Unifying Semirings and Monads in Probabilistic Programming Languages

Semirings are a fundamental concept in computation, providing a way to represent and combine multiple values or effects in a systematic manner. In essence, semirings are algebraic structures that generalize the notion of a set with two binary operations (usually called multiplication and addition). These operations must satisfy certain properties, such as associativity and commutativity, which make them useful for modeling various computational phenomena.
One key application of semirings is in probability theory, where they provide a way to represent unnormalized measures or opinions. For instance, the non-negative reals (R+) are a common example of a semiring used in Bayesian statistics and other optimization scenarios. Semirings can also be applied to input/output messages, where they represent a set of possible inputs or outputs that can be combined using logical operations.
The paper proposes two key methods for working with semirings: tensors and convexity classes. Tensors allow for calculating the commutative combination of different effects, which is particularly useful in probability theory. Convexity classes provide a way to generalize these combinations to more complex scenarios, such as iteration and state. These methods have been shown to be effective in demystifying complex concepts in computation and providing new insights into how semirings can be used to model various phenomena.
In summary, semirings are a powerful tool for understanding and modeling computational phenomena. By providing a way to represent and combine multiple values or effects in a systematic manner, they have applications in probability theory, input/output messages, and more. The proposed methods of tensors and convexity classes offer new insights into how semirings can be used to model complex scenarios, making them an important area of research in the field of computation.