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Probabilistic automatic complexity of finite strings

Kenneth Gill

TL;DR

This work introduces $A_P(x)$, the probabilistic automatic complexity of a finite string, defined via PFAs as the smallest number of states that yield a unique most-likely accepted string of a given length; it also defines a gap-parameterized variant $A_{P,δ}(x)$. The authors establish tight bounds relating $A_P$ to $A_N$ and $A_D$, prove δ-computability of $A_{P,δ}$ for all $δ$, and present a complete binary-string classification for strings with $A_P=2$, using an affine iterated function system (IFS) correspondence to analyze witnesses. They develop a computable-analysis framework proving uniform computability of $A_{P,δ}$ across inputs and noting a countable set of discontinuities tied to maximal gaps; they also explore extensions, variations, and potential refinements via GPFA generalizations, gap-structure functions, witness-bit-length, and measures over witnesses. Together, these results illuminate the algorithmic content and computability of probabilistic automata-based string complexity, and suggest rich avenues for further refinement and applications in formal language and complexity theory.

Abstract

We introduce a new complexity measure for finite strings using probabilistic finite-state automata (PFAs), in the same spirit as existing notions employing DFAs and NFAs, and explore its properties. The PFA complexity $A_P(x)$ is the least number of states of a PFA for which $x$ is the most likely string of its length to be accepted. The variant $A_{P,δ}(x)$ adds a real-valued parameter $δ$ specifying a required lower bound on the gap in acceptance probabilities between $x$ and other strings. We prove $A_{P,δ}$ is $δ$-computable for all $δ$, relate $A_P$ to the DFA and NFA complexities, and obtain a complete classification of binary strings with $A_P=2$. Finally, we discuss several other variations on $A_P$ with a view to obtaining additional desirable properties.

Probabilistic automatic complexity of finite strings

TL;DR

This work introduces , the probabilistic automatic complexity of a finite string, defined via PFAs as the smallest number of states that yield a unique most-likely accepted string of a given length; it also defines a gap-parameterized variant . The authors establish tight bounds relating to and , prove δ-computability of for all , and present a complete binary-string classification for strings with , using an affine iterated function system (IFS) correspondence to analyze witnesses. They develop a computable-analysis framework proving uniform computability of across inputs and noting a countable set of discontinuities tied to maximal gaps; they also explore extensions, variations, and potential refinements via GPFA generalizations, gap-structure functions, witness-bit-length, and measures over witnesses. Together, these results illuminate the algorithmic content and computability of probabilistic automata-based string complexity, and suggest rich avenues for further refinement and applications in formal language and complexity theory.

Abstract

We introduce a new complexity measure for finite strings using probabilistic finite-state automata (PFAs), in the same spirit as existing notions employing DFAs and NFAs, and explore its properties. The PFA complexity is the least number of states of a PFA for which is the most likely string of its length to be accepted. The variant adds a real-valued parameter specifying a required lower bound on the gap in acceptance probabilities between and other strings. We prove is -computable for all , relate to the DFA and NFA complexities, and obtain a complete classification of binary strings with . Finally, we discuss several other variations on with a view to obtaining additional desirable properties.
Paper Structure (17 sections, 23 theorems, 43 equations, 3 figures)

This paper contains 17 sections, 23 theorems, 43 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. First results on $A_P$
  4. Classification of binary strings with $A_P=2$
  5. The iterated function system approach
  6. Proof of \ref{['classification2']}
  7. Proof of \ref{['classification2-reverse']}
  8. Further remarks
  9. Computability of probabilistic complexity
  10. Background in computable analysis
  11. Proofs of \ref{['apdeltacomp']} and \ref{['apcomp']}
  12. Remarks
  13. Other approaches to probabilistic complexity
  14. Relaxing the definition of a PFA
  15. Gap structure function
  16. ...and 2 more sections

Key Result

Theorem 1.3

For $w \in \{i,j\}^\ast$, we have $A_P(w)=2$ if and only if $w$ is of the form for some $n,m\geq0$.

Figures (3)

  • Figure 1: An example of a PFA. Numbers in parentheses are transition probabilities, so that the PFA starts in state $s_1$ with probability 1. $s_3$ is the unique accepting state.
  • Figure 2: An NFA witnessing that $A_N(0001101)=4$.
  • Figure 3: Subcases for $f_0$, $f_1$ with positive slope (Case 2)

Theorems & Definitions (32)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Theorem 1.3
  • Corollary 1.3
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3: Shallit and Wang SW01
  • Definition 2.4: Hyde H13
  • Remark 2.5
  • ...and 22 more