axiom design

This work addresses the algebraic classification and tractability boundary of constraint satisfaction problems (CSPs). Focusing on conservative CSP templates, we develop a unified theoretical framework linking absorptive algebras to local consistency solvability—the first systematic characterization of their deep interplay. Using subpower-finite algebraic modeling, absorptive subalgebra analysis, and polynomial-time reductions, we rigorously establish a complexity dichotomy theorem for conservative CSPs: every template is either solvable in polynomial time via local consistency algorithms or NP-complete. Our result provides a complete tractability criterion for conservative CSPs and strengthens the expressive power of algebraic methods in CSP classification. Moreover, it furnishes a rigorous theoretical foundation and decision framework for consistency-based solving approaches.

Optimization and feasibility checking for algebraic functions involving radicals, rational expressions, and similar non-polynomial terms pose significant challenges due to the “curse of dimensionality” inherent in conventional layered auxiliary-variable lifting. Method: This paper introduces a compact polynomial reconstruction framework that replaces each algebraic function with a single new variable, coupled with an integrated algorithmic pipeline grounded in real algebraic geometry: implicitization, branch-isolating inequality generation, and Positivstellensatz/SOS-based feasibility verification. Contribution/Results: By avoiding redundant variable proliferation, the method substantially improves modeling efficiency and scalability of polynomial programming formulations. On enhanced classical benchmarks, it achieves up to 50× speedup over naive reconstruction. The approach provides both theoretical guarantees and a practical toolkit for efficiently translating algebraic programs into polynomial optimization problems.

This work addresses the satisfiability checking and quantifier elimination problems for nonlinear real arithmetic (NRA) formulas featuring both Boolean structure and quantifiers. We present the first extension of cylindrical algebraic coverings (CAC) to full first-order logic formula verification. Our approach introduces a novel CAC variant that integrates CAD-based covering construction, hierarchical quantifier handling, explicit Boolean structure incorporation, and adaptive splitting and pruning heuristics. Unlike conventional methods, our framework avoids constructing a complete cylindrical algebraic decomposition (CAD), thereby substantially reducing computational complexity. Experimental evaluation on diverse nonlinear quantified benchmarks demonstrates that our method outperforms state-of-the-art SMT solvers—including Z3 and CVC5—as well as specialized quantifier elimination tools such as QEPCAD and Redlog, in both solution accuracy and runtime efficiency. The gains are particularly pronounced on high-dimensional, sparse constraint instances.

This work addresses the automatic inference of closed-form bounds for recursively defined functions—such as operator fixed points or solutions to functional equations—arising in program cost analysis, loop acceleration, and hybrid system verification. We introduce the *B-bound abstract domain*, which approximates numerical functions via conjunctions of predefined bounding functions, enabling synthesis of highly nonlinear invariants. To systematically lift Galois connections from value domains to function spaces, we design an *abstract domain functor*. Our approach integrates constraint-driven abstract construction, higher-order abstract interpretation, operator fixed-point theory, and symbolic-numerical dimensionality reduction. Experiments demonstrate that the framework efficiently handles multivariate, piecewise, and non-discrete functions; significantly improves nonlinear invariant inference; simplifies transition function design; and achieves end-to-end automation across diverse verification and analysis tasks.

This paper addresses the challenge of jointly modeling cardinality restrictions and concrete domains in description logics. We introduce a novel logic, $mathcal{ALCOSCC}(mathfrak{D})$, which integrates cardinality constraints—expressed via set-based comparisons of role successors—with concrete domain constraints—such as comparisons over feature values. It is the first formalism unifying the set-comparison capabilities of $mathcal{ALCSCC}$ with the concrete domain reasoning of $mathcal{ALC}(mathfrak{D})$, supporting joint reasoning over named individuals, sets of role successors, and numerical constraints. Assuming $mathfrak{D}$ is an $omega$-admissible concrete domain, we prove that knowledge base consistency is ExpTime-complete—matching the complexity of basic $mathcal{ALC}$—and rigorously delineate the decidability frontier: several natural extensions render the logic undecidable. Our work provides a compact syntax and sound, optimal-complexity reasoning support for highly expressive knowledge representation.

This work proposes a novel method to overcome the limitations of the classical absolute positivity criterion, which fails to handle nonlinear polynomial constraints involving universal quantifiers. Specifically, the approach addresses ∃∀ inequalities over the natural numbers by integrating monotonic algebra with well-founded order theory, thereby dispensing with the absolute positivity assumption. This advancement substantially broadens the class of constructible nonlinear polynomial interpretations. Experimental results demonstrate that the technique successfully solves constraint instances previously intractable to existing methods, thus extending the applicability of polynomial interpretations in termination and complexity analysis of term rewriting systems.

Traditional coequational methods suffer from limitations in expressiveness and usability. This work proposes “equational path constraints” as an algebraic alternative: by assigning a pair of values to each path in a coalgebra and enforcing their equality, it algebraically characterizes finite-behavior properties, thereby enabling an axiomatic definition of covarieties and the construction of final coalgebras. The approach establishes a connection with coequations in the setting of monads and provides an upper bound on the number of colors required. Combining category theory, coalgebraic techniques, and Adámek–Barr final sequence constructions, the method is successfully validated across multiple case studies—including automata commutativity, differential equations, bi-infinite streams, and modal frame conditions—demonstrating its effectiveness and broad applicability.

This work addresses the problem of deciding robust safety for dynamical systems governed by polynomial differential equations over bounded time horizons. By reducing δ-robust safety to the sound axiomatization of polynomial invariants, the authors construct the first complete logical proof system and integrate it with a computable algorithm to decide safety for arbitrary perturbation parameters δ. Innovatively leveraging subanalytic geometry, the approach enables inductive safety proofs and approximate decidability for general hybrid dynamical systems without requiring positive separation between initial and postconditions. This paper establishes, for the first time, a complete axiomatization framework for robust safety and provides an effective symbolic verification algorithm.

This study addresses the inefficiency and poor interpretability of traditional cylindrical algebraic decomposition (CAD) methods in quantifier elimination problems involving multiple equality constraints, particularly in characterizing the relationship between parameters and unknowns. The authors propose a refined partitioning strategy for the parameter space that explicitly distinguishes between cases yielding finitely many versus infinitely many solutions. Under specific conditions, this approach substantially simplifies the equality projection steps in CAD, thereby overcoming limitations inherent in existing theoretical frameworks. The method not only enables an explicit description of parameter-dependent solution structures but also significantly enhances computational efficiency and result interpretability in applications such as approximation theory, classification of robotic singularities, and modeling of biochemical systems.

This work addresses the construction of most general solutions for parameterized constraint formulas of the form ∃x₁…∃xₙ φ(x₁,…,xₙ,y₁,…,yₘ) within theories 𝒯 that admit elimination of specific existential quantifiers, where φ is a quantifier-free conjunction of literals and the yᵢ are parameters. By introducing conditional function symbols that capture “if-then-else” constructs, the authors generalize existing results on the existence of most general unifiers in discriminator clusters. Integrating parameterized constraint solving with algebraic semantic characterizations, they establish a unified framework for constructing most general solutions. This approach substantially broadens the scope of applicability compared to prior methods, and its effectiveness and generality are demonstrated through illustrative examples.