Modular Runtime Complexity Analysis of Probabilistic While Programs

1 Preliminaries

Multisets

A (possibly infinite) multiset over a set $A$ is a mapping $M:A→\N$ . The union $⨄_{i \in I} M_{i}$ of a countably many multisets $M_{i}$ is defined by

\pa[]⨄i∈IMi(a)\defeq∑i∈IMi(a)

which forms a multiset if and only if $\sum_{i \in I} M_{i} (a)$ is finite for every $a \in A$ . The sum of a multiset $M$ with respect to $f:A→\Realpos$ is defined by

∑a∈Mf(a)\defeq∑a∈AM(a)⋅f(a)

We use set-like notations for multisets: $\emptyset$ denotes the empty multiset $∅(a)\defeq0$ , $\msetai∣i∈I$ is the multiset $M$ with $M(a)=|\seti∈I∣ai=a|$ , and $\mseta1,…,an$ is its special case where $I=\set1,…,n$ is finite.

Weighted Abstract Reduction Systems

A weighted ARS over state space $A$ is a ternary relation $→⊆A×A×\Realnonneg$ . We write $a→\ATcb$ meaning $\tpa,b,c∈→$ . We define the ARS $\to^{c} \subseteq A \times A$ induced by $\to$ at cost $c∈\Realnonneg$ , by the following inference rules:

\infera→0a\infera→cba→\ATcb\infera→c+dba→ca′a′→db

We say $a \in A$ is a normal form with respect to $\to$ if there exists no $b \in A$ with $a \to^{+} b$ . The set of normal forms with respect to $\to$ is denoted by $\NF(→)$ . The potential of $a \in A$ with respect to $\to$ is defined by $\pot[→]a\defeqsup\setc∣∃b. a→cb$ . The weighted ARS $\to$ is called strongly bounded on $S \subseteq A$ , $\SB→(S)$ , if for every $a \in S$ , there exists $p∈\Realnonneg$ such that $a \to^{c} b$ implies $c \leq p$ ; This is equivalent to saying that $\pot[→]a≤∞$ for every $a \in S$ .

Weighted Probabilistic Abstract Reduction Systems

A multidistribution on a set $A$ is a multiset $\mdist$ of pairs of $a \in A$ and $0 < p \leq 1$ , written $p : a$ , satisfying $\sz\mdist\defeq∑p:a∈\mdistp ≤ 1$ . We denote the set of multidistributions on $A$ by $\MDistA$ . Multidistributions are closed under convex multiset unions $⨄i∈Ipi⋅\mdisti\defeq∑i∈Ipi⋅\sz\mdisti≤1$ for every finite or countable infinite index set $I$

and probabilities

p_{i} > 0

with

\sum_{i \in I} p_{i} \leq 1

, where scalar multiplication is defined by

p⋅\msetqi:ai∣i∈I\defeq\msetp⋅qi:ai∣i∈I

for

0 < p \leq 1

The restriction of a multidistribution $\mdist∈\MDistA$ to a set $P \subseteq A$ is defined by $\restrict\mdistP\defsym\prmsp:a∣p:a∈\mdist,a∈P$ . For a function $f : A \to B$ , we denote by $\fmapf$ its homomorphic extension $\fmapf:\MDistA→\MDistB$ defined by

\fmapf(\prmspi:ai∣i∈I)\defsym\prmspi:f(ai)∣i∈I\tpkt

For $\mdist∈\MDistA$ , we define the expectation of a function $f\ofdomA→\Realposinf$ as $\E\mdist(f)\defsym∑p:a∈\mdistp⋅f(a)$ . Notice that $\E⨄i∈Ipi⋅\mdisti(f)=∑i∈Ipi⋅\E\mdisti(f)$ .

Definition 1.

A weighted probabilistic ARS over $A$ is a set $→⊆A×\MDistA×\Realnonneg$ . As before, we may write $a→\ATcμ$ for $\tpa,μ,c∈→$ . We define the weighted ARS $\wars→$ over $\MDistA$ induced by $\to$ as follows:

For a weighted probabilistic ARS $\to$ , let us define $\pot[→]a\defeq\pot[\wars→]\mset1:a$ . A weighted probabilistic ARS $\to$ over $A$ is strongly bounded on a set $S \subseteq A$ if $\SB\wars→(\set\mset1:a∣a∈S)$ , i.e., $\pot[→]a<∞$ for all $S \in A$ .

2 Probabilistic While

We consider an imperative language $\lang$ in the spirit of Dijkstra’s Guarded Command Language [Dijkstra75], endowed with primitives for sampling from discrete distributions as well as non-deterministic and probabilistic choice. Let $\Var$ denote a finite set of integer-valued variables $\var,\vartwo,…$ . We denote by $\Store\defsym\Var→\Z$ the set of stores, that associate variables with their integer contents. The syntax of program commands $\Cmd$ over $\Var$ is given by the following grammar.

$(\Cmd)$	$\cmd,\cmdtwo$	$\bnfdef\skipc$	effectless operation
$∣\tickcr$	resource consumption
$∣\haltc$	termination
$∣\var\passign\expd$	probabilistic assignment
$∣$

Modular Runtime Complexity Analysis of Probabilistic While Programs

Automatic Complexity Analysis of Integer Programs via Triangular Weakly Non-Linear Loops

Inferring Expected Runtimes of Probabilistic Integer Programs Using Expected Sizes

Runtime Analysis of Quantum Programs: A Formal Approach

Cost Analysis of Nondeterministic Probabilistic Programs

Automated Amortised Resource Analysis for Term Rewrite Systems

C3: Lightweight Incrementalized MCMC for Probabilistic Programs using Continuations and Callsite Caching

Slicing of Probabilistic Programs based on Specifications

1 Preliminaries

Multisets

Weighted Abstract Reduction Systems

Weighted Probabilistic Abstract Reduction Systems

Definition 1.

2 Probabilistic While

Modular Runtime Complexity Analysis of Probabilistic While Programs

Related Research

Automatic Complexity Analysis of Integer Programs via Triangular Weakly Non-Linear Loops

Inferring Expected Runtimes of Probabilistic Integer Programs Using Expected Sizes

Runtime Analysis of Quantum Programs: A Formal Approach

Cost Analysis of Nondeterministic Probabilistic Programs

Automated Amortised Resource Analysis for Term Rewrite Systems

C3: Lightweight Incrementalized MCMC for Probabilistic Programs using Continuations and Callsite Caching

Slicing of Probabilistic Programs based on Specifications

1 Preliminaries

Multisets

Weighted Abstract Reduction Systems

Weighted Probabilistic Abstract Reduction Systems

Definition 1.

2 Probabilistic While