The Tensor-Plus Calculus

2025-12-26

AI Summary

Graphical languages conventionally require explicit syntactic annotations—such as tapes or world labels—to distinguish multiplicative (synchronizing) from additive (branching) parallelism, undermining concision and mathematical elegance. Method: We introduce a *colored PROP* framework wherein multiplicative and additive monoidal structures are implicitly distinguished by contextual coloring, eliminating the need for auxiliary syntax. Semantics are uniformly assigned via parameterization over commutative semirings, enabling integrated modeling of nondeterministic, probabilistic, and quantum computation. Contribution/Results: This work achieves, for the first time in a colored PROP setting, the implicit separation of two monadic structures; it establishes an internal language for semiadditive categories and proves both semantic completeness and equational completeness of its associated theory. The resulting framework provides a universal, mathematically rigorous, and syntactically streamlined graphical foundation for parallel computation.

Technology Category

• Machine Learning: Probabilistic Circuits and Graphical Models
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Reasoning under Uncertainty: Probabilistic Programming

Application Category

• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Data transparency and provenance

Abstract

We propose a graphical language that accommodates two monoidal structures: a multiplicative one for pairing and an additional one for branching. In this colored PROP, whether wires in parallel are linked through the multiplicative structure or the additive structure is implicit and determined contextually rather than explicitly through tapes, world annotations, or other techniques, as is usually the case in the literature. The diagrams are used as parameter elements of a commutative semiring, whose choice is determined by the kind of computation we want to model, such as non-deterministic, probabilistic, or quantum. Given such a semiring, we provide a categorical semantics of diagrams and show the language as universal for it. We also provide an equational theory to identify diagrams that share the same semantics and show that the theory is sound and complete and captures semantical equivalence. In categorical terms, we design an internal language for semiadditive categories (C,+,0) with a symmetric monoidal structure (C,x,1) distributive over it, and such that the homset C(1,1) is isomorphic to a given commutative semiring, e.g., the semiring of non-negative real numbers for the probabilistic case.

Problem

• Designs a graphical language for monoidal structures
• Provides categorical semantics for semiring-based computations
• Develops equational theory for semantic diagram equivalence

Innovation

• Graphical language with two monoidal structures
• Implicit wiring through contextual determination
• Diagrams as parameter elements of semiring