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.

Abstract Answer Set Programming (ASP) provides a powerful declarative paradigm for knowledge representation and reasoning. Recently, counting answer sets has emerged as an important computational problem with applications in probabilistic reasoning, network reliability analysis, and other domains. This has motivated significant research into designing efficient ASP counters. While substantial progress has been made for normal logic programs, the development of practical counters for disjunctive logic programs remains challenging. We present $\mathsf{sharpASP}$ - $\mathcal{SR}$ , a novel framework for counting answer sets of disjunctive logic programs based on subtractive reduction to projected propositional model counting. Our approach introduces an alternative characterization of answer sets that enables efficient reduction while ensuring the intermediate representations remain polynomial in size. This allows $\mathsf{sharpASP}$ - $\mathcal{SR}$ to leverage recent advances in projected model counting technology. Through extensive experimental evaluation on diverse benchmarks, we demonstrate that $\mathsf{sharpASP}$ - $\mathcal{SR}$ significantly outperforms existing counters on instances with large answer set counts. Building on these results, we develop a hybrid counting approach that combines enumeration techniques with $\mathsf{sharpASP}$ - $\mathcal{SR}$ to achieve state-of-the-art performance across the full spectrum of disjunctive programs. The extended version of the paper is available at: https://arxiv.org/abs/2507.11655 .

Answer Set Programming (ASP) has emerged as a promising paradigm in knowledge representation and automated reason- ing owing to its ability to model hard combinatorial problems from diverse domains in a natural way. Building on advances in propositional SAT solving, the past two decades have wit- nessed the emergence of well-engineered systems for solv- ing the answer set satisfiability problem, i.e., finding mod- els or answer sets for a given answer set program. In re- cent years, there has been growing interest in problems be- yond satisfiability, such as model counting, in the context of ASP. Akin to the early days of propositional model count- ing, state-of-the-art exact answer set counters do not scale well beyond small instances. Exact ASP counters struggle with handling larger input formulas. The primary contribu- tion of this paper is a new ASP counting framework, called sharpASP, which counts answer sets avoiding larger input formulas. This relies on an alternative way of defining answer sets that allows lifting of key techniques developed in the con- text of propositional model counting. Our extensive empirical analysis over 1470 benchmarks demonstrates significant per- formance gain over current state-of-the-art exact answer set counters. Specifically, by using sharpASP, we were able to solve 1062 benchmarks with PAR2 score of 3082 whereas using prior state-of-the-art, we could only solve 895 bench- marks with PAR2 score of 4205, all other experimental con- ditions being the same.

Abstract Minimal models of a Boolean formula play a pivotal role in various reasoning tasks. While previous research has primarily focused on qualitative analysis over minimal models; our study concentrates on the quantitative aspect, specifically counting of minimal models. Exact counting of minimal models is strictly harder than $\#\mathsf{P}$ , prompting our investigation into establishing a lower bound for their quantity, which is often useful in related applications. In this paper, we introduce two novel techniques for counting minimal models, leveraging the expressive power of answer set programming: the first technique employs methods from knowledge compilation, while the second one draws on recent advancements in hashing-based approximate model counting. Through empirical evaluations, we demonstrate that our methods significantly improve the lower bound estimates of the number of minimal models, surpassing the performance of existing minimal model reasoning systems in terms of runtime.

The quantification of system reliability is fundamental to the assessment of a system’s safety and resilience, and has been of interest to decision-makers. Since quantifying the system reliability is shown to be computationally intractable, researchers aim to find approximations. Existing approaches to approximate reliability either suffer from poor scalability or lack of correctness guarantees. Answer Set Programming (ASP) is a powerful tool for knowledge representation that can specify complex combinatorial problems. In recent years, the new applications of ASP have propelled the emergence of well-engineered ASP systems. This paper proposes a new ASP counting based framework, RelNet-ASP, to approximate or estimate the reliability of a system or network. The framework reduces the problem of reliability estimation to an approximate model counting problem on ASP programs, offering formal guarantees of the estimated reliability. The experimental evaluation demonstrates that RelNet-ASP outperforms state-of-the-art techniques in terms of both runtime performance and accuracy.

Answer Set Programming (ASP) is a framework in artificial intelligence and knowledge representation for declarative modeling and problem solving. Modern ASP solvers focus on the computation or enumeration of answer sets. However, a variety of probabilistic applications in reasoning or logic programming require counting answer sets. While counting can be done by enumeration, simple enumeration becomes immediately infeasible if the number of solutions is high. On the other hand, approaches to exact counting are of high worst-case complexity. In fact, in propositional model counting, exact counting becomes impractical. In this work, we present a scalable approach to approximate counting for answer set programming. Our approach is based on systematically adding XOR constraints to ASP programs, which divide the search space. We prove that adding random XOR constraints partitions the answer sets of an ASP program. In practice, we use a Gaussian elimination-based approach by lifting ideas from SAT to ASP and integrating it into a state of the art ASP solver, which we call ApproxASP. Finally, our experimental evaluation shows the scalability of our approach over the existing ASP systems.

MOTIVATION: Researchers and practitioners use a number of popular sequence comparison tools that use many alignment-based techniques. Due to high time and space complexity and length-related restrictions, researchers often seek alignment-free tools. Recently, some interesting ideas, namely, Minimal Absent Words (MAW) and Relative Absent Words (RAW), have received much interest among the scientific community as distance measures that can give us alignment-free alternatives. This drives us to structure a framework for analysing biological sequences in an alignment-free manner. RESULTS: In this application note, we present Alignment-free Dissimilarity Analysis & Comparison Tool (ADACT), a simple web-based tool that computes the analogy among sequences using a varied number of indexes through the distance matrix, species relation list and phylogenetic tree. This tool basically combines absent word (MAW or RAW) computation, dissimilarity measures, species relationship and thus brings all required software in one platform for the ease of researchers and practitioners alike in the field of bioinformatics. We have also developed a restful API. AVAILABILITY AND IMPLEMENTATION: ADACT has been hosted at http://research.buet.ac.bd/ADACT/. SUPPLEMENTARY INFORMATION: Supplementary data are available at Bioinformatics online.