The read once formula of a series–parallel network

doi:10.1016/j.dam.2022.06.002

Discrete Applied Mathematics

Volume 320, 30 October 2022, Pages 304-310

https://doi.org/10.1016/j.dam.2022.06.002 Get rights and content

Abstract

For any two-terminal resistive series–parallel network $N$ with $n$ resistors, let $R : {[0, \infty]}^{n} \to [0, \infty]$ denote the resistance function of $N$ . Let $R |ˋ {0, \infty}^{n}$ be the ${0, \infty}$ -valued function obtained from $R$ by only allowing the resistors in $N$ to be open/short-circuited. $R |ˋ {0, \infty}^{n}$ is a boolean function that can be coded by a boolean formula without repeated variables and no negation symbols—for short, a positive read-once formula. Let both networks $N_{1}$ and $N_{2}$ have $n$ resistors with resistances $r_{1, j} = r_{2, j} \in [0, \infty]$ for each $j = 1, \dots, n$ . We prove that if $R_{1} |ˋ {0, \infty}^{n} = R_{2} |ˋ {0, \infty}^{n}$ then $R_{1} = R_{2}$ . We extend this result to the Wheatstone bridge.

Introduction

A series–parallel network (“network” for short) $N$ is a two-terminal multigraph with a source node $⊤_{N}$ and a sink node $⊥_{N}$ , whose inductive definition is as follows:

A single edge $E = (⊤_{E}, ⊥_{E})$ is a network.
If $H$ and $K$ are networks with disjoint sets of nodes, then so is their series composition
, where $⊥_{H}$ is identified with $⊤_{K}$ . It is then stipulated that
and
.
If $H$ and $K$ are networks with disjoint sets of nodes, then so is their parallel composition $H ∥ K$ , where the source nodes of $H$ and $K$ are identified, and so are their sink nodes.

Edges are traditionally renamed resistors, and nodes are also said to be vertices or junctions. Let $E_{1}, \dots, E_{n}$ be the resistors of a network $N$ . Once each $E_{j}$ is independently assigned a resistance $r_{j} \in [0, \infty]$ , the (equivalent) resistance function $R (r_{1}, \dots, r_{n}) : {[0, \infty]}^{n} \to [0, \infty]$ of $N$ is defined inductively by stipulating that resistance is additive for series compositions, and reciprocal resistance is additive for parallel compositions. To take care of the open/short-circuit values $\infty$ and 0, for all $x, y \in [0,\infty]$ we set:

with a self-explanatory notation. Induction on

n

shows that for any

{j_{1}, \dots, j_{m}} \subseteq {1, \dots, n}

the value of the resistance function

R

N

for

r_{j_{1}} = \dots = r_{j_{m}} = \infty

equals

{lim}_{r_{j_{1}} \to \infty} \dots {lim}_{r_{j_{m}} \to \infty} R,

independently of the order of these limits.

Let

be the

{0, \infty}

-valued function obtained from

R

by only allowing the resistors in

N

to be open/short-circuited.

is a boolean function which can be coded by a boolean formula without repeated variables and no negation symbols—for short, a positive read-once formula. In [3], [4] Gurvich gave a characterization of these functions that does not mention their coding boolean formulas. Also see [2, Chapter 10, Theorem 10.6] and [6, Theorem 1.1].

The author was unable to find a proof of the following result in the literature.

Theorem 1.1

Let $E_{1, 1}, \dots, E_{1, n}$ and $E_{2, 1}, \dots, E_{2, n}$ be the resistors of two networks $N_{1}$ and $N_{2}$ . For each $j = 1, \dots, n$ , let $E_{1, j}$ and $E_{2, j}$ be assigned the same resistance $r_{1, j} = r_{2, j} = r_{j} \in [0, \infty]$ . If the positive read-once formulas

and
coincide, then the resistance functions

R_{1}

and

R_{2}

coincide.

Thus in particular, the open/short circuit resistance of a network $N$ uniquely determines the resistance function of $N$ .

Section snippets

Proof of Theorem 1.1

We let $R_{> 0}$ and $R_{\geq 0}$ be shorthand for ${x \in R ∣ x > 0}$ and ${x \in R ∣ x \geq 0}$ .

Kirchhoff’s analysis [7], [8] of general electric circuits yields in particular:

Lemma 2.1

Let $N$ be a network whose resistors $E_{1}, \dots, E_{n}$ have resistances $r_{1}, \dots, r_{n} \in [0,\infty]$ . To avoid trivialities, suppose $R (r_{1}, \dots, r_{n}) \neq 0$ . For all $V \in R_{\geq 0}$ , Kirchhoff’s equations assign to each $E_{j}$ of $N$

(i)
a real number $i_{j} = i_{E_{j}} \in R_{\geq 0}$ , known as the current flowing through $E_{j}$ , given the voltage (also known as “electromotive force”, or “potential”) $V$ between the source node $⊤_{N}$ and the sink

Extending Theorem 1.1 to the wheatstone bridge

A Wheatstone bridge is a two-terminal graph $W$ obtained by deleting one edge $η$ of the complete graph $K_{4}$ and calling the vertices of $η$ the positive and the negative “node”, or “terminal” of $W$ . For each $i = 0, \dots, 4$ , the arc (or “resistor”) $r_{i}$ of $W$ is labeled by a “resistance” $R_{i} \in [0, \infty]$ . The only resistor $r_{0}$ adjacent to all other resistors is known as the “bridge” resistor of $W$ .

The (“equivalent”, or “total”) resistance function $R : {[0, \infty]}^{5} \to [0, \infty]$ of $W$ is obtainable from Kirchhoff’s equations.²

Acknowledgments

The author is grateful to the referees for their careful reading of this paper and their valuable suggestions for improvement. Their remarks motivated the extension of Theorem 1.1 to the Wheatstone bridge—a non-series–parallel network that is commonly found in books on electric networks.

References (8)

KarchmerM. et al.
Combinatorial characterization of read-once formulae
Discrete Math.
(1993)
M.C. Golumbic, V. Gurvich, Read-Once Functions, pp. 519–560, Chapter 10, In...
GurvichV.A.
On repetition-free Boolean functions
Uspekhi Mat. Nauk.
(1977)

There are more references available in the full text version of this article.

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NoteThe read once formula of a series–parallel network

Abstract

Introduction

Section snippets

Proof of Theorem 1.1

Extending Theorem 1.1 to the wheatstone bridge

Acknowledgments

Discrete Math.

On repetition-free Boolean functions

Uspekhi Mat. Nauk.

Note
The read once formula of a series–parallel network