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Accepting Normalization via Markov Magmoids

Elena Di Lavore, Mario Román

TL;DR

This work develops a synthetic, category-free approach to normalization in probabilistic semantics by introducing distributive sesquilaws, which capture multiplicativity up to an idempotent and justify a non-associative yet structurally rich normalization magmoid. It shows that normalized channels form a unital magmoid equipped with a monoidal structure and interacts with subdistributions via a distributive sesquilaw, yielding an action of Subd on Norm and a coherent theory of guarded choice through tricocycloids. The paper also clarifies the relation between normalized kernels and stochastic relations, arguing that stochastic relations are best modeled as a magmoid rather than a category, with support providing a morphism to May-Must and Dijkstra-type relation frameworks. Overall, it provides a unifying algebraic framework that captures normalization, guarded choice, and stochastic-relational analogies, connecting Markov-cat like semantics with relational abstractions. This advances probabilistic programming semantics by offering a principled way to reason about normalization without forcing strict categorical associativity.

Abstract

Normalization of probability distributions is not a distributive law. We introduce distributive sesquilaws to abstract normalization and, from their axioms, we derive some of its properties. In particular, normalized channels form a unital magmoid with an action of the category of subdistributions; its possibilistic analogue is the action from may-must relations into the category of relations. We argue that the magmoid of normalized channels is the stochastic analog of the category of relations.

Accepting Normalization via Markov Magmoids

TL;DR

This work develops a synthetic, category-free approach to normalization in probabilistic semantics by introducing distributive sesquilaws, which capture multiplicativity up to an idempotent and justify a non-associative yet structurally rich normalization magmoid. It shows that normalized channels form a unital magmoid equipped with a monoidal structure and interacts with subdistributions via a distributive sesquilaw, yielding an action of Subd on Norm and a coherent theory of guarded choice through tricocycloids. The paper also clarifies the relation between normalized kernels and stochastic relations, arguing that stochastic relations are best modeled as a magmoid rather than a category, with support providing a morphism to May-Must and Dijkstra-type relation frameworks. Overall, it provides a unifying algebraic framework that captures normalization, guarded choice, and stochastic-relational analogies, connecting Markov-cat like semantics with relational abstractions. This advances probabilistic programming semantics by offering a principled way to reason about normalization without forcing strict categorical associativity.

Abstract

Normalization of probability distributions is not a distributive law. We introduce distributive sesquilaws to abstract normalization and, from their axioms, we derive some of its properties. In particular, normalized channels form a unital magmoid with an action of the category of subdistributions; its possibilistic analogue is the action from may-must relations into the category of relations. We argue that the magmoid of normalized channels is the stochastic analog of the category of relations.

Paper Structure

This paper contains 24 sections, 14 theorems, 33 equations, 3 figures.

Table of Contents

  1. Introduction
  2. The Quest for Normalized Kernel Composition
  3. (a) Work up-to-scalar
  4. (b) Work up-to-parameterized-scalar
  5. (c) Work up-to-failure
  6. Accepting Normalization
  7. Normalization
  8. Normalization
  9. The normalization magmoid
  10. Distributive Laws
  11. Subdistributions
  12. Partial distributions
  13. Distributive Sesquilaws
  14. Almost distributive laws
  15. Distributive Sesquilaws
  16. ...and 9 more sections

Key Result

proposition 1

Normalization of two distributions is the normalization of their joint independent distribution, $\mathsf{n}(f ⊗ g) = \mathsf{n}(f) ⊗ \mathsf{n}(g)$.

Figures (3)

  • Figure 1: Renormalization equation.
  • Figure 2: Proof of the multiplicativity of the action induced by a distributive sesquilaw.
  • Figure 3: Definition of the action.

Theorems & Definitions (40)

  • Definition 1: Normalization
  • proposition 1: Normalization is monoidal
  • proof
  • Definition 2: Unital magmoid
  • proposition 2: Normalization magmoid
  • proof
  • Definition 3: Associating morphisms of a magmoid
  • proof
  • Definition 4: Strict monoidal magmoid
  • Remark 5: Coherence for monoidal magmoids
  • ...and 30 more