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

Authors

Inferring Expected Runtimes of Probabilistic Integer Programs Using Expected Sizes

Runtime Analysis of Quantum Programs: A Formal Approach

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

Cost Analysis of Nondeterministic Probabilistic Programs

Automated Amortised Resource Analysis for Term Rewrite Systems

An Approach to Incremental and Modular Context-sensitive Analysis of Logic Programs

Runtime Complexity Analysis of Logically Constrained Rewriting

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

Authors

Inferring Expected Runtimes of Probabilistic Integer Programs Using Expected Sizes

Runtime Analysis of Quantum Programs: A Formal Approach

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

Cost Analysis of Nondeterministic Probabilistic Programs

Automated Amortised Resource Analysis for Term Rewrite Systems

An Approach to Incremental and Modular Context-sensitive Analysis of Logic Programs

Runtime Complexity Analysis of Logically Constrained Rewriting

This week in AI

1 Preliminaries

Multisets

Weighted Abstract Reduction Systems

Weighted Probabilistic Abstract Reduction Systems

Definition 1.

2 Probabilistic While