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Power of Counting by Nonuniform Families of Polynomial-Size Finite Automata

Tomoyuki Yamakami

TL;DR

Power of Counting by Nonuniform Families of Polynomial-Size Finite Automata develops counting-based frameworks (1# and 1Gap) for nonuniform polynomial-size automata and their promise problems, and grounds them in a broad landscape of counting classes (1U, 1⊕, 1C_{=}, 1SP, 1P) tied to nonuniform automata. It formalizes 1# and 1Gap via witnessable function families (including 1F, 1F_{N}, 1F_{Z}) and analyzes their relationships to 1D/1N and pushdown variants (1DPD, 1NPD), achieving several major separations (e.g., 1N ⊊ co-1C_{=}, parity vs 1P) using a novel linkage to one-tape linear-time machines with linear-size advice. A key methodological advance is exploiting the one-tape linear-time connection (via lemmas such as property-1cequal and property-onep) to derive separations without unproven hardness assumptions, and then exploring closure properties and non-closure under functional operations for 1# to illuminate the structure of counting in nonuniform models. The work also relates counting frameworks to nonuniform pushdown automata, showing incomparabilities between 1N, 1DPD, and 1NPD, and discusses the implications of closure properties to broader complexity landscapes, while outlining open problems and potential extensions to two-way and weighted automata. Overall, the paper maps a rich, nuanced counting-theoretic landscape in nonuniform polynomial-state models with both theoretical and methodological innovations that sharpen our understanding of nondeterminism and counting in restricted computational paradigms.

Abstract

Lately, there have been intensive studies on strengths and limitations of nonuniform families of promise decision problems solvable by various types of polynomial-size finite automata families, where ``polynomial-size'' refers to the polynomially-bounded state complexity of a finite automata family. In this line of study, we further expand the scope of these studies to families of partial counting and gap functions, defined in terms of nonuniform families of polynomial-size nondeterministic finite automata, and their relevant families of promise decision problems. Counting functions have an ability of counting the number of accepting computation paths produced by nondeterministic finite automata. With no unproven hardness assumption, we show numerous separations and collapses of complexity classes of those partial counting and gap function families and their induced promise decision problem families. We also investigate their relationships to pushdown automata families of polynomial stack-state complexity.

Power of Counting by Nonuniform Families of Polynomial-Size Finite Automata

TL;DR

Power of Counting by Nonuniform Families of Polynomial-Size Finite Automata develops counting-based frameworks (1# and 1Gap) for nonuniform polynomial-size automata and their promise problems, and grounds them in a broad landscape of counting classes (1U, 1⊕, 1C_{=}, 1SP, 1P) tied to nonuniform automata. It formalizes 1# and 1Gap via witnessable function families (including 1F, 1F_{N}, 1F_{Z}) and analyzes their relationships to 1D/1N and pushdown variants (1DPD, 1NPD), achieving several major separations (e.g., 1N ⊊ co-1C_{=}, parity vs 1P) using a novel linkage to one-tape linear-time machines with linear-size advice. A key methodological advance is exploiting the one-tape linear-time connection (via lemmas such as property-1cequal and property-onep) to derive separations without unproven hardness assumptions, and then exploring closure properties and non-closure under functional operations for 1# to illuminate the structure of counting in nonuniform models. The work also relates counting frameworks to nonuniform pushdown automata, showing incomparabilities between 1N, 1DPD, and 1NPD, and discusses the implications of closure properties to broader complexity landscapes, while outlining open problems and potential extensions to two-way and weighted automata. Overall, the paper maps a rich, nuanced counting-theoretic landscape in nonuniform polynomial-state models with both theoretical and methodological innovations that sharpen our understanding of nondeterminism and counting in restricted computational paradigms.

Abstract

Lately, there have been intensive studies on strengths and limitations of nonuniform families of promise decision problems solvable by various types of polynomial-size finite automata families, where ``polynomial-size'' refers to the polynomially-bounded state complexity of a finite automata family. In this line of study, we further expand the scope of these studies to families of partial counting and gap functions, defined in terms of nonuniform families of polynomial-size nondeterministic finite automata, and their relevant families of promise decision problems. Counting functions have an ability of counting the number of accepting computation paths produced by nondeterministic finite automata. With no unproven hardness assumption, we show numerous separations and collapses of complexity classes of those partial counting and gap function families and their induced promise decision problem families. We also investigate their relationships to pushdown automata families of polynomial stack-state complexity.
Paper Structure (22 sections, 21 theorems, 2 figures)

This paper contains 22 sections, 21 theorems, 2 figures.

Table of Contents

  1. Background and New Approach
  2. Nonuniform Automata Families and Counting
  3. Exploitation of Close Connections to One-Tape Linear-Time Machines
  4. Counting by Nonuniform Automata Families
  5. Main Contributions of This Work
  6. Preparation: Notions and Notation
  7. Numbers, Strings, and Promise Decision Problems
  8. Various Types of One-Way Finite Automata
  9. Nonuniform Families of Promise Problems and Finite Automata
  10. Introduction of Counting Functions and Counting Complexity Classes
  11. Definitions of Counting and Gap Functions
  12. Basic Relationships among 1F, 1#, and 1Gap
  13. Definitions of Counting Complexity Classes
  14. Relationships among Counting Complexity Classes
  15. Basic Closure Properties
  16. ...and 7 more sections

Key Result

Lemma 2.1

Let $M$ be any 1nfa solving a promise problem $(L^{(+)},L^{(-)})$. (1) There exists another 1nfa $N$ in a branching normal form such that $\#M(x)=\#N(x)$ for all valid $x$ and $sc(N)\leq 3 sc(M)$. (2) There exists another 1nfa $N'$ in a branching normal form such that $\#M(x)-\#\overline{M}(x) = \#N

Figures (2)

  • Figure 1: Containments and separations of "advised" one-tape linear-time complexity classes shown in TYL10Yam10Yam11Yam16. The term "ALL" refers to the family of all languages. The notation $1\hbox{-}\mathrm{BPLIN}$ is the bounded-error variant of $1\hbox{-}\mathrm{PLIN}$TYL10. The complexity classes $\mathrm{REG}$/n TYL10 and $\mathrm{CFL}$/n Yam10 are the collections of all languages recognized respectively by 1dfa's and 1npda's with size-$n$ advice. The suffixes "/Rn" and "/Rlin" indicate the use of "randomized" advice, which is a distribution of advice strings, opposed to "deterministic" advice.
  • Figure 2: Containments and separations of nonuniform polynomial state complexity classes shown in this work, marked by red lines and red dotted lines, except for $1\mathrm{D}\neq1\mathrm{U}\neq1\mathrm{N}$Yam22bYam22c, $\mathrm{co}\hbox{-}1\mathrm{U}\nsubseteq1\mathrm{N}$Yam22c, and $1\mathrm{D}\neq1\mathrm{DPD}$Yam21a marked by black lines and black dotted lines.

Theorems & Definitions (26)

  • Lemma 2.1: branching normal form
  • Lemma 2.2
  • Definition 3.1
  • Example 3.2
  • Example 3.3
  • Lemma 3.4
  • Lemma 3.5
  • Lemma 3.6
  • Lemma 3.7
  • Definition 3.8
  • ...and 16 more