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On the Hierarchies for Deterministic, Nondeterministic and Probabilistic Ordered Read-k-times Branching Programs

Kamil Khadiev

TL;DR

This work analyzes hierarchy theorems for deterministic, nondeterministic, and probabilistic ordered read-$k$-times branching programs, showing that increasing $k$ (even when non-constant) can increase model power, and that hierarchies extend to widths beyond polynomial, including subexponential and sublinear regimes. It develops two complementary approaches: a functional-decomposition approach that expresses $k$-OBDD/NOBDD computations as OR-of-ANDs of width-$w$ components, and a communication-complexity framework that maps BP computations to automata-like protocols and matrices to bound the number of subfunctions. Central to the results are the lower bounds for explicit hard functions EQS$_d$ and SAF$_{k,w}$, which drive separations between $ ext{OBDD}$ and $ ext{NOBDD}$ (and their probabilistic versions) across width classes. The paper thereby extends Bollig–Sauerhoff–Sieling–Wegener-type hierarchies to wider $k$ and width regimes, including sublinear widths, and provides a coherent methodology for deriving both polynomial and superpolynomial/subexponential width hierarchies via functional-description and communication-complexity techniques with implications for the structure of read-$k$-times branching programs.

Abstract

The paper examines hierarchies for nondeterministic and deterministic ordered read-$k$-times Branching programs. The currently known hierarchies for deterministic $k$-OBDD models of Branching programs for $ k=o(n^{1/2}/\log^{3/2}n)$ are proved by B. Bollig, M. Sauerhoff, D. Sieling, and I. Wegener in 1998. Their lower bound technique was based on communication complexity approach. For nondeterministic $k$-OBDD it is known that, if $k$ is constant then polynomial size $k$-OBDD computes same functions as polynomial size OBDD (The result of Brosenne, Homeister and Waack, 2006). In the same time currently known hierarchies for nondeterministic read $k$-times Branching programs for $k=o(\sqrt{\log{n}}/\log\log{n})$ are proved by Okolnishnikova in 1997, and for probabilistic read $k$-times Branching programs for $k\leq \log n/3$ are proved by Hromkovic and Saurhoff in 2003. We show that increasing $k$ for polynomial size nodeterministic $k$-OBDD makes model more powerful if $k$ is not constant. Moreover, we extend the hierarchy for probabilistic and nondeterministic $k$-OBDDs for $ k=o(n/ \log n)$. These results extends hierarchies for read $k$-times Branching programs, but $k$-OBDD has more regular structure. The lower bound techniques we propose are a "functional description" of Boolean function presented by nondeterministic $k$-OBDD and communication complexity technique. We present similar hierarchies for superpolynomial and subexponential width nondeterministic $k$-OBDDs. Additionally we expand the hierarchies for deterministic $k$-OBDDs using our lower bounds for $ k=o(n/ \log n)$. We also analyze similar hierarchies for superpolynomial and subexponential width $k$-OBDDs.

On the Hierarchies for Deterministic, Nondeterministic and Probabilistic Ordered Read-k-times Branching Programs

TL;DR

This work analyzes hierarchy theorems for deterministic, nondeterministic, and probabilistic ordered read--times branching programs, showing that increasing (even when non-constant) can increase model power, and that hierarchies extend to widths beyond polynomial, including subexponential and sublinear regimes. It develops two complementary approaches: a functional-decomposition approach that expresses -OBDD/NOBDD computations as OR-of-ANDs of width- components, and a communication-complexity framework that maps BP computations to automata-like protocols and matrices to bound the number of subfunctions. Central to the results are the lower bounds for explicit hard functions EQS and SAF, which drive separations between and (and their probabilistic versions) across width classes. The paper thereby extends Bollig–Sauerhoff–Sieling–Wegener-type hierarchies to wider and width regimes, including sublinear widths, and provides a coherent methodology for deriving both polynomial and superpolynomial/subexponential width hierarchies via functional-description and communication-complexity techniques with implications for the structure of read--times branching programs.

Abstract

The paper examines hierarchies for nondeterministic and deterministic ordered read--times Branching programs. The currently known hierarchies for deterministic -OBDD models of Branching programs for are proved by B. Bollig, M. Sauerhoff, D. Sieling, and I. Wegener in 1998. Their lower bound technique was based on communication complexity approach. For nondeterministic -OBDD it is known that, if is constant then polynomial size -OBDD computes same functions as polynomial size OBDD (The result of Brosenne, Homeister and Waack, 2006). In the same time currently known hierarchies for nondeterministic read -times Branching programs for are proved by Okolnishnikova in 1997, and for probabilistic read -times Branching programs for are proved by Hromkovic and Saurhoff in 2003. We show that increasing for polynomial size nodeterministic -OBDD makes model more powerful if is not constant. Moreover, we extend the hierarchy for probabilistic and nondeterministic -OBDDs for . These results extends hierarchies for read -times Branching programs, but -OBDD has more regular structure. The lower bound techniques we propose are a "functional description" of Boolean function presented by nondeterministic -OBDD and communication complexity technique. We present similar hierarchies for superpolynomial and subexponential width nondeterministic -OBDDs. Additionally we expand the hierarchies for deterministic -OBDDs using our lower bounds for . We also analyze similar hierarchies for superpolynomial and subexponential width -OBDDs.

Paper Structure

This paper contains 18 sections, 28 theorems, 83 equations, 1 table.

Table of Contents

  1. Preliminaries and Results
  2. Decomposition and Simulation of Nondeterministic Ordered Read-$k$-times Branching Programs
  3. Decomposition of Nondeterministic Ordered Read-$k$-times Branching Programs
  4. Constructing Nondeterministic Ordered Read-ones Branching Program by Decomposition of Nondeterministic Ordered Read-$k$-times Branching Program
  5. Lower bound for Boolean Function $EQS_d$.
  6. Lower Bounds for Nondeterministic and Deterministic Ordered Read-$k$-times Branching Programs. Communication Complexity Technique
  7. The Proof of Theorem \ref{['th-main2-small']}
  8. Lower bound for Boolean Function $SAF_{k,w}$.
  9. Function $SAF_{k,w}$.
  10. Lower Bounds for Probabilistic Ordered Read-$k$-times Branching Programs. Communication Complexity Technique.
  11. The Proof of Theorem \ref{['th-main3-small']}
  12. The Proof of Lemma \ref{['pl2']}
  13. Measure of Matrix Closeness
  14. Final Phase of The Proof of Lemma \ref{['pl2']}
  15. Lower bound for Boolean Function $SAF_{k,w}$.
  16. ...and 3 more sections

Key Result

Lemma 1

For integer $k,w$, such that $k\log w<n$, the following statement is true: $k{\bf\hbox{-}NOBDD}_w\subseteq {\bf F}(w^{k-1}, k, w)$.

Theorems & Definitions (32)

  • Lemma 1
  • Theorem 1
  • Lemma 2
  • Theorem 2
  • Theorem 3
  • Definition 1: Nondeterministic Automata protocol
  • Remark 1
  • Lemma 3
  • Lemma 4
  • Lemma 5
  • ...and 22 more