Higher-order causal theories are models of BV-logic

• URL: http://arxiv.org/abs/2205.11219v1
• Date: Mon, 23 May 2022 11:57:48 GMT
• Title: Higher-order causal theories are models of BV-logic
• Abstract summary: Caus[-] construction takes a compact closed category of basic processes and yields a *-autonomous category of higher-order processes. This paper looks at instances where the base category C satisfies additional properties yielding an affine-linear structure on Caus[C] and a substantially richer internal logic.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: The Caus[-] construction takes a compact closed category of basic processes and yields a *-autonomous category of higher-order processes obeying certain signalling/causality constraints, as dictated by the type system in the resulting category. This paper looks at instances where the base category C satisfies additional properties yielding an affine-linear structure on Caus[C] and a substantially richer internal logic. While the original construction only gave multiplicative linear logic, here we additionally obtain additives and a non-commutative, self-dual sequential product yielding a model of Guglielmi's BV logic. Furthermore, we obtain a natural interpretation for the sequential product as "A can signal to B, but not vice-versa", which sits as expected between the non-signalling tensor and the fully-signalling (i.e. unconstrained) par. Fixing matrices of positive numbers for C recovers the BV category structure of probabilistic coherence spaces identified by Blute, Panangaden, and Slavnov, restricted to normalised maps. On the other hand, fixing the category of completely positive maps gives an entirely new model of BV consisting of higher order quantum channels, encompassing recent work in the study of quantum and indefinite causal structures.

Related papers

• Symplectic H2 Model Reduction for High-Dimensional Linear Quantum Systems [3.7887687619674413]
  A symplectic Petrov-Galerkin framework is presented, in which reduced-order models automatically satisfy the PR identities by construction.<n>A symplectic variant of the iterative rational Krylov algorithm is developed and referred to as Quantum IRKA (Q-IRKA)<n>Results show that Q-IRKA is effective for large-scale linear quantum systems.
  arXiv  Detail & Related papers  (2026-05-08T02:38:57Z)
• Higher-Order Quantum Objects are Strong Profunctors [0.0]
  We construct a functor F : Caus(C) -> StProf(C1) from higher-order causal categories to the category of strong profunctors over first-order causal processes.<n>We conclude that insofar as compositional constraints can be used to express causality constraints, the profunctorial approach generalises higher-order quantum theory to a construction over general symmetric monoidal categories.
  arXiv  Detail & Related papers  (2026-03-11T18:35:59Z)
• Random-Matrix-Induced Simplicity Bias in Over-parameterized Variational Quantum Circuits [72.0643009153473]
  We show that expressive variational ansatze enter a Haar-like universality class in which both observable expectation values and parameter gradients concentrate exponentially with system size.<n>As a consequence, the hypothesis class induced by such circuits collapses with high probability to a narrow family of near-constant functions.<n>We further show that this collapse is not unavoidable: tensor-structured VQCs, including tensor-network-based and tensor-hypernetwork parameterizations, lie outside the Haar-like universality class.
  arXiv  Detail & Related papers  (2026-01-05T08:04:33Z)
• Limitation of Stoquastic Quantum Annealing: A Structural Perspective [0.0]
  We provide a structural explanation for the anti-crossing arising from the competition between the energies associated with a set of degenerate local minima.<n>This paper serves as a supplementary companion to our main work on the DIC-DAC-DOA algorithm.
  arXiv  Detail & Related papers  (2025-09-18T04:39:48Z)
• Learning Identifiable Structures Helps Avoid Bias in DNN-based Supervised Causal Learning [56.22841701016295]
  Supervised Causal Learning (SCL) is an emerging paradigm in this field. Existing Deep Neural Network (DNN)-based methods commonly adopt the "Node-Edge approach"
  arXiv  Detail & Related papers  (2025-02-15T19:10:35Z)
• Algorithmic syntactic causal identification [0.8901073744693314]
  Causal identification in causal Bayes nets (CBNs) is an important tool in causal inference. Most existing formulations of causal identification using techniques such as d-separation and do-calculus are expressed within the mathematical language of classical probability theory. We show that this restriction can be lifted by replacing the use of classical probability theory with the alternative axiomatic foundation of symmetric monoidal categories.
  arXiv  Detail & Related papers  (2024-03-14T17:14:53Z)
• A complete logic for causal consistency [0.0]
  We introduce graph types as a tool to examine causal structures over graphs in this model. The properties of graph types are then used to prove completeness for causal consistency of a new causal logic that conservatively extends pomset logic. Using the fact that causal logic conservatively extends pomset logic, we finish by giving a physically-meaningful interpretation of a separating statement between pomset and BV.
  arXiv  Detail & Related papers  (2024-03-14T11:36:53Z)
• Stable Nonconvex-Nonconcave Training via Linear Interpolation [51.668052890249726]
  This paper presents a theoretical analysis of linearahead as a principled method for stabilizing (large-scale) neural network training. We argue that instabilities in the optimization process are often caused by the nonmonotonicity of the loss landscape and show how linear can help by leveraging the theory of nonexpansive operators.
  arXiv  Detail & Related papers  (2023-10-20T12:45:12Z)
• Towards Training Without Depth Limits: Batch Normalization Without Gradient Explosion [83.90492831583997]
  We show that a batch-normalized network can keep the optimal signal propagation properties, but avoid exploding gradients in depth. We use a Multi-Layer Perceptron (MLP) with linear activations and batch-normalization that provably has bounded depth. We also design an activation shaping scheme that empirically achieves the same properties for certain non-linear activations.
  arXiv  Detail & Related papers  (2023-10-03T12:35:02Z)
• Entanglement of Sections: The pushout of entangled and parameterized quantum information [0.0]
  Recently Freedman & Hastings asked for a mathematical theory that would unify quantum entanglement/tensor-structure with parameterized/-structure. We make precise a form of the relevant pushout diagram in monoidal category theory. We show how this model category serves as categorical semantics for the linear-multiplicative fragment of Linear Homotopy Type Theory.
  arXiv  Detail & Related papers  (2023-09-13T18:28:43Z)
• Unextendibility, uncompletability, and many-copy indistinguishable ensembles [49.1574468325115]
  We show that the complement of any bipartite pure entangled state is spanned by product states which form a nonorthogonal unextendible product basis (nUPB) of maximum cardinality. We also report a class of multipartite many-copy indistinguishable ensembles for which local indistinguishability property increases with decreasing number of mixed states.
  arXiv  Detail & Related papers  (2023-03-30T16:16:41Z)
• Finding Alignments Between Interpretable Causal Variables and Distributed Neural Representations [62.65877150123775]
  Causal abstraction is a promising theoretical framework for explainable artificial intelligence. Existing causal abstraction methods require a brute-force search over alignments between the high-level model and the low-level one. We present distributed alignment search (DAS), which overcomes these limitations.
  arXiv  Detail & Related papers  (2023-03-05T00:57:49Z)
• Controlling the Complexity and Lipschitz Constant improves polynomial nets [55.121200972539114]
  We derive new complexity bounds for the set of Coupled CP-Decomposition (CCP) and Nested Coupled CP-decomposition (NCP) models of Polynomial Nets. We propose a principled regularization scheme that we evaluate experimentally in six datasets and show that it improves the accuracy as well as the robustness of the models to adversarial perturbations.
  arXiv  Detail & Related papers  (2022-02-10T14:54:29Z)
• Topological and Algebraic Structures of the Space of Atanassov's Intuitionistic Fuzzy Values [8.518591988944358]
  We show that the space of intuitionistic fuzzy values (IFVs) with the linear order based on a score function and an accuracy function has the same algebraic structure as the one induced by the linear order based on a similarity function and an accuracy function. By introducing a new operator for IFVs via the linear order based on a score function and an accuracy function, we present that such an operator is a strong negation on IFVs.
  arXiv  Detail & Related papers  (2021-11-17T06:43:02Z)
• Categories of Br\`egman operations and epistemic (co)monads [0.0]
  We construct a categorical framework for nonlinear postquantum inference, with embeddings of convex closed sets of suitable reflexive Banach spaces as objects. It provides a nonlinear convex analytic analogue of Chencov's programme of study of categories of linear positive maps between spaces of states. We show that the bregmanian approach provides some special cases of this setting.
  arXiv  Detail & Related papers  (2021-03-13T23:10:29Z)
• Models of zero-range interaction for the bosonic trimer at unitarity [91.3755431537592]
  We present the construction of quantum Hamiltonians for a three-body system consisting of identical bosons mutually coupled by a two-body interaction of zero range. For a large part of the presentation, infinite scattering length will be considered.
  arXiv  Detail & Related papers  (2020-06-03T17:54:43Z)
• Kernel and Rich Regimes in Overparametrized Models [69.40899443842443]
  We show that gradient descent on overparametrized multilayer networks can induce rich implicit biases that are not RKHS norms. We also demonstrate this transition empirically for more complex matrix factorization models and multilayer non-linear networks.
  arXiv  Detail & Related papers  (2020-02-20T15:43:02Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.