steps, metastable states, commitment points, and subsequent irreversible steps together with molecular signatures associated with cell fate decision and differentiation. Formation of cyclic transits identified by transition invariants in mutants that are locked in a proliferative state is remarkable considering the view that oncogenic alterations may cause the formation of cancer attractors. We conclude that the Petri net approach is useful to probe the Waddington landscape of cellular reprogramming, to disentangle developmental routes for the reconstruction of the gene regulatory network, and to understand how genetic alterations or physiological conditions reshape the landscape eventually creating new basins of attraction. Unraveling the complexity of pathogenesis, disease progression, drug response or the analysis of attractor landscapes in other complex systems of uncertain structure might be additional fields of application.

IsiKnock: in silico knockouts in signaling pathways.

isiKnock is a new software that automatically conducts in silico knockouts for mathematical models of signaling pathways. The software allows for the prediction of the behavior of biological systems after single or multiple knockout. The implemented algorithm applies transition invariants and the novel concept of Manatee invariants. A knockout matrix visualizes the results. The tool enables the analysis of dependencies, for example, in signal flows from the receptor activation to the cell response at steady state. isiKnock is an open-source tool, freely available at http://www.bioinformatik.uni-frankfurt.de/tools/isiKnock/. It requires at least Java 8 and runs under Microsoft Windows, Linux, and Mac OS.

Modeling the Metabolism of Arabidopsis thaliana: Application of Network Decomposition and Network Reduction in the Context of Petri Nets.

Motivation: Arabidopsis thaliana is a well-established model system for the analysis of the basic physiological and metabolic pathways of plants. Nevertheless, the system is not yet fully understood, although many mechanisms are described, and information for many processes exists. However, the combination and interpretation of the large amount of biological data remain a big challenge, not only because data sets for metabolic paths are still incomplete. Moreover, they are often inconsistent, because they are coming from different experiments of various scales, regarding, for example, accuracy and/or significance. Here, theoretical modeling is powerful to formulate hypotheses for pathways and the dynamics of the metabolism, even if the biological data are incomplete. To develop reliable mathematical models they have to be proven for consistency. This is still a challenging task because many verification techniques fail already for middle-sized models. Consequently, new methods, like decomposition methods or reduction approaches, are developed to circumvent this problem.Methods: We present a new semi-quantitative mathematical model of the metabolism of Arabidopsis thaliana. We used the Petri net formalism to express the complex reaction system in a mathematically unique manner. To verify the model for correctness and consistency we applied concepts of network decomposition and network reduction such as transition invariants, common transition pairs, and invariant transition pairs.Results: We formulated the core metabolism of Arabidopsis thaliana based on recent knowledge from literature, including the Calvin cycle, glycolysis and citric acid cycle, glyoxylate cycle, urea cycle, sucrose synthesis, and the starch metabolism. By applying network decomposition and reduction techniques at steady-state conditions, we suggest a straightforward mathematical modeling process. We demonstrate that potential steady-state pathways exist, which provide the fixed carbon to nearly all parts of the network, especially to the citric acid cycle. There is a close cooperation of important metabolic pathways, e.g., the de novo synthesis of uridine-5-monophosphate, the γ-aminobutyric acid shunt, and the urea cycle. The presented approach extends the established methods for a feasible interpretation of biological network models, in particular of large and complex models.

Quasi-Steady-State Analysis based on Structural Modules and Timed Petri Net Predict System's Dynamics: The Life Cycle of the Insulin Receptor.

The insulin-dependent activation and recycling of the insulin receptor play an essential role in the regulation of the energy metabolism, leading to a special interest for pharmaceutical applications. Thus, the recycling of the insulin receptor has been intensively investigated, experimentally as well as theoretically. We developed a time-resolved, discrete model to describe stochastic dynamics and study the approximation of non-linear dynamics in the context of timed Petri nets. Additionally, using a graph-theoretical approach, we analyzed the structure of the regulatory system and demonstrated the close interrelation of structural network properties with the kinetic behavior. The transition invariants decomposed the model into overlapping subnetworks of various sizes, which represent basic functional modules. Moreover, we computed the quasi-steady states of these subnetworks and demonstrated that they are fundamental to understand the dynamic behavior of the system. The Petri net approach confirms the experimental results of insulin-stimulated degradation of the insulin receptor, which represents a common feature of insulin-resistant, hyperinsulinaemic states.

Petri nets in systems biology

Petri nets are used in many areas. This article discusses the application of Petri nets in systems biology. Using an example from biochemistry, concepts for the automatic decomposition of biochemical systems are introduced. The article focuses on those concepts that fulfill steady-state conditions. Interestingly, all the concepts are based on minimal, semi-positive transition invariants. The article describes, which new definitions for network decomposition can be derived and how they can be interpreted in the context of biology. This is illustrated with the example of the citric acid cycle, for which a new metabolic pathway could be predicted with the help of such an analysis.

A New DES Control Synthesis Approach Based on Structural Model Properties

This work proposes a novel approach for controller synthesis on the shop-floor level for discrete-event systems. The approach is based on a modular modeling formalism and structural properties of models of uncontrolled plant behavior that are designed using this modeling formalism. The approach takes advantage of the transition invariants of the underlying Petri net components of the model and uses the event interconnections of transitions that are part of the transition invariants. The result is the so-called transition invariant graph that represents a new structural property of the modeled system. The synthesis procedure takes models of the uncontrolled plant behavior and formal specifications of the desired cyclic plant behavior under control and forbidden states as well. From these ingredients, the transition invariant graph is computed from which the admissible trajectories are extracted. This is performed by partial reachability analysis. An example taken from a lab-scaled manufacturing system illustrates the methodology and shows the application. The complexity of the synthesis procedure is significantly reduced. That makes it feasible to be used for even larger systems of real industrial scale.

On functional module detection in metabolic networks.

Functional modules of metabolic networks are essential for understanding the metabolism of an organism as a whole. With the vast amount of experimental data and the construction of complex and large-scale, often genome-wide, models, the computer-aided identification of functional modules becomes more and more important. Since steady states play a key role in biology, many methods have been developed in that context, for example, elementary flux modes, extreme pathways, transition invariants and place invariants. Metabolic networks can be studied also from the point of view of graph theory, and algorithms for graph decomposition have been applied for the identification of functional modules. A prominent and currently intensively discussed field of methods in graph theory addresses the Q-modularity. In this paper, we recall known concepts of module detection based on the steady-state assumption, focusing on transition-invariants (elementary modes) and their computation as minimal solutions of systems of Diophantine equations. We present the Fourier-Motzkin algorithm in detail. Afterwards, we introduce the Q-modularity as an example for a useful non-steady-state method and its application to metabolic networks. To illustrate and discuss the concepts of invariants and Q-modularity, we apply a part of the central carbon metabolism in potato tubers (Solanum tuberosum) as running example. The intention of the paper is to give a compact presentation of known steady-state concepts from a graph-theoretical viewpoint in the context of network decomposition and reduction and to introduce the application of Q-modularity to metabolic Petri net models.

Loop summarization using state and transition invariants

This paper presents algorithms for program abstraction based on the principle of loop summarization, which, unlike traditional program approximation approaches (e.g., abstract interpretation), does not employ iterative fixpoint computation, but instead computes symbolic abstract transformers with respect to a set of abstract domains. This allows for an effective exploitation of problem-specific abstract domains for summarization and, as a consequence, the precision of an abstract model may be tailored to specific verification needs. Furthermore, we extend the concept of loop summarization to incorporate relational abstract domains to enable the discovery of transition invariants, which are subsequently used to prove termination of programs. Well-foundedness of the discovered transition invariants is ensured either by a separate decision procedure call or by using abstract domains that are well-founded by construction. We experimentally evaluate several abstract domains related to memory operations to detect buffer overflow problems. Also, our light-weight termination analysis is demonstrated to be effective on a wide range of benchmarks, including OS device drivers.

Reduction techniques for network validation in systems biology

The rapidly increasing amount of experimental biological data enables the development of large and complex, often genome-scale models of molecular systems. The simulation and analysis of these computer models of metabolism, signal transduction, and gene regulation are standard applications in systems biology, but size and complexity of the networks limit the feasibility of many methods. Reduction of networks provides a hierarchical view of complex networks and gives insight knowledge into their coarse-grained structural properties. Although network reduction has been extensively studied in computer science, adaptation and exploration of these concepts are still lacking for the analysis of biochemical reaction systems. Using the Petri net formalism, we describe two local network structures, common transition pairs and minimal transition invariants. We apply these two structural elements for network reduction. The reduction preserves the CTI-property (covered by transition invariants), which is an important feature for completeness of biological models. We demonstrate this concept for a selection of metabolic networks including a benchmark network of Saccharomyces cerevisiae whose straightforward treatment is not yet feasible even on modern supercomputers.

An abstract interpretation framework for termination

Proof, verification and analysis methods for termination all rely on two induction principles: (1) a variant function or induction on data ensuring progress towards the end and (2) some form of induction on the program structure. The abstract interpretation design principle is first illustrated for the design of new forward and backward proof, verification and analysis methods for safety. The safety collecting semantics defining the strongest safety property of programs is first expressed in a constructive fixpoint form. Safety proof and checking/verification methods then immediately follow by fixpoint induction. Static analysis of abstract safety properties such as invariance are constructively designed by fixpoint abstraction (or approximation) to (automatically) infer safety properties. So far, no such clear design principle did exist for termination so that the existing approaches are scattered and largely not comparable with each other. For (1), we show that this design principle applies equally well to potential and definite termination. The trace-based termination collecting semantics is given a fixpoint definition. Its abstraction yields a fixpoint definition of the best variant function. By further abstraction of this best variant function, we derive the Floyd/Turing termination proof method as well as new static analysis methods to effectively compute approximations of this best variant function. For (2), we introduce a generalization of the syntactic notion of struc- tural induction (as found in Hoare logic) into a semantic structural induction based on the new semantic concept of inductive trace cover covering execution traces by segments, a new basis for formulating program properties. Its abstractions allow for generalized recursive proof, verification and static analysis methods by induction on both program structure, control, and data. Examples of particular instances include Floyd's handling of loop cutpoints as well as nested loops, Burstall's intermittent assertion total correctness proof method, and Podelski-Rybalchenko transition invariants.

Exhaustive Analysis of the Modular Structure of the Spliceosomal Assembly Network: A Petri Net Approach

Spliceosomes are macro-complexes involving hundreds of proteins with many functional interactions. Spliceosome assembly belongs to the key processes that enable splicing of mRNA and modulate alternative splicing. A detailed list of factors involved in spliceosomal reactions has been assorted over the past decade, but, their functional interplay is often unknown and most of the present biological models cover only parts of the complete assembly process. It is a challenging task to build a computational model that integrates dispersed knowledge and combines a multitude of reaction schemes proposed earlier.Because for most reactions involved in spliceosome assembly kinetic parameters are not available, we propose a discrete modeling using Petri nets, through which we are enabled to get insights into the system's behavior via computation of structural and dynamic properties. In this paper, we compile and examine reactions from experimental reports that contribute to a functional spliceosome. All these reactions form a network, which describes the inventory and conditions necessary to perform the splicing process. The analysis is mainly based on system invariants. Transition invariants (T-invariants) can be interpreted as signaling routes through the network. Due to the huge number of T-invariants that arise with increasing network size and complexity, maximal common transition sets (MCTS) and T-clusters were used for further analysis. Additionally, we introduce a false color map representation, which allows a quick survey of network modules and the visual detection of single reactions or reaction sequences, which participate in more than one signaling route. We designed a structured model of spliceosome assembly, which combines the demands on a platform that i) can display involved factors and concurrent processes, ii) offers the possibility to run computational methods for knowledge extraction, and iii) is successively extendable as new insights into spliceosome function are reported by experimental reports. The network consists of 161 transitions (reactions) and 140 places (reactants). All reactions are part of at least one of the 71 T-invariants. These T-invariants define pathways, which are in good agreement with the current knowledge and known hypotheses on reaction sequences during spliceosome assembly, hence contributing to a functional spliceosome. We demonstrate that present knowledge, in particular of the initial part of the assembly process, describes parallelism and interaction of signaling routes, which indicate functional redundancy and reflect the dependency of spliceosome assembly initiation on different cellular conditions. The complexity of the network is further increased by two switches, which introduce alternative routes during A-complex formation in early spliceosome assembly and upon transition from the B-complex to the C-complex. By compiling known reactions into a complete network, the combinatorial nature of invariant computation leads to pathways that have previously not been described as connected routes, although their constituents were known. T-clusters divide the network into modules, which we interpret as building blocks in spliceosome maturation. We conclude that Petri net representations of large biological networks and system invariants, are well-suited as a means for validating the integration of experimental knowledge into a consistent model. Based on this network model, the design of further experiments is facilitated.

Applying infinite state model checking and other analysis techniques to tabular requirements specifications of safety-critical systems

Although it is most often applied to finite state models, in recent years, symbolic model checking has been extended to infinite state models using symbolic representations that encode infinite sets. This paper investigates the application of an infinite state symbolic model checker called Action Language Verifier (ALV) to formal requirements specifications of safety-critical systems represented in the SCR (Software Cost Reduction) tabular notation. After reviewing the SCR method and tools, the Action Language for representing state machine models, and the ALV infinite state model checker, the paper presents experimental results of formally analyzing two SCR specifications using ALV. The application of ALV to verify or falsify (by generating counterexample behaviors) the state and transition invariants of SCR specifications and to check Disjointness and Coverage properties is described. The results of formal analysis with ALV are then compared with the results of formal analysis using techniques that have been integrated into the SCR toolset. Based on the experimental results, strengths and weaknesses of infinite state model checking with respect to other formal analysis approaches such as explicit and finite state model checking and theorem proving are discussed.

Non-Fourier motion analysis.

It has been realized for some time that the visual system performs at least two general sorts of motion processing. First-order motion processing applies some variant of standard motion analysis (i.e. spatiotemporal Fourier energy analysis) directly to stimulus luminance, whereas second-order motion processing applies standard motion analysis to one or another grossly non-linear transformation of stimulus luminance. We have developed a method for disentangling the different sorts of mechanisms that may operate in human vision to detect second-order motion. This method hinges on an empirical condition called transition invariance that may or may not be satisfied by a family psi of textures. Any failure of this condition indicates that more than one mechanism is involved in detecting the motion of stimuli composed of the textures in psi. We have shown that the family of sinusoidal gratings oriented orthogonally to the direction of motion and varying in contrast and spatial frequency is transition invariant. We modelled the results in terms of a single-channel motion computation. We have new results indicating that a specific class of textures differing in texture element density and texture element contrast decisively fails the test of transition invariance. These findings suggest that in addition to the single second-order motion channel required by our earlier results there exists at least one other second-order motion channel. We argue that the preprocessing transformation used by this channel is a pointwise non-linearity that maps stimulus contrasts of absolute value less than some relatively high threshold tau onto 0, but increases with magnitude of c-tau for contrasts. c of absolute value greater than tau.

Formal semantics and analysis of control flow in WS-BPEL

Web service composition refers to the creation of new (Web) services by combining functionalities provided by existing ones. A number of domain-specific languages for service composition have been proposed, with consensus being formed around a process-oriented language known as WS-BPEL (or BPEL). The kernel of BPEL consists of simple communication primitives that may be combined using control-flow constructs expressing sequence, branching, parallelism, synchronization, etc. We present a comprehensive and rigorously defined mapping of BPEL constructs onto Petri net structures, and use this for the analysis of various dynamic properties related to unreachable activities, conflicting messages, garbage collection, conformance checking, and deadlocks and lifelocks in interaction processes. We use a mapping onto Petri nets because this allows us to use existing theoretical results and analysis tools. Unlike approaches based on finite state machines, we do not need to construct the state space, and can use structural analysis (e.g., transition invariants) instead. We have implemented a tool that translates BPEL processes into Petri nets and then applies Petri-net-based analysis techniques. This tool has been tested on different examples, and has been used to answer a variety of questions.

Verifying Workflows with Cancellation Regions and OR-joins: An Approach Based on Relaxed Soundness and Invariants

YAWL (Yet Another Workflow Language) workflow language supports the most frequent control-flow patterns found in the current workflow practice. As a result, most workflow languages can be mapped onto YAWL without the loss of control-flow details, even languages allowing for advanced constructs such as cancellation regions and OR-joins. Hence, a verification approach for YAWL is desirable, because such an approach could be used for any workflow language that can be mapped onto YAWL. Unfortunately, cancellation regions and OR-joins are ‘non-local’ properties, and in general we cannot even decide whether the desired final state is reachable if both patterns are present. This paper proposes a verification approach based on (i) an abstraction of the OR-join semantics; (ii) the relaxed soundness property; and (iii) transition invariants. This approach is correct (errors reported are really errors), but not necessarily complete (not every error might get reported). This incompleteness can be explained because, on the one hand, the approach abstracts from the OR-join semantics and on the other hand, it may use only transition invariants, which are structural properties. Nevertheless, our approach can be used to successfully detect errors in YAWL models. Moreover, the approach can be easily transferred to other workflow languages allowing for advanced constructs such as cancellations and OR-joins.