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

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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 Follow this and additional works at: https://corescholar.libraries.wright.edu/knoesis Part of the Bioinformatics Commons, Communication Technology and New Media Commons, Databases and Information Systems Commons, OS and Networks Commons, and the Science and Technology Studies Commons 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 This Article is brought to you for free and open access by the The Ohio Center of Excellence in Knowledge-Enabled Computing (Kno.e.sis) at CORE Scholar. It has been accepted for inclusion in Kno.e.sis Publications by an authorized administrator of CORE Scholar. For more information, please contact [email protected]. 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 in 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 1 Patnaik and Immermans DynFO is very similar though dierent from our foies 2 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 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 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 k FOIES k were achieved through queries whose input relations are k ary k 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 B 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 1k d 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 B 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 RK Now we consider m parity functions x x i m Let N mn and i i in X fx j i m j ng ij Q x The following In the algebra R each parity function has the form of ij i j 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 N functions h X i m log m Then such that jC j i k k N Pro of Let d N and c log m Supp ose jC j eN where eN will b e B determined later Then we have m p olynomials p X B R of degree N and an ideal i N 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 i canonical representation for each p i X Y S p f h X i j i j S S mc S where deg f N and
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