Centrala begrepp
Probabilistic automatic complexity of finite strings is characterized by specific forms, revealing structural properties.
Sammanfattning
The content introduces a new complexity measure for finite strings using probabilistic finite-state automata (PFAs). It explores the properties of PFA complexity, including a variant with a real-valued parameter. The article relates PFA complexity to deterministic and nondeterministic finite-state automata complexities. It provides a complete classification of binary strings with complexity 2 and discusses the computability of the PFA complexity. The content also delves into the concept of probabilistic automatic complexity with a lower bound on the gap between acceptance probabilities. It presents theorems, propositions, and proofs related to the classification of binary strings with complexity 2, highlighting the unique structural properties of these strings. The article further discusses the connection between PFAs and iterated function systems, demonstrating how PFAs can be used to generate fractal images. It concludes with a detailed proof of the classification of binary strings with complexity 2.
Statistik
The PFA complexity AP(x) is defined as the least number of states of a PFA for which x is the most likely string of its length to be accepted.
The variant AP,δ(x) adds a real-valued parameter δ specifying a lower bound on the gap in acceptance probabilities between x and other strings.
Theorem 4.1 states that for a binary string w, AP(w) = 2 if and only if w is of specific forms: injm, injmi, in(ji)m, or inj(ij)m for some n, m ≥ 0, where i, j ∈ {0, 1}.
Corollary 4.13 highlights that the quantity AN(w) - AP(w) may be arbitrarily large among binary w.
Theorem 5.1 shows that AP,δ(w) is computable for all w and almost all δ ≠ 0.
Citat
"AP does appear to capture some intuitive structural properties of strings."
"The study of AN has been continued by Kjos-Hanssen."