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Width Hierarchy for k-OBDD of Small Width

Kamil Khadiev

TL;DR

This work establishes a strict width hierarchy for $k$-OBDDs on small width by introducing the Shuffled Address Function $SAF_{k,w}$ as a hard target. It defines the models and complexity classes, then proves lower and upper bounds on the number of distinct subfunctions under partitions to separate narrow-$w$ from wider-$w$ representations, leveraging a $2k$-OBDD construction for SAF and a known lower bound from ak13, establishing a hierarchy analogous to OBDD hierarchies. The result provides a granular width-based separation for restricted read-once, oblivious branching programs under parameter regime $2kw(2w+\lceil \log k\rceil+\lceil \log 2w\rceil)<n$, $k\ge 2$, $w\ge 64$. This contributes to understanding the expressive power of restricted branching programs and informs complexity-theoretic gaps between width-bounded representations.

Abstract

In this paper was explored well known model k-OBDD. There are proven width based hierarchy of classes of boolean functions which computed by k-OBDD. The proof of hierarchy is based on sufficient condition of Boolean function's non representation as k-OBDD and complexity properties of Boolean function SAF. This function is modification of known Pointer Jumping (PJ) and Indirect Storage Access (ISA) functions.

Width Hierarchy for k-OBDD of Small Width

TL;DR

This work establishes a strict width hierarchy for -OBDDs on small width by introducing the Shuffled Address Function as a hard target. It defines the models and complexity classes, then proves lower and upper bounds on the number of distinct subfunctions under partitions to separate narrow- from wider- representations, leveraging a -OBDD construction for SAF and a known lower bound from ak13, establishing a hierarchy analogous to OBDD hierarchies. The result provides a granular width-based separation for restricted read-once, oblivious branching programs under parameter regime , , . This contributes to understanding the expressive power of restricted branching programs and informs complexity-theoretic gaps between width-bounded representations.

Abstract

In this paper was explored well known model k-OBDD. There are proven width based hierarchy of classes of boolean functions which computed by k-OBDD. The proof of hierarchy is based on sufficient condition of Boolean function's non representation as k-OBDD and complexity properties of Boolean function SAF. This function is modification of known Pointer Jumping (PJ) and Indirect Storage Access (ISA) functions.

Paper Structure

This paper contains 2 sections, 6 theorems, 15 equations, 1 figure, 1 table.

Table of Contents

  1. Preliminaries
  2. Proof of Theorem \ref{['h-kobdd']}

Key Result

Theorem 1

For integer $k=k(n),w=w(n)$ such that $2kw(2w + \lceil \log k \rceil + \lceil \log 2w \rceil)<n, k\geq 2, w\geq 64$ we have $\mathsf{k-OBDD_{\lfloor w/16 \rfloor-3}}\subsetneq\mathsf{k-OBDD_{w}}$.

Figures (1)

  • Figure 1: $p$-th block of layer $2t-1$

Theorems & Definitions (6)

  • Theorem 1
  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • Theorem 2: ak13