Kuntions

Kuntions are a theoretical construct used in abstract mathematics and theoretical computer science to model generalized function-like processes that yield probabilistic outcomes rather than a single result. In this framework, a kuntion from a set A to a set B assigns to each input a probability distribution over B, rather than a single value. Deterministic functions correspond to degenerate kuntions where each input has a single output with probability one.

Formal definition and composition

A kuntion f from A to B can be described as a family {f(a)}_{a in A} of probability distributions over B. Equivalently, f can be viewed as a function f: A -> Dist(B), where Dist(B) denotes the set of probability distributions on B. The probability that f(a) outputs b is denoted p_f(a, b), with sum_{b in B} p_f(a, b) = 1 for all a in A.

Given kuntions f: A -> B and g: B -> C, their composite h = g ∘ f is defined by h(a, c) = sum_{b in B} p_f(a, b) * p_g(b, c). This yields a distribution over C for each a. The identity kuntion id_A: A -> A is defined by p_id(a, a) = 1, giving a deterministic outcome equal to the input.

Applications and context

Kuntions are used in discussions of nondeterministic processes, probabilistic programming semantics, and certain toy models in category theory where function-like structure is enriched with probability. They illustrate how composition preserves probabilistic structure, and how deterministic functions appear as special cases.

Examples

A simple example uses A = {0,1} and B = {0,1}. If f(0) = {0:0.7, 1:0.3} and f(1) = {0:0.2, 1:0.8}, and g(0) = {0:0.5, 1:0.5}, g(1) = {0:0.9, 1:0.1}, then h = g ∘ f assigns to each input a resulting distribution on C with probabilities computed by the composition rule.