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On universality of regular realizability problems

Alexander Rubtsov, Michael Vyalyi

TL;DR

The universality of the regular realizability problem for several classes of filters is proved with respect to the disjunctive reduction in polynomial time for unary relations and in polynomial space for invariant binary relations.

Abstract

We prove the universality of the regular realizability problems for several classes of filters. The filters are encodings of finite relations on the set of non-negative integers in the format proposed by P. Wolf and H. Fernau. The universality has proven up to disjunctive truth table polynomial reductions for unary relations and polynomial space reductions for invariant binary relations. Stronger reductions correspond to the results of P. Wolf and H. Fernau about decidability of regular realizability problems for many graph-theoretic properties.

On universality of regular realizability problems

TL;DR

The universality of the regular realizability problem for several classes of filters is proved with respect to the disjunctive reduction in polynomial time for unary relations and in polynomial space for invariant binary relations.

Abstract

We prove the universality of the regular realizability problems for several classes of filters. The filters are encodings of finite relations on the set of non-negative integers in the format proposed by P. Wolf and H. Fernau. The universality has proven up to disjunctive truth table polynomial reductions for unary relations and polynomial space reductions for invariant binary relations. Stronger reductions correspond to the results of P. Wolf and H. Fernau about decidability of regular realizability problems for many graph-theoretic properties.
Paper Structure (12 sections, 17 theorems, 32 equations)

This paper contains 12 sections, 17 theorems, 32 equations.

Table of Contents

  1. Introduction
  2. Definitions and Basic Constructions
  3. Automata and transducers
  4. Reductions
  5. Finite Relation Encodings
  6. Automata over Infinite Alphabets
  7. A Framework for Universality Proofs
  8. Paths on Labeled Graphs
  9. Universality of Unary Relation Descriptions
  10. Universality of Invariant Binary Relation Descriptions
  11. Universality of Invariant Unary Relation Encodings
  12. Concluding Remarks

Key Result

Lemma 1

For $k = O(1)$, a set $L_{q_1q_2}$ is a union of the Cartesian products of semilinear sets, and its encoding has the size $\mathop{\mathrm{poly}}\nolimits(\mathop{\mathrm{size}} (\mathcal{A}))$, where $\mathop{\mathrm{size}}(\mathcal{A})$ is the size of an encoding of $\mathcal{A}$.

Theorems & Definitions (37)

  • Example 1: Unary relations
  • Example 2
  • Example 3
  • Example 4
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Corollary 3
  • proof
  • ...and 27 more