Boolean Nets Research Articles

On the Complexity of Techniques That Make Transition Systems Implementable by Boolean Nets

Let us consider some class of (Petri) nets. The corresponding Synthesis problem consists in deciding whether a given labeled transition system (TS) A can be implemented by a net N of that class. In case of a negative decision, it may be possible to convert A into an implementable TS B by applying various modification techniques, like relabeling edges that previously had the same label, suppressing edges/states/events, etc. It may however be useful to limit the number of such modifications to stay close to the original problem, or optimize the technique. In this paper, we show that most of the corresponding problems are NP-complete if the considered class corresponds to so-called flip-flop nets or some flip-flop net derivatives.

On the parameterized complexity of the synthesis of Boolean nets with restricted place environments

Though constructing a sequential system model may be technically uncomplicated and rather ‘straightforward’, its result is often cumbersome and difficult to analyze since all possible interleavings of system's actions have to be present. This naturally raises the interest to the possibility of automatic synthesis of models that are able to grasp behavioral parallelism of the system. Boolean Petri nets are a classical formalism that enjoys generic concepts of concurrency, synchronization and conflict, and allows to obtain a concise yet expressive system representation. Let τ be a Boolean type of nets. The problem of τ-synthesis consists in deciding whether a given transition system A is isomorphic to the reachability graph of a Boolean Petri net N of type τ (a τ-net, for short). In case of a positive decision, N should be constructed. Another instance of the synthesis problem takes the dependencies of places into account: A place p of a τ-net depends on a transition t if p can influence t, that is, a marking of p can inhibit the firing of t, or if p can be influenced by t, that is, the firing of t can change a marking of p. The problem τ-synthesis of Boolean nets with restricted dependencies (τ-SOBNWRD) consists in deciding for a given transition system A and natural number d whether there exists a Boolean Petri net N of type τ such that, firstly, the reachability graph of N and A are isomorphic and, secondly, every place p of N depends on at most d transitions. For many Boolean types that allow independence, the (unrestricted) τ-synthesis problem is NP-complete, and for all of them τ-SOBNWRD inherits the NP-completeness. In this paper, we enhance our understanding of τ-SOBNWRD from a parameterized point of view: We show for 27 of the 128 thinkable Boolean types that allow independence that τ-SOBNWRD parameterized by d is W[1]-hard, and for 75 of them that the problem is W[2]-hard.

Integrated Information in Process-Algebraic Compositions.

Integrated Information Theory (IIT) is most typically applied to Boolean Nets, a state transition model in which system parts cooperate by sharing state variables. By contrast, in Process Algebra, whose semantics can also be formulated in terms of (labeled) state transitions, system parts—“processes”—cooperate by sharing transitions with matching labels, according to interaction patterns expressed by suitable composition operators. Despite this substantial difference, questioning how much additional information is provided by the integration of the interacting partners above and beyond the sum of their independent contributions appears perfectly legitimate with both types of cooperation. In fact, we collect statistical data about —integrated information—relative to pairs of boolean nets that cooperate by three alternative mechanisms: shared variables—the standard choice for boolean nets—and two forms of shared transition, inspired by two process algebras. We name these mechanisms , and . Quantitative characterizations of all of them are obtained by considering three alternative execution modes, namely synchronous, asynchronous and “hybrid”, by exploring the full range of possible coupling degrees in all three cases, and by considering two possible definitions of based on two alternative notions of distribution distance.

Enhancing compound safety assessment using “Multitask” deep neural nets

This paper presents Fuzzy Boolean Nets (FBN), a nature inspired Boolean neural model, in the sense that it exhibits topologic similarities with natural systems, as well as learning dynamic activity at low level and high immunity to individual errors or deletions of neurons or synaptic connections. It is shown that one can interpret reasoning in FBN as a particular case of fuzzy qualitative reasoning. It is proven that the model is a Universal Approximator since it can be interpreted as a Parzen Window probability density estimator. Also, conditions for learning a discrete set of rules without cross influence are investigated, being proven that this goal can be achieved with an appropriate net parameters relationship. Moreover, adequate parameter choices for learning continuous functions with interpolation capabilities are deduced.

Incremental dataflow execution, resource efficiency and probabilistic guarantees with Fuzzy Boolean nets

Currently, there is a strong need for organizations to analyze and process ever-increasing volumes of data in order to answer to real-time processing demands. Such continuous and data-intensive processing is often achieved through the composition of complex data-intensive workflows (i.e., dataflows).Dataflow management systems typically enforce strict temporal synchronization across the various processing steps. Non-synchronous behavior often has to be explicitly programmed on an ad-hoc basis, which requires additional lines of code in programs and thus the possibility of errors. More so, in a large set of scenarios for continuous and incremental processing, the output of dataflow applications at each execution can suffer almost no difference when comparing to the previous execution, and therefore resources, energy and computational power are unknowingly wasted.To face such lack of efficiency, transparency, and generality, we introduce the notion of Quality-of-Data (QoD), which describes the level of changes required on a data store that cause the triggering of processing steps. This, so that the dataflow (re-)execution is reduced until its outcome would reach a significant and meaningful variation, which is inside a specified freshness limit.Based on the QoD notion, we propose a novel dataflow model, with framework (Fluxy), for orchestrating data-intensive processing steps, which communicate data via a NoSQL storage, and whose triggering semantics is driven by dynamic QoD constraints automatically defined for different datasets by means of Fuzzy Boolean Nets. These nets give probabilistic guarantees about the prediction of the cumulative error between consecutive dataflow executions. With Fluxy, we demonstrate how dataflows can be leveraged to respond to quality boundaries (that can be seen as SLAs) to deliver controlled and augmented performance, rationalization of resources, and task prioritization.

Fuzzy Boolean Nets – a nature inspired model for learning and reasoning

This paper presents Fuzzy Boolean Nets (FBN), a nature inspired Boolean neural model, in the sense that it exhibits topologic similarities with natural systems, as well as learning dynamic activity at low level and high immunity to individual errors or deletions of neurons or synaptic connections. It is shown that one can interpret reasoning in FBN as a particular case of fuzzy qualitative reasoning. It is proven that the model is a Universal Approximator since it can be interpreted as a Parzen Window probability density estimator. Also, conditions for learning a discrete set of rules without cross influence are investigated, being proven that this goal can be achieved with an appropriate net parameters relationship. Moreover, adequate parameter choices for learning continuous functions with interpolation capabilities are deduced.

An integrated recycling approach for GFRP pultrusion wastes: recycling and reuse assessment into new composite materials using Fuzzy Boolean Nets

In this study, efforts were made in order to put forward an integrated recycling approach for the thermoset based glass fibre reinforced polymer (GPRP) rejects derived from the pultrusion manufacturing industry. Both the recycling process and the development of a new cost-effective end-use application for the recyclates were considered. For this purpose, i) among the several available recycling techniques for thermoset based composite materials, the most suitable one for the envisaged application was selected (mechanical recycling); and ii) an experimental work was carried out in order to assess the added-value of the obtained recyclates as aggregates and reinforcement replacements into concrete-polymer composite materials. Potential recycling solution was assessed by mechanical behaviour of resultant GFRP waste modified concrete-polymer composites with regard to unmodified materials. In the mix design process of the new GFRP waste based composite material, the recyclate content and size grade, and the effect of the incorporation of an adhesion promoter were considered as material factors and systematically tested between reasonable ranges. The optimization process of the modified formulations was supported by the Fuzzy Boolean Nets methodology, which allowed finding the best balance between material parameters that maximizes both flexural and compressive strengths of final composite. Comparing to related end-use applications of GFRP wastes in cementitious based concrete materials, the proposed solution overcome some of the problems found, namely the possible incompatibilities arisen from alkalis-silica reaction and the decrease in the mechanical properties due to high watercement ratio required to achieve the desirable workability. Obtained results were very promising towards a global cost-effective waste management solution for GFRP industrial wastes and end-of-life products that will lead to a more sustainable composite materials industry.

Step semantics of boolean nets

Boolean nets are a family of Petri net models with very simple markings which are sets of places. We investigate several classes of boolean nets distinguished by different kinds of individual connections between places and transitions, as well as different ways in which these connections are combined in order to specify the effect of executing steps of transitions. The latter aspect can be captured by connection monoids. A key advantage of using connection monoids is that by describing the step semantics of a class of Petri nets in terms of a connection monoid, one can apply results developed within a general theory of Petri net synthesis. In this paper, we provide an extensive classification of boolean nets which can be described by connection monoids. This classification is based on the realisation that the different ways of interpreting combinations of connections can be made explicit using a higher level monoid. Moreover, we demonstrate that connection monoids can capture other behavioural properties of boolean nets, such as structural conflicts between transitions.

A qualitative description of soil parameters variation due to a prescribed fire in Portuguese northwestern forests using Fuzzy Boolean Nets — The case study of Cabreira mountain

In this paper we study the modifications that occurred in some forest soil properties after a prescribed fire. The research focused on the alterations of soil pH, soil moisture and soil organic matter content during a two-year span, from 2008 to 2009. The study site is located in Anjos, Vieira do Minho municipality, a forest site that has suffered from recurrent wildfires for several decades. Furze (Ulex, sp.), broom (Cytisus, sp.), gorse (Chamaespartum tridentatum) and a very few disperse adult pine (Pinus sylvestris) are the predominant vegetation type in the study area. The average height of this shrub vegetation is around 1.5m. The prescribed fire was conducted by the National Forestry Authority (AFN) in November 2008.Fuzzy Boolean Nets (FBN) were used to evaluate the alteration in soil parameters when compared with adjacent spots where: i) no fire occurrence was registered since 1998; ii) fire occurrence was registered in 2008; and iii) vegetation pruning by mechanical cut was done in Spring six months prior to the prescribed fire event.Results suggest that in the particular case of the studied site, Anjos, the observed soil properties alterations cannot be related with the prescribed fire.

Prescribed burning impact on forest soil properties–A Fuzzy Boolean Nets approach

The Portuguese northern forests are often and severely affected by wildfires during the Summer season. These occurrences significantly affect and negatively impact all ecosystems, namely soil, fauna and flora. In order to reduce the occurrences of natural wildfires, some measures to control the availability of fuel mass are regularly implemented. Those preventive actions concern mainly prescribed burnings and vegetation pruning. This work reports on the impact of a prescribed burning on several forest soil properties, namely pH, soil moisture, organic matter content and iron content, by monitoring the soil self-recovery capabilities during a one year span. The experiments were carried out in soil cover over a natural site of Andaluzitic schist, in Gramelas, Caminha, Portugal, which was kept intact from prescribed burnings during a period of four years. Soil samples were collected from five plots at three different layers (0–3, 3–6 and 6–18) 1 day before prescribed fire and at regular intervals after the prescribed fire. This paper presents an approach where Fuzzy Boolean Nets (FBN) and Fuzzy reasoning are used to extract qualitative knowledge regarding the effect of prescribed fire burning on soil properties. FBN were chosen due to the scarcity on available quantitative data. The results showed that soil properties were affected by prescribed burning practice and were unable to recover their initial values after one year.

Universal Boolean Systems

Boolean interaction systems and hard interaction systems define nets of interacting cells. They are based on the same local interaction principle between two cells as interaction nets but do not allow that the structure of nets may evolve. With boolean nets, it is not possible to create or destroy cells or links between existing cells. They are very similar to hardware circuits but based on an implicit rendez-vous information exchange mechanism.If we want to implement such systems using hardware circuits, it is important to define a set of universal combinators that reduces this task to the implementation of a fixed number of known agents. Here, we show how we can simulate every hard interaction system by a universal boolean interaction system composed of three combinators: a duplicator, a NAND gate and a three-state input/output channel.

Random Boolean nets and features of language

Describes an attempt to cast several abstract properties of natural languages in the framework of Kauffman's (1993, 1995) random Boolean nets (RBN). The properties are complexity, interconnectedness, stability, diversity, and underdeterminedness. A language is modeled as a Boolean net attractor. (Groups of) net nodes are linguistic principles or parameters as posited by Chomskyan theory, according to which the language learner sets parameters to appropriate values on the basis of very limited experience of the language. The setting of one parameter can have a complex effect on the settings of others. A RBN is generated to find an attractor. A state from this attractor is degraded, which represents the degenerate input of language to the learner, and this state is then input to a net with the same connectivity and activation functions as the original net to see whether it converges on the same attractor. Many nets degenerate into attractors representing complete uncertainty. Others settle at intermediate levels of uncertainty, and some manage to overcome the incompleteness of input and converge on attractors identical to that from which the original inputs were (de)generated. Finally, an attempt was made to select a population of such successful nets, using a genetic algorithm where fitness was correlated with an ability to acquire several different languages faithfully. This has so far proved impossible, supporting the Chomskyan suggestion that the human language acquisition capacity is not the outcome of natural selection.

DYNAMICAL BEHAVIOR OF KAUFFMAN NETWORKS WITH AND-OR GATES

We study the parallel dynamics of a class of Kauffman boolean nets such that each vertex has a binary state machine {AND, OR} as local transition function. We have called this class of nets AON. In a finite, connected and undirected graph, the transient length, attractors and its basins of attraction are completely determined in the case of only OR (AND) functions in the net. For finite, connected and undirected AON, an exact linear bound is given for the transient time using a Lyapunov functional. Also, a necessary and sufficient condition is given for the diffusion problem of spreading a one all over the net, which generalizes the primitivity notion on graphs. This condition also characterizes its architecture. For finite, strongly connected and directed AON a non-polynomial time bound is given for the transient time and for the period on planar graphs, together with an example where this transient time and period are attained. Furthermore, on infinite but finite connected, directed and non planar AON we simulate an universal two-register machine, which allows us to exhibit universal computing capabilities.

Phase transitions in random networks: Simple analytic determination of critical points

The standard method for determining the critical points in random networks (such as Boolean nets) is based on the Derrida approximation. This method leads to the critical points based on the so-called annealed approximation. In this paper, we present a very simple annealed approximation based on the study of damage spreading when single elements are modified. Several examples are analyzed.

Median attractor and transients in random boolean nets

Variants of a random boolean network of binary gates called the Kauffman net are studied. Each gate receives two random inputs from the other gates in the network and is assigned a transition function from a non-uniform distribution. Fine upper bounds on the attractor and transient lengths as the behavioral characteristics of the network are determined for large net size, N, and different transition function distributions. The effects of these distributions on the localization property of the net is shown. Analysis of nets with N > 1000 was shown to be very difficult. We also determine the scaling properties of attractors and transients with the net size. Our results exhibit strong “antichaos” behavior even for very large N for some of the networks.

The learning property in size-variable Boolean nets

We report in this paper the construction of a size-variable Boolean net and the investigation of the learning property. The net is constructed so that it allows a change in size with the increase of the input-output cases to be learned. Adaptability to the dynamic environment is considered to be the characteristic flexibility of living organisms. We have constructed a learning model in which the Boolean net size can be increased, and the adaptation is realized as the net size is increased. The learning model is more efficient compared with fixed size nets. However, the flexibility of living organisms should not be expressed as fixed operations, but should be considered to include the deviation from the global tendency of adaptation. A reverse operation is introduced to realize and to show the deviation. We attained higher learning velocity with the learning model having the reverse operation.

Evolution of Optimal Behaviour in Networks of Boolean Automata

This paper describes the response to selection for optimal behaviour in networks of Boolean automata. The networks considered are "open" in the sense that they receive an input and produce an output. The task that the nets are selected to solve is to maximize the mean "height" attained in a fixed number of iterations in "land-scapes" that vary in their ruggedness. This is analogous to an organism with a fixed time budget that attempts to maximize its food intake in an environment where the food concentration varies in space. The results suggest that it is possible to select simple Boolean nets to respond adaptively to their inputs. Selection on simple environments produces a near "optimal" response. The response to complex environments is worse. The results suggest that nets selected on complex environments perform better on environments of different degrees of complexity than those selected on simple environments. Although nets with larger numbers of automata do not respond markedly better to selection, the variability in the response to selection is reduced. It appears to be difficult to predict the movements of nets from a study of the attractors of nets with sensory units clamped at particular values. The cycle of bits in clamped nets can vary greatly with small changes in the inputs to the nets. The actual movement of nets does not, however, reflect this instability. It is concluded that networks of Boolean automata may be useful models with which to investigate different genetic algorithms. Furthermore, perturbations in such systems may give insights into the nature of pleiotropy and epistasis.

The Behaviour of Affine Boolean Sequential Networks

Abstract Whereas a significant amount of empirical work has been carried out in the area of logical neural networks, the fundamental theoretical basis is underdeveloped. This paper provides an analysis of recurrent nets of affine Boolean functions and defines the relation between their behaviour and that of linear Boolean nets. It is shown that the only effect on the unlabelled state structure of adding any number of inverters to a linear Boolean net is a restricted modification of cycle lengths.

Attractor dominance patterns in sparsely connected boolean nets

An attractor “dominates” those states in phase space that lie in its basin of attraction. This paper examines attractor dominance as a function of distance from the attractor. When trajectories of sparsely connected Boolean nets are aggregated suitably, attractor dominance is a reasonably smooth function of distance. The shape of the dominance function varies widely, depending on the automaton rule in place. In this paper I order the dominance functions of the rules of a family of sparsely connected Boolean nets. These nets are structurally generalized cellular automata. Two classes contain most of the variability in attractor dominance patterns. Within these classes, two dimensions locate dominance functions and reveal changes in phase space dominance with lattice size. Increasing lattice size appears to move most, but, importantly, not all dominance patterns toward the extremes shown by the three trivial rules. Implications of these findings are discussed.

Universality two-dimensional Kauffman model for parallel and sequential updating

Kauffman's random Boolean nets are studied on the square lattice by computer simulation comparing parallel and sequential updating of the automata. At the phase transition the fractal dimensions of time and actual damage are found to be independent of the updating method.