Monotonic Algebras
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Related Articles from SNS
Passive Learning of Symbolic Automata over Monotonic Algebras
arXiv:2606.06050v1 Announce Type: new Abstract: Symbolic automata extend classical finite-state automata to handle large or infinite alphabets by labeling transitions by predicates coming from a boolean algebra. Many results from automata theory have been lifted to this model, and it has proved its usefulness for example in multiple software verification applications. Here, we tackle the passive learning problem of identification in the limit, i.e. learning a model from a sample without...
Convex algebras on an interval with semicontinuous monotone operations
arXiv:2603.14955v2 Announce Type: replace Abstract: In a recent work of Matteo Mio on compact quantitative equational theories (here compact means that all its consequences are derivable by means of finite proofs) convex algebras on the carrier set [0,1] whose operations are monotone and satisfy certain semicontinuity properties occurred. We fully classify those algebraic structures by giving an explicit construction of all possible convex operations on [0,1] possessing the mentioned...
Kronecker products and iterated matrix multiplication
arXiv:2606.08363v1 Announce Type: new Abstract: We observe that the Kronecker product of tensors is the operation that converts the determinant polynomial into Cayley's first hyperdeterminant. We apply the Kronecker product to iterated matrix multiplication, which results in the hypercomputant, a VNP-complete and VW[1]-complete polynomial whose hardness we prove via the equivariance of the Kronecker product. The construction works over arbitrary commutative semirings and also for the tensor...
Hessian-recovery-based C0 finite element methods for non-divergence form elliptic equations
arXiv:2606.03276v1 Announce Type: new Abstract: A Hessian-recovery-based C0 finite element framework is proposed for second-order elliptic equations in non-divergence form. The construction is based on a direct approximation of the strong non-divergence operator: the Hessian D2u is replaced by a recovered Hessian Hhuh, so that A : D2u is approximated by A : Hhuh. The resulting discretizations include a nodal formulation and a Galerkin-type formulation for general Lagrange finite element...
Evaluating Relational Reasoning in LLMs with REL
arXiv:2604.12176v2 Announce Type: replace Abstract: Relational reasoning is the ability to infer relations that jointly bind multiple entities, attributes, or variables. This ability is central to scientific reasoning, but existing evaluations of relational reasoning in large language models often focus on structured inputs such as tables, graphs, or synthetic tasks, and do not isolate the difficulty introduced by higher-arity relational binding. We study this problem through the lens of...
State-Dependent Lyapunov Analysis of Rank-1 Matrix Factorization
Announce Type: replace Abstract: We study gradient descent for rank-1 matrix factorization through a state-dependent Lyapunov perspective. The central object is a parameterized quadratic certificate $I(\delta;\,\cdot)$ whose boundary-inward property induces a monotone state parameter $\delta_t$, thereby certifying that the trajectory is confined to a shrinking family of level sets. For certified initializations below the critical step size, this mechanism proves convergence to global minimizers.
Variational Free Energy Pivot Selection for Pivoted Cholesky
arXiv:2606.01821v1 Announce Type: new Abstract: Pivoted Cholesky factorizations construct low-rank approximations of symmetric positive definite matrices by sequentially selecting pivots from the residual diagonal. Classical greedy and randomized rules, such as randomly pivoted Cholesky, target the algebraic trace-norm error of the residual. In many applications, however, the matrix enters a nonlinear matrix functional whose value, not the trace-norm error, determines solution quality, and...