asp encoding

Existing hybrid Answer Set Programming (ASP) solvers—such as CLINGCON and CLINGO[DL]—handle real-world problems with numerical constraints but lack a unified, rigorous semantic foundation, leading to imprecise modeling and compromised solver reliability. Method: We introduce Constraint-Augmented Here-and-There logic (HTₐ), the first equilibrium-logic-based formal semantics for hybrid ASP, systematically integrating nonmonotonic reasoning with numerical constraint solving. HTₐ provides verifiable semantics for hybrid rules, supports constraint embedding, and enables sound model checking. Contribution/Results: Our framework theoretically unifies constraint satisfaction, HT logic, and ASP architecture. It yields correctness-preserving design principles for hybrid solvers and demonstrates expressive power and practical utility in real-world applications—particularly product configuration—thereby significantly improving modeling precision and solution trustworthiness for hybrid problems.

This work addresses the lack of a unified logical semantic foundation in existing hybrid Answer Set Programming (ASP) solvers when handling linear constraints, which leads to inconsistent system behavior. It proposes a multi-sorted variant of Bound-founded Here-and-There logic (HTb), establishing for the first time a uniform semantic framework for ASP with difference constraints and formally defining the notion of foundedness for numeric variables. Grounded in equilibrium model theory and a semantic analysis of constraint atoms, the approach not only uncovers the root causes of semantic discrepancies among systems such as clingo[DL], clingcon, and flingo, but also provides a foundational platform capable of uniformly characterizing the semantics of diverse hybrid ASP systems. This framework further supports program simplification and facilitates the integration of future semantic principles.

This work addresses the limited expressiveness of current Constraint Answer Set Programming (CASP) solvers when handling numerical constraints, which often forfeits key features of Answer Set Programming (ASP), such as default reasoning, undefined attributes, non-deterministic choices, and aggregates. To bridge this gap, the paper introduces FLINGO, a novel language that, for the first time, integrates ASP-style declarative modeling of numerical attributes into linear integer constraints. FLINGO achieves this by employing a syntactic translation mechanism that compiles its constructs into standard CLINGCON input. The approach preserves the efficiency of state-of-the-art numerical constraint solving while substantially enhancing modeling expressiveness. A prototype implementation demonstrates the effectiveness and practicality of the proposed method across several illustrative examples.

This work addresses the inefficiency of linear constraint solving in Constraint Answer Set Programming (CASP). We propose a translation-based solving framework that uniformly compiles CASP programs into the FlatZinc intermediate representation, enabling seamless integration with diverse constraint programming (CP) and integer programming (IP) backends. The framework supports a rich set of linear arithmetic constraints as well as common global constraints—including `alldifferent` and `cumulative`—thereby enhancing modeling expressiveness and solving flexibility. Experimental evaluation on ASP competition benchmarks shows competitive performance relative to state-of-the-art ASP solvers. On representative CASP instances, our approach achieves superior solving speed and scalability compared to the current best-performing system, clingcon. These results demonstrate both the effectiveness and competitiveness of the proposed framework in advancing CASP solving technology.

This work addresses the limitation of Answer Set Programming (ASP) solvers that rely on symbolic engines (e.g., ASP/SAT). We propose the first fully numerical, end-to-end ASP solving framework. Our method reformulates logic programs, the Lin-Zhao stable model theorem, and problem constraints as a differentiable cost function in vector space, enabling direct gradient-based search for stable models. To reduce computational complexity, we introduce program pre-compression and algebraic encoding of loop formulas. Technically, we adopt matrix-based program representation and vectorized modeling, supporting efficient GPU and multi-core parallelization. Experimental evaluation on benchmark problems—including 3-coloring and Hamiltonian cycle—demonstrates effectiveness, scalability, and competitive performance. Crucially, the framework is inherently differentiable, enabling end-to-end neuro-symbolic training and seamless integration with deep learning pipelines.

This work addresses the challenge of efficiently integrating constraints and logical reasoning in complex combinatorial search by proposing a scalable translation-based Constraint Answer Set Programming (CASP) framework. The approach seamlessly combines Answer Set Programming with Satisfiability Modulo Theories (SMT), leveraging mature SMT solvers—such as CVC5, Yices, and Z3—to enable efficient reasoning over mixed integer and real-valued constraints. The framework introduces a more expressive input language that supports weak constraint optimization and provides a unified architecture for incorporating novel constraint types. Experimental results demonstrate that the system substantially outperforms state-of-the-art solvers, including CLINGCON, CLINGO[DL], and CLINGO[LP], achieving significant improvements in both expressiveness and solving efficiency on standard benchmarks.

Traditional discrete Event Calculus suffers from combinatorial explosion and imprecise representation when handling continuous change and large-scale, dense temporal and fluent domains, while base-free approaches risk non-termination. This work proposes Hybrid Event Calculus (Hybrid EC), which unifies discrete and continuous change through functional fluents and an abstract time-step mechanism. By integrating hybrid ASP solvers such as Clingcon and clingo-lpx, Hybrid EC encodes fluents and time over dense domains as linear constraints delegated to external solvers. The approach guarantees termination while enabling precise modeling of continuous dynamics. Experimental results demonstrate that its performance remains insensitive to the scale of the value domain, and the use of rational numbers does not compromise scalability.

This work addresses the lack of effective static analysis techniques for goal-directed Answer Set Programming (ASP), which hinders compile-time verification and optimization. It introduces abstract interpretation to this paradigm for the first time, proposing a top-down analysis algorithm based on PLAI fixpoints and designing a novel Shared-Constraints abstract domain to precisely capture variable relationships induced by constraints. Implemented as a preprocessor in Ciao Prolog and integrated into the s(CASP) system, the approach enables efficient detection of spurious odd loops, optimized forall evaluation, and global constraint simplification. These capabilities substantially enhance both the compile-time analyzability and runtime performance of goal-directed ASP programs.

This work investigates the computational complexity and efficient solving techniques for second-order Quantified Answer Set Programming with weak constraints (2-ASP(Q)ʷ). It provides the first complete characterization of the complexity of its primary reasoning tasks within the Δ₃^P level of the polynomial hierarchy. The paper introduces a Counterexample-Guided Abstraction Refinement (CEGAR) solving strategy specifically tailored to the semantics of ASP(Q), effectively addressing non-trivial cases not handled by prior approaches. Implemented in the Casper system, the proposed method demonstrates significant performance improvements over existing solvers on multiple challenging benchmark instances, thereby confirming its effectiveness and practical applicability.