Van-Giang Trinh

Van-Giang Trinh, Sylvain Soliman, François Fages et al.

The logical semantics of normal logic programs has traditionally been based on the notions of Clark's completion and two-valued or three-valued canonical models, including supported, stable, regular, and well-founded models. Two-valued interpretations can also be seen as states evolving under a program's update operator, producing a transition graph whose fixed points and cycles capture stable and oscillatory behaviors, respectively. We refer to this view as dynamical semantics since it characterizes the program's meaning in terms of state-space trajectories, as first introduced in the stable (supported) class semantics. Recently, we have established a formal connection between Datalog^\neg programs (i.e., normal logic programs without function symbols) and Boolean networks, leading to the introduction of the trap space concept for Datalog^\neg programs. In this paper, we generalize the trap space concept to arbitrary normal logic programs, introducing trap space semantics as a new approach to their interpretation. This new semantics admits both model-theoretic and dynamical characterizations, providing a comprehensive approach to understanding program behavior. We establish the foundational properties of the trap space semantics and systematically relate it to the established model-theoretic semantics, including the stable (supported), stable (supported) partial, regular, and L-stable model semantics, as well as to the dynamical stable (supported) class semantics. Our results demonstrate that the trap space semantics offers a unified and precise framework for proving the existence of supported classes, strict stable (supported) classes, and regular models, in addition to uncovering and formalizing deeper relationships among the existing semantics of normal logic programs.

Van-Giang Trinh, Belaid Benhamou

Answer Set Programming (ASP) is a declarative problem solving paradigm that can be used to encode a combinatorial problem as a logic program whose stable models correspond to the solutions of the considered problem. ASP has been widely applied to various domains in AI and beyond. The question "What can be said about stable models of a logic program from its static information?" has been investigated and proved useful in many circumstances. In this work, we dive into this direction more deeply by making the connection between a logic program and a Boolean network, which is a prominent modeling framework with applications to various areas. The proposed connection can bring the existing results in the rich history on static analysis of Boolean networks to explore and prove more theoretical results on ASP, making it become a unified and powerful tool to further study the static analysis of ASP. In particular, the newly obtained insights have the potential to benefit many problems in the field of ASP.

Samuel Pastva, Van-Giang Trinh

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 ADFs. Our approach is inspired by the recently discovered equivalence between Boolean Networks (BNs) and ADFs by Heyninck et al. (2024) and Azpeitia et al. (2024), 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.

Van-Giang Trinh, Belaid Benhamou, Sylvain Soliman et al.

The regular models of a normal logic program are a particular type of partial (i.e. 3-valued) models which correspond to stable partial models with minimal undefinedness. In this paper, we explore graphical conditions on the dependency graph of a finite ground normal logic program to analyze the existence, unicity and number of regular models for the program. We show three main results: 1) a necessary condition for the existence of non-trivial (i.e. non-2-valued) regular models, 2) a sufficient condition for the unicity of regular models, and 3) two upper bounds for the number of regular models based on positive feedback vertex sets. The first two conditions generalize the finite cases of the two existing results obtained by You and Yuan (1994) for normal logic programs with well-founded stratification. The third result is also new to the best of our knowledge. Key to our proofs is a connection that we establish between finite ground normal logic programs and Boolean network theory.

Van-Giang Trinh, Belaid Benhamou, Sylvain Soliman et al.

Datalog$^\neg$ is a central formalism used in a variety of domains ranging from deductive databases and abstract argumentation frameworks to answer set programming. Its model theory is the finite counterpart of the logical semantics developed for normal logic programs, mainly based on the notions of Clark's completion and two-valued or three-valued canonical models including supported, stable, regular and well-founded models. In this paper we establish a formal link between Datalog$^\neg$ and Boolean network theory first introduced for gene regulatory networks. We show that in the absence of odd cycles in a Datalog$^\neg$ program, the regular models coincide with the stable models, which entails the existence of stable models, and in the absence of even cycles, we prove the uniqueness of stable partial models and regular models. This connection also gives new upper bounds on the numbers of stable partial, regular, and stable models of a Datalog$^\neg$ program using the cardinality of a feedback vertex set in its atom dependency graph. Interestingly, our connection to Boolean network theory also points us to the notion of trap spaces. In particular we show the equivalence between subset-minimal stable trap spaces and regular models.