Postselecting probabilistic finite state recognizers and verifiers

TL;DR

This paper explores the computational capabilities of postselecting probabilistic finite automata, demonstrating their ability to recognize and verify complex languages using rational and real-valued transitions, including uncountably many languages.

Contribution

It introduces the power of bounded-error postselecting probabilistic automata with rational and real-valued transitions, showing their ability to recognize and verify complex languages beyond regular ones.

Findings

PostPFAs with rational transitions can verify some nonregular unary languages.

Real-valued transitions enable recognition of uncountably many binary languages.

PostPFAs can verify uncountably many unary languages with a prover.

Abstract

In this paper, we investigate the computational and verification power of bounded-error postselecting realtime probabilistic finite state automata (PostPFAs). We show that PostPFAs using rational-valued transitions can do different variants of equality checks and they can verify some nonregular unary languages. Then, we allow them to use real-valued transitions (magic-coins) and show that they can recognize uncountably many binary languages by help of a counter and verify uncountably many unary languages by help of a prover. We also present some corollaries on probabilistic counter automata.