Probabilistic Arithmetic Automata and their Applications

TL;DR

This paper introduces probabilistic arithmetic automata (PAAs), a versatile framework for modeling and analyzing probabilistic computations across various applications like pattern matching, peptide analysis, and sequencing, with exact distribution calculation algorithms.

Contribution

It presents a unifying framework of PAAs for diverse probabilistic problems, introduces deterministic automata for constructing PAAs, and demonstrates multiple applications in computational biology.

Findings

PAAs can model various probabilistic processes in biology and computer science.

Algorithms for exact distribution calculations are provided.

The framework is flexible and applicable to multiple domains.

Abstract

We present probabilistic arithmetic automata (PAAs), a general model to describe chains of operations whose operands depend on chance, along with two different algorithms to exactly calculate the distribution of the results obtained by such probabilistic calculations. PAAs provide a unifying framework to approach many problems arising in computational biology and elsewhere. Here, we present five different applications, namely (1) pattern matching statistics on random texts, including the computation of the distribution of occurrence counts, waiting time and clump size under HMM background models; (2) exact analysis of window-based pattern matching algorithms; (3) sensitivity of filtration seeds used to detect candidate sequence alignments; (4) length and mass statistics of peptide fragments resulting from enzymatic cleavage reactions; and (5) read length statistics of 454 sequencing reads. The diversity of these applications indicates the flexibility and unifying character of the presented framework. While the construction of a PAA depends on the particular application, we single out a frequently applicable construction method for pattern statistics: We introduce deterministic arithmetic automata (DAAs) to model deterministic calculations on sequences, and demonstrate how to construct a PAA from a given DAA and a finite-memory random text model. We show how to transform a finite automaton into a DAA and then into the corresponding PAA.