Decision Questions for Probabilistic Automata on Small Alphabets

TL;DR

The paper investigates emptiness and $\lambda$-reachability for unary and binary probabilistic finite automata, focusing on how ambiguity and alphabet size affect computational complexity. It develops a decomposition framework and leverages Jordan form representations to express acceptance probabilities as $\mathcal{P}(a^k)=c+\sum_i p_i(k)\lambda_i^k$, enabling $EXPTIME$ decisions for unary polynomially ambiguous PFAs and $NP$-time decisions under binary and commuting-matrix constraints. The authors prove NP-hardness for several restricted cases, including unary finitely ambiguous PFAs with a $\{0,1\}$-matrix and binary polynomially ambiguous PFAs with fixed commuting matrices (dimensions $9$, $37$, and $40$ for $\lambda$-reachability and emptiness variants), tying results to binary quadratic Diophantine equations and the Skolem problem. They also discuss connections to linear recurrence sequences and outline open questions, notably the exact complexity for polynomially ambiguous unary PFAs, offering a path toward tighter characterizations and potential further hardness results.

Abstract

We study the emptiness and $λ$-reachability problems for unary and binary Probabilistic Finite Automata (PFA) and characterise the complexity of these problems in terms of the degree of ambiguity of the automaton and the size of its alphabet. Our main result is that emptiness and $λ$-reachability are solvable in EXPTIME for polynomially ambiguous unary PFA and if, in addition, the transition matrix is binary, we show they are in NP. In contrast to the Skolem-hardness of the $λ$-reachability and emptiness problems for exponentially ambiguous unary PFA, we show that these problems are NP-hard even for finitely ambiguous unary PFA. For binary polynomially ambiguous PFA with fixed and commuting transition matrices, we prove NP-hardness of the $λ$-reachability (dimension 9), nonstrict emptiness (dimension 37) and strict emptiness (dimension 40) problems.

Figures (2)

• Figure 1: EDA Property
• Figure 2: IDA$_d$

Theorems & Definitions (28)

• Lemma 2.1
• Theorem 2.2: IR86Re77WS91
• Theorem 3.1
• proof
• Theorem 3.2
• Lemma 3.3
• proof
• Lemma 3.4
• proof
• Lemma 3.5