Separating Auxiliary Arity Hierarchy of First-Order Incremental Evaluation Using (3+1)-Ary Input Relations

Wright State University CORE Scholar

The Ohio Center of Excellence in Knowledge- Kno.e.sis Publications Enabled Computing (Kno.e.sis)

2000

Separating Auxiliary Arity Hierarchy of First-Order Incremental Evaluation Using (3+1)-ary Input Relations

Guozhu Dong Wright State University - Main Campus, [email protected]

Louxin Zhang

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Repository Citation Dong, G., & Zhang, L. (2000). Separating Auxiliary Arity Hierarchy of First-Order Incremental Evaluation Using (3+1)-ary Input Relations. International Journal of Foundations of Computer Science, 11 (4), 573-578. https://corescholar.libraries.wright.edu/knoesis/405

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Separating Auxiliary Arity Hierarchy

of FirstOrder Incremental Evaluation

Using k ary Input Relations

Guozhu Dong Louxin Zhang

Univ of Melb ourne Inst of Systems Science

August

Key words Arity hierarchy online evaluation incremental evaluation database view

maintenance rstorder logic dynamic inductive denition help bit auxiliary rela

tion

The Main Result

A rstorder incremental evaluation system foies uses rstorder queries to maintain a

database view which is dened by a nonrstorder query some auxiliary relations views

may also need to b e maintained similarly

In it was shown that using k ary auxiliary relations is strictly less p owerful

than using k ary ones using a reduction to a result by Cai on help bits However that

separation was achieved using queries having input relations of arity at least k In this note

we improve the separation result by reducing the arity of the input relations of the queries

to k Some necessary background knowledge is given in the next section

This work was supp orted in part by a research grant from the Australian Research Council

Part of work by the second author was done while visiting University of Melb ourne

Guozhu Dongs address Department of Computer Science University of Melb ourne Parkville Vic

Australia Email dongcsmuozau

Louxin Zhangs address Institute of Systems Science Heng Mui Keng Terrace Singap ore Email

lxzhangissnussg

Theorem For each integer k FOIES FOIES In fact there is a query over

k k

input relations of arity k in FOIES FOIES

k k

As will b e discussed in the next section this result can b e viewed as one step towards

answering the question of whether graph problems such as transitive closure of directed

graphs have foies or whether some PTIMEcomplete graph problems have foies By reducing

the arity of queries needed in separating FOIES and FOIES we hop e to contribute

k k

towards the understanding of the p ower of foies A further research problem is to improve

the result of this pap er by reducing k to k or k or even a constant

We will prove this by mo difying Cais result and by mo difying the reduction used

Section provides a brief review the notion of rstorder incremental evaluation sys

tems Section establishes a necessary technical lemma which is a variant of Cais theorem

Section gives the pro of of the ab ove theorem

FirstOrder Incremental Evaluation Systems

It is common knowledge that many useful database queries such as transitive closure of

undirected graphs and parity whether the numb er of tuples in a relation is even cannot

b e expressed in rstorder logic Interestingly many materialized database views dened

by such queries Q can b e maintained using rstorder queries

Roughly sp eaking such maintenance for the view dened by Q is carried out through

a set of rstorder queries xed for Q which is called a rstorder incremental evaluation

system or foies for short One of these queries directly maintains the answer to Q while

the others maintain some auxiliary relations views they are used to derive the new views

Q or auxiliary after each p ermissible up date to the database Thus each of these rst

order queries has as input the old database the old answer view the old auxiliary views

and the up date

Being able to maintain non rstorder views using rstorder queries is desirable for two

main reasons i Such maintenance can b e implemented in all relational database systems

since rstorder queries are available in every relational database system even though the

views themselves cannot b e dened in rst order ii Firstorder maintenance algorithms

Patnaik and Immermans DynFO is very similar though dierent from our foies

Permissible up dates are those up dates whose sizes are b ounded by a constant dep endent only on the

query Q and which transforms the old database in the domain of Q to a new database in the domain of Q

have great p otential to b e adapted for parallel execution since they have constant parallel

complexity Such maintenance may also b e of interest from a mathematical logic and

descriptive complexity p ersp ective

There have b een attempts to ascertain these two research problems whether the transi

tive closure of arbitrary directed graphs can b e maintained in rstorder after edge deletions

and whether the same generation query can b e maintained after edge insertions Clearly

these problems can b e settled in the p ositive by nding rstorder maintenance queries

which only use auxiliary relations of arity k for some k and can b e settled in the negative

by showing that there is no such k

It is thus interesting to understand the p ower of foies using auxiliary relations of xed

arities Starting from the maximum arity of the auxiliary relations has b een used as a

measure of how hard it is to maintain a query using foies Observe that with maximal arity

k the auxiliary relations can hold at most O n tuples where n is the numb er of constants

in the input database

For each natural numb er k let FOIES denote the class of queries having foies using

k ary auxiliary relations Obviously FOIES FOIES k In it was shown that

k k

FOIES FOIES for all k The separation for the small arities were achieved using

k k

queries whose input relations are unary or binary However the separation of FOIES

FOIES k were achieved through queries whose input relations are k ary

We now briey review some previous results on the maintenance of the transitive clo

sure of graphs of various kinds In some foies using binary auxiliary relations for

insertion only were given for generalized transitive closure of lab elled graphs For the tran

sitive closure of acyclic directed graphs gave a foies with no auxiliary relations For

undirected graphs there is a foies using ternary auxiliary relations it maintains an undi

rected spanning forest for the undirected graphs from which the reachability relation can b e

extracted There is also a foies using binary auxiliary relations it maintains a directed

spanning forest of the undirected graphs plus some approximation of a total order on the

no des in the graph it was shown that there is no foies using unary auxiliary relations for

transitive closure of undirected graphs

Lemma on Help Bits

For indeterminate sets X and B dene

R Z X x x j x X and

R Z X B x x b b j x X b B

jX j

Lemma R is a commutative algebra of dimension

jX j

For any ideal K in R dimRK dimK

Let F b e the class of Bo olean functions on X We have an injection from F to R

satisfying

x x

f g f g f g

f g f g

f f

Thus each Bo olean function takes a form in R

Consider a multioutput Bo olean circuit C of and gates where and gates

have unb ounded fanin We will use jC j to denote the numb er of gates in C Assume C

has input variables X B and m outputs f f f where B fb b b g will b e

m s

considered as help bits The following lemma is a consequence of a result of Razb orov

see also for its generalization which app eared in

Lemma Let e d b e real numb ers and C any circuit of depth k with jC j e

Let h i s b e any Bo olean functions and f i m b e the outputs of C when

i i

jX j

substituting h X for b i s Then there exist an ideal K of dimK e and

i i

p olynomials p X B R of degree d such that f X p X h X h X is in

i i i s

Now we consider m parity functions x x i m Let N mn and

i i in

X fx j i m j ng

x The following In the algebra R each parity function has the form of

ij i

result is a variant of theorem in replacing Cais m mn by m n and

has a similar pro of For completeness we give a full pro of

Lemma Let and m n Supp ose a Bo olean unb ounded fanin circuit C of

depth k computes all m parity functions x x x with the help of m log m

i i i in

functions h X i m log m Then such that jC j

i k

Pro of Let d N and c log m Supp ose jC j eN where eN will b e

determined later Then we have m p olynomials p X B R of degree N and an ideal

K of dimension eN such that X p X h X h X h X is in RK

i i mc

for i m

We observe that each h is f gvalued thus h h for all i Hence there is a

i i

canonical representation for each p

X Y

p f h X

i j

j S

S mc

where deg f N and m c denotes the set of integers b etween and m c

Consider the parity basis of R For each X let fx x g for each i

i i in

Then

Y Y Y

x x

ij i

x by If j j n we replace

ij i

Y X Y

x g X h X

ij i j

x j S

S mc

i ij

fx x g Note that g has degree in RK see for details where

i i in i

n N in fx x g and N for all other x Thus we have that

i in

X Y Y

x p X h X

j S

S mc

where each p X has degree n mN in each group of variables fx x g

i in

Let M denote the numb er of monomials in X satisfying this restriction Then

nmN

n n n n

Since and for each n i mN we further get

i i

m n

mN M

nmN

n m

m n nm

mN using the Stirling formula n n

n e

m m

o since m n

h X and since for each such term Since there are at most terms of the form

j S

Q Q

x there are at most M choices for the asso ciated p X h X in

x j S

mc N

dimRK M o

c c

Let eN o Since c log m dimR dimRK a contradiction 2

Pro of of the Theorem

The pro of uses a query which enco des the multiple parity problem similarly to

Pro of Consider the database schema fR R R g where each R is a k ary relation

Let k Mo d b e the query dened for each database instance I of this schema by

k Mo dI fx x j N I mo d g

k x x

where N I jfy y j R x x y y gj

x x k i k k

Then k Mo d FOIES To maintain the answers to k Mo d one just maintains

k ary auxiliary relations S S S where S is dened to contain all x x tuples in D

i k

such that N I mo d i It is easy to see that two auxiliary relations say S and

x x

S suce b ecause they and the query answer allow us to derive S

To prove that k Mo d is not in FOIES supp ose to the contrary that k Mo d has

a foies F using at most k ary auxiliary relations We rst rewrite F into another foies

F that uses only k ary auxiliary relations and supp ose F has auxiliary relations

say S S We pick two constants a and b a p ositive integer and a set D of

constants including a and b

k k

Let P b e a given instance of the multiple parity problem with m and n

We will show how to solve the multiple parity problem using an AC circuit through the foies

F for k Mo d Let t t b e an enumeration of D and s s b e an enumeration

m n

of D Let I b e the following database instance

k k

I R D fag

I R I R ft s j x is true in the instance P g

i j ij

Observe that for each i i m if x x x or then N I is mo d

i i in t

or mo d resp ectively Thus k Mo dI

For each i i m consider the insertion of the set ft b bg into R

Since F is a foies for k Mo d there is a rstorder query I I which computes the

i aux

new answer observe that the current answer is empty From the denition of the query

k Mo d and the database I it is clear that the new answer will b e either the same as the

current answer or the current answer plus the tuple t By building the up date into

i i

and using the construction of I we can easily obtain a rstorder formula I I which

i aux

returns true i x x x

i i in

We can now construct an AC circuit C to compute the multiple parity problem with

m log m help bits We can represent each auxiliary relation S by bits Therefore

the auxiliary relations S S can b e represented by m log m bits which

will b e the help bits Now for each i i m we construct an AC circuit C from to

i i

compute the parity of x x x Since C is an AC circuit it has a p olynomial size

i i in

5 1

k k

k 5k

in m Set of Lemma to m Lemma Then n

now implies that C must b e of size exp onential in m a contradiction

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