We present BAss (BDD-based ADF symbolic solver), a novel analysis tool for Abstract Dialectical Frameworks (ADFs) based on Binary Decision Diagrams (BDDs). It supports the fully-symbolic computation of all admissible, complete, and preferred interpretations, as well as two-valued and stable models of an ADF. Our approach is inspired by the recently discovered equivalence between Boolean Networks (BNs) and ADFs, significantly extending current BDD-based tools bioLQM, aeon, and adf-bdd. We conducted experiments on a large-scale collection of real-world models from both the BN and ADF communities. Our results show that BAss dramatically outperforms previous BDD-based tools and is competitive (even significantly better in some cases) with state-of-the-art SAT/ASP-based methods, particularly in scenarios involving large solution spaces. Notably, BAss is able to enumerate all fixed points or minimal trap spaces of certain biological networks beyond the reach of existing tools, thereby enabling new analysis and case studies in systems biology. These results highlight the practical relevance of symbolic reasoning for complex real-world applications, particularly in systems biology and formal argumentation.

Uninterpreted functions are a key modeling tool for systems with unknown or abstracted components. Certain domains, such as systems biology, additionally impose monotonicity constraints on these components, requiring specific inputs to have a consistently positive or negative effect on the output. In this paper, we tackle the model inference problem for biological systems by applying the theory of uninterpreted functions with monotonicity constraints. We compare the performance of naive quantified encodings of the problem and the performance of the existing approach based on eager quantifier instantiation, which is based on the fact that a finite set of quantifier-free monotonicity lemmas is sufficient to encode the monotonicity of uninterpreted functions. Additionally, we consider a lazy variant of the approach that introduces the monotonicity lemmas on demand. We evaluate the SMT-based approach to model inference using a large collection of systems biology benchmarks. The results demonstrate that the instantiation-based encodings significantly outperform quantified encodings, which typically struggle with large function arities and complex instances. As the key result, we show that our approach based on SMT with uninterpreted functions and monotonicity constraints significantly outperforms state-of-the-art domain-specific tools used in systems biology, such as the ASP-based Bonesis and the BDD-based AEON.

MOTIVATION: System control can be used to provide new insights into the dynamics of biological systems. A key application is the identification of therapeutic targets in silico, which requires an executable model of the system's dynamics. However, such models are typically underspecified due to incomplete mechanistic knowledge. RESULTS: We introduce a novel computational framework that employs control-guided model refinement, predicting informative perturbation experiments to reduce knowledge gaps. The approach is based on partially specified Boolean networks (PSBNs), which enable direct integration of uncertain or incomplete information into executable models. We further extend the framework to handle oscillatory phenotypes as explicit control targets. The applicability of the method is demonstrated on receptor-tyrosine kinase (RTK) signaling, with a focus on fibroblast growth factor signaling in the context of skeletal dysplasias and cancer. We obtain several new insights into modelling of the FGFR3-MAPK pathway. AVAILABILITY AND IMPLEMENTATION: Code and datasets are available at https://doi.org/10.5281/zenodo.16886813.

Boolean Networks (BNs) serve as a fundamental modeling framework for capturing complex dynamical systems across various domains, including systems biology, computational logic, and artificial intelligence. A crucial property of BNs is the presence of trap spaces - subspaces of the state space that, once entered, cannot be exited. Minimal trap spaces, in particular, play a significant role in analyzing the long-term behavior of BNs, making their efficient enumeration and counting essential. The fixed points in BNs are a special case of minimal trap spaces. In this work, we formulate several meaningful counting problems related to minimal trap spaces and fixed points in BNs. These problems provide valuable insights both within BN theory (e.g., in probabilistic reasoning and dynamical analysis) and in broader application areas, including systems biology, abstract argumentation, and logic programming. To address these computational challenges, we propose novel methods based on approximate answer set counting, leveraging techniques from answer set programming. Our approach efficiently approximates the number of minimal trap spaces and the number of fixed points without requiring exhaustive enumeration, making it particularly well-suited for large-scale BNs. Our experimental evaluation on an extensive and diverse set of benchmark instances shows that our methods significantly improve the feasibility of counting minimal trap spaces and fixed points, paving the way for new applications in BN analysis and beyond.

MOTIVATION: Boolean networks are popular dynamical models of cellular processes in systems biology. Their attractors model phenotypes that arise from the interplay of key regulatory subcircuits. A succession diagram (SD) describes this interplay in a discrete analog of Waddington's epigenetic attractor landscape that allows for fast identification of attractors and attractor control strategies. Efficient computational tools for studying SDs are essential for the understanding of Boolean attractor landscapes and connecting them to their biological functions. RESULTS: We present a new approach to SD construction for asynchronously updated Boolean networks, implemented in the biologist's Boolean attractor landscape mapper, biobalm. We compare biobalm to similar tools and find a substantial performance increase in SD construction, attractor identification, and attractor control. We perform the most comprehensive comparative analysis to date of the SD structure in experimentally-validated Boolean models of cell processes and random ensembles. We find that random models (including critical Kauffman networks) have relatively small SDs, indicating simple decision structures. In contrast, nonrandom models from the literature are enriched in extremely large SDs, indicating an abundance of decision points and suggesting the presence of complex Waddington landscapes in nature. AVAILABILITY AND IMPLEMENTATION: The tool biobalm is available online at https://github.com/jcrozum/biobalm. Further data, scripts for testing, analysis, and figure generation are available online at https://github.com/jcrozum/biobalm-analysis and in the reproducibility artefact at https://doi.org/10.5281/zenodo.13854760.

Boolean Networks (BNs) are widely used as a modeling formalism in several domains, notably systems biology and computer science. A fundamental problem in BN analysis is the enumeration of trap spaces, which are hypercubes in the state space that cannot be escaped once entered. Several methods have been proposed for enumerating trap spaces, however they often suffer from scalability and efficiency issues, particularly for large and complex models. To our knowledge, the most efficient and recent methods for the trap space enumeration all rely on Answer Set Programming (ASP), which has been widely applied to the analysis of BNs. Motivated by these considerations, our work proposes a new method for enumerating trap spaces in BNs using ASP. We evaluate the method on a mix of 250+ real-world and 400+ randomly generated BNs, showing that it enables analysis of models beyond the capabilities of existing tools (namely pyboolnet, mpbn, trappist, and trapmvn).

MOTIVATION: The problem of model inference is of fundamental importance to systems biology. Logical models (e.g. Boolean networks; BNs) represent a computationally attractive approach capable of handling large biological networks. The models are typically inferred from experimental data. However, even with a substantial amount of experimental data supported by some prior knowledge, existing inference methods often focus on a small sample of admissible candidate models only. RESULTS: We propose Boolean network sketches as a new formal instrument for the inference of Boolean networks. A sketch integrates (typically partial) knowledge about the network's topology and the update logic (obtained through, e.g. a biological knowledge base or a literature search), as well as further assumptions about the properties of the network's transitions (e.g. the form of its attractor landscape), and additional restrictions on the model dynamics given by the measured experimental data. Our new BNs inference algorithm starts with an 'initial' sketch, which is extended by adding restrictions representing experimental data to a 'data-informed' sketch and subsequently computes all BNs consistent with the data-informed sketch. Our algorithm is based on a symbolic representation and coloured model-checking. Our approach is unique in its ability to cover a broad spectrum of knowledge and efficiently produce a compact representation of all inferred BNs. We evaluate the method on a non-trivial collection of real-world and simulated data. AVAILABILITY AND IMPLEMENTATION: All software and data are freely available as a reproducible artefact at https://doi.org/10.5281/zenodo.7688740.

MOTIVATION: Boolean networks are simple but efficient mathematical formalism for modelling complex biological systems. However, having only two levels of activation is sometimes not enough to fully capture the dynamics of real-world biological systems. Hence, the need for multi-valued networks (MVNs), a generalization of Boolean networks. Despite the importance of MVNs for modelling biological systems, only limited progress has been made on developing theories, analysis methods, and tools that can support them. In particular, the recent use of trap spaces in Boolean networks made a great impact on the field of systems biology, but there has been no similar concept defined and studied for MVNs to date. RESULTS: In this work, we generalize the concept of trap spaces in Boolean networks to that in MVNs. We then develop the theory and the analysis methods for trap spaces in MVNs. In particular, we implement all proposed methods in a Python package called trapmvn. Not only showing the applicability of our approach via a realistic case study, we also evaluate the time efficiency of the method on a large collection of real-world models. The experimental results confirm the time efficiency, which we believe enables more accurate analysis on larger and more complex multi-valued models. AVAILABILITY AND IMPLEMENTATION: Source code and data are freely available at https://github.com/giang-trinh/trap-mvn.

SUMMARY: AEON.py is a Python library for the analysis of the long-term behaviour in very large asynchronous Boolean networks. It provides significant computational improvements over the state-of-the-art methods for attractor detection. Furthermore, it admits the analysis of partially specified Boolean networks with uncertain update functions. It also includes techniques for identifying viable source-target control strategies and the assessment of their robustness with respect to parameter perturbations. AVAILABILITY AND IMPLEMENTATION: All relevant results are available in Supplementary Materials. The tool is accessible through https://github.com/sybila/biodivine-aeon-py. SUPPLEMENTARY INFORMATION: Supplementary data are available at Bioinformatics online.

BACKGROUND: Boolean networks (BNs) provide an effective modelling formalism for various complex biochemical phenomena. Their long term behaviour is represented by attractors-subsets of the state space towards which the BN eventually converges. These are then typically linked to different biological phenotypes. Depending on various logical parameters, the structure and quality of attractors can undergo a significant change, known as a bifurcation. We present a methodology for analysing bifurcations in asynchronous parametrised Boolean networks. RESULTS: In this paper, we propose a computational framework employing advanced symbolic graph algorithms that enable the analysis of large networks with hundreds of Boolean variables. To visualise the results of this analysis, we developed a novel interactive presentation technique based on decision trees, allowing us to quickly uncover parameters crucial to the changes in the attractor landscape. As a whole, the methodology is implemented in our tool AEON. We evaluate the method's applicability on a complex human cell signalling network describing the activity of type-1 interferons and related molecules interacting with SARS-COV-2 virion. In particular, the analysis focuses on explaining the potential suppressive role of the recently proposed drug molecule GRL0617 on replication of the virus. CONCLUSIONS: The proposed method creates a working analogy to the concept of bifurcation analysis widely used in kinetic modelling to reveal the impact of parameters on the system's stability. The important feature of our tool is its unique capability to work fast with large-scale networks with a relatively large extent of unknown information. The results obtained in the case study are in agreement with the recent biological findings.

Edge-coloured directed graphs provide an essential structure for modelling and analysis of complex systems arising in many scientific disciplines (e.g. feature-oriented systems, gene regulatory networks, etc.). One of the fundamental problems for edge-coloured graphs is the detection of strongly connected components, or SCCs. The size of edge-coloured graphs appearing in practice can be enormous both in the number of vertices and colours. The large number of vertices prevents us from analysing such graphs using explicit SCC detection algorithms, such as Tarjan's, which motivates the use of a symbolic approach. However, the large number of colours also renders existing symbolic SCC detection algorithms impractical. This paper proposes a novel algorithm that symbolically computes all the monochromatic strongly connected components of an edge-coloured graph. In the worst case, the algorithm performs $O(p \cdot n \cdot log~n)$ symbolic steps, where $p$ is the number of colours and $n$ is the number of vertices. We evaluate the algorithm using an experimental implementation based on binary decision diagrams (BDDs). Specifically, we use our implementation to explore the SCCs of a large collection of coloured graphs (up to $2^{48}$) obtained from Boolean networks -- a modelling framework commonly appearing in systems biology.

Abstract Detection of bottom strongly connected components (BSCC) in state-transition graphs is an important problem with many applications, such as detecting recurrent states in Markov chains or attractors in dynamical systems. However, these graphs’ size is often entirely out of reach for algorithms using explicit state-space exploration, necessitating alternative approaches such as the symbolic one. Symbolic methods for BSCC detection often show impressive performance, but can sometimes take a long time to converge in large graphs. In this paper, we provide a symbolic state-space reduction method for labelled transition systems, calledinterleaved transition guided reduction(ITGR), which aims to alleviate current problems of BSCC detection by efficiently identifying large portions of the non-BSCC states. We evaluate the suggested heuristic on an extensive collection of 125 real-world biologically motivated systems. We show that ITGR can easily handle all these models while being either the only method to finish, or providing at least an order-of-magnitude speedup over existing state-of-the-art methods. We then use a set of synthetic benchmarks to demonstrate that the technique also consistently scales to graphs with more than $$2^{1000}$$ 21000 vertices, which was not possible using previous methods.

Abstract Problems arising in many scientific disciplines are often modelled using edge-coloured directed graphs. These can be enormous in the number of both vertices and colours. Given such a graph, the original problem frequently translates to the detection of the graph’s strongly connected components, which is challenging at this scale. We propose a new, symbolic algorithm that computes all the monochromatic strongly connected components of an edge-coloured graph. In the worst case, the algorithm performs $$O(p\cdot n\cdot \log n)$$ O ( p · n · log n ) symbolic steps, where p is the number of colours and n the number of vertices. We evaluate the algorithm using an experimental implementation based on Binary Decision Diagrams (BDDs) and large (up to $$2^{48}$$ 2 48 ) coloured graphs produced by models appearing in systems biology.

Boolean networks (BNs) provide an effective modelling tool for various phenomena from science and engineering. Any long-term behaviour of a BN eventually converges to a so-called attractor. Depending on various logical parameters, the structure and quality of attractors can undergo a significant change, known as a bifurcation. We present a tool for analysing bifurcations in asynchronous parametrised Boolean networks. To fight the state-space and parameter-space explosion problem the tool uses a parallel semi-symbolic algorithm.

Formal analysis of non-linear continuous and hybrid systems is a hot topic. A common approach builds on computing a suitable finite discrete abstraction of the continuous system. In this paper, we propose a facetal abstraction which eliminates certain drawbacks of existing abstractions. The states of our abstraction are built primarily from facets of a polytopal partitioning of the system's state space taking thus into account the flow of the continuous dynamics and leading to global over-approximation. The transition system construction is based on queries solved by a δ-decision SMT-solver. The method is evaluated on several case studies.

Digital bifurcation analysis is a new algorithmic method for exploring how the behaviour of a parameter-dependent computer system varies with a change in its parameters and, in particular, for identification of bifurcation points where such variation becomes dramatic. We have developed the method in an analogy with the traditional bifurcation theory and have it successfully applied to models taken from systems biology. In this case study paper, we demonstrate the appropriateness and usefulness of the digital bifurcation analysis as a push-button alternative to the classical approaches as traditionally used for analysing the stability of TCP/IP protocols. We consider two typical examples (congestion control and buffer sizes throughput influence) and show that the method provides the same results as obtained with classical non-automatic analytical and numerical methods.

Equivalence and model checking problems can be encoded into computing fixed points on dependency graphs. Dependency graphs represent causal dependencies among the nodes of the graph by means of hyper-edges. We suggest to extend the model of dependency graphs with so-called negation edges in order t o increase their applicability. The graphs (as well as the verification problems) suffer from the state space explosion problem. To combat this issue, we design an on-the-fly algorithm for efficiently computing fixed points on extended dependency graphs. Our algorithm supplements previous approaches with the possibility to back-propagate, in certain scenarios, the domain value 0, in addition to the standard back-propagation of the value 1. Finally, we design a distributed version of the algorithm, implement it in our open-source tool TAPAAL, and demonstrate the efficiency of our general approach on the benchmark of Petri net models and CTL queries from the annual Model Checking Contest.