By Dirk Draheim
This e-book takes a foundational method of the semantics of probabilistic programming. It elaborates a rigorous Markov chain semantics for the probabilistic typed lambda calculus, that is the typed lambda calculus with recursion plus probabilistic choice.
The booklet starts off with a recapitulation of the elemental mathematical instruments wanted in the course of the ebook, particularly Markov chains, graph idea and area idea, and in addition explores the subject of inductive definitions. It then defines the syntax and establishes the Markov chain semantics of the probabilistic lambda calculus and, additionally, either a graph and a tree semantics. according to that, it investigates the termination habit of probabilistic courses. It introduces the notions of termination measure, bounded termination and direction stoppability and investigates their mutual relationships. finally, it defines a denotational semantics of the probabilistic lambda calculus, in line with non-stop services over chance distributions as domains.
The paintings in most cases appeals to researchers in theoretical laptop technological know-how targeting probabilistic programming, randomized algorithms, or programming language theory.
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Extra resources for Semantics of the Probabilistic Typed Lambda Calculus: Markov Chain Semantics, Termination Behavior, and Denotational Semantics
Example text
36) The fact that hitting probabilities are characterized by the equations in Eqns. 35) is important because it allows us to determine concrete hitting probability values. It is also important because the equations are needed to prove propositions about Markov chains and the system they model. 31. 31 explains how a bounded hitting probability can be calculated from the bounded hitting probabilities of a fewer number of steps. 39) t∈S Proof. Let us start with Eqn. 37), which follows from the definition of bounded hitting probabilities in Def.
29. The characterization as a linear equation system is important for us, because, based on it, we can exploit the results and algorithms of linear algebra to determine and argue about reduction probabilities of the probabilistic lambda calculus. 29 (Linear Equation System of Hitting Probabilities) Given a Markov chain X = (Xn : Ω −→ S)n 0 with transition matrix p : S × S → [0, 1], a start state s and a set of target states T ⊆ S. The vector of hitting probabilities (ηX i, T )i∈S is the least solution of the following equation system: ηX s, T = 1 ∀s ∈ T .
36 (Nodes and Sizes in Digraphs) Given a digraph G = VG , EG . , κ(G) = VG . , |G| = |VG |. Given a walk w ∈ ⊕WG , we denote the set of its nodes by κ((wi )i∈I ) = {wi | i ∈ I}, its size by |w| = |κ(w)| and its length by l(w) = #(w) − 1. Given a set of walks U ⊆ ⊕WG , we denote the set of its nodes by κ(U ) = ∪{κ(u)| u ∈ U } and its size by |U | = |κ(U )|. A walk consists of at least one node; compare with Eqn. 53). Therefore, walks of length zero also consist of one node. We distinguish between the 36 2 Preliminary Mathematics length l(w) of a walk w and its sequence length #(w).