DynFO A Parallel Dynamic Complexity Class y Neil Immerman SushantPatnaik Computer Science Dept Computer Science Dept University of Massachusetts University of Massachusetts Amherst MA Amherst MA patnaikbearcom immermancsumassedu Abstract Traditionally computational complexity has considered only static problems Clas sical Complexity Classes suchasNCP and NP are dened in terms of the complexity of checking up on presentation of an entire input whether the input satises a certain prop erty For many applications of computers it is more appropriate to mo del the pro cess as a dynamic one There is a fairly large ob ject b eing worked on over a p erio d of time The ob ject is rep eatedly mo died by users and computations are p erformed Wedevelop a theory of Dynamic ComplexityWe study the new complexity class Dynamic FirstOrder Logic DynFO This is the set of prop erties that can b e main tained and queried in rstorder logic ie relational calculus on a relational database We showthatmanyinteresting prop erties are in DynFO including multiplication graph connectivity bipartiteness and the computation of minimum spanning trees Note that none of these problems is in static FO and this fact has b een used to justify increasing the p ower of query languages b eyond rstorder It is thus striking that these prob lems are indeed dynamic rstorder and thus were computable in rstorder database languages all along We also dene b oundedexpansion reductions which honor dynamic complexity classes Weprove that certain standard complete problems for static complexity classes suchas REACH for P remain complete via these new reductions On the other hand a weprove that other such problems including REACH for NL and REACH for L are d no longer complete via b oundedexpansion reductions Furthermore weshowthata version of REACH calledREACH is not in DynFO unless all of P is contained a a in parallel linear time Intro duction Traditional complexity classes are not completely appropriate for database systems Unfor tunately appropriate Database Complexity Classes havenotyet b een dened This pap er makes a step towards correcting this situation Research of b oth authors supp orted by NSF grants CCR and CCR y Current address Bear Stearns Park Avenue New York NY In our view the main two dierences b etween database complexity and traditional com plexity are Databases are dynamic The work to b e done consists of a long sequence of small up dates and queries to a large database Each up date and query should b e p erformed very quickly in comparison to the size of the database Computations on databases are for the most part disk access b ound The cost of computing a request is usually tied closely to the numb er of disk pages that must b e read or written to fulll the request Of course a signicant p ercentage of all uses of computers have the ab ovetwo features In this pap er we fo cus on the rst issue Dynamic complexity is quite relevantinmostday to day tasks For example texing a le compiling a program pro cessing a visual scene p erforming a complicated calculation in Mathematica etc Yet an adequate theory of dynamic complexity is lacking Recently there have b een some signicantcontributions in this direction eg MSV Note that dynamic complexity is dierent although somewhat related to online complexitywhich is receiving a great deal of attention lately We will dene the complexity class DynFO to b e the set of dynamic problems that can b e expressed in rstorder logic What this means is that we maintain a database of relevant information so that the action invoked byeach insert delete and query is rstorder ex pressible This is very natural in the database setting In fact DynFO is really the set of queries that are computable in a traditional rstorder query language Manyinteresting queries such as connectivity for undirected graphs are not rstorder when considered as static queries This has led to muchwork on database query languages such as Datalog that are more expressive than rstorder logic We show the surprising fact that a wealth of problems including connectivity are in DynFO Thus considered as dynamic problems and that is what database problems are these problems are already rstorder computable The problems we showtobein DynFO include reachability in undirected graphs maintaining a minimum spanning forest k edge connectivity and bipartiteness All regular languages are shown to b e in DynFO In P it is shown that some NPcomplete problems admit DynFO approximation algo rithms Dong and Su DS showed that reachability in directed acyclic graphs and in function graphs is in DynFO The static versions of all these problems are not rstorder Related work on dynamic complexity app ears in MSV In DST rstorder incremen tal evaluation system FOIES are dened A problem has an FOIES i it is in DynFO In TY Tarjan and Yao prop ose a dynamic mo del whose complexity measure is the num b er of prob es into a data structure and any other computation is free A log n log log n lower b ound on a dynamic prex multiplication problem was proved in FS Other lower b ounds M R have b een proved using these metho ds Other work on dynamic complexity for databases includes the theory of maintaining materi alized views up on up dates J GMS Io and in integrity constraint simplication LST N The design of dynamic algorithms is an active eld See for example E E b R CT F F among many others There is also a large amountofwork in the programming language community on incremental computation see for example RR LT This pap er is organized as follows In Section we b egin with some background on Descrip tive Complexity In Section for any static complexityclass C we dene the corresp onding dynamic class DynC The class DynFO is the case we emphasize In Section we present the ab ovementioned DynFO algorithms In Section we describ e and investigate reduc tions honoring dynamic complexity Finallywe suggest some future directions for the study of dynamic complexity Descriptive Complexity Background and Denitions In this section we recall the notation of Descriptive Complexity See I for a survey and IL for an extensive study of rstorder reductions In the development of descriptive complexity it has turned out that natural complexity classes have natural descriptivecharacterizations For example space corresp onds to number of variables and parallel time is linearly related to quantierdepth Sequential time on the other hand do es not seem to have a natural descriptivecharacterization We liketothinkofthisasyet another indication that sequential time is not a natural notion simply an artifact of the socalled VonNeumann b ottleneck As another example the class P consisting of those problems that can b e p erformed in a p olynomial amountofwork has an extremely natural characterization as FOLFP rstorder logic closed under the ability to make inductive denitions It is reassuring that our notions of naturalness in logic corresp ond so nicely with naturalness in complexity theory In the presentwork weventure into the terrain of dynamic complexity What is natural is not yet clear We use the intuitions gained from the descriptive approach to aid us in our search We will co de all inputs as nite logical structures ie relational databases A vocabulary a a r c c i is a tuple of input relation and constantsymb ols A structure hR R s r A A A A with vo cabulary is a tuple A hjAjR R c c i whose universe is the nonempty r s A set jAjFor each relation symbol R of arity a in A has a relation R of arity a dened i i i i A a i on jAj ie R jAj For each constantsymbol c A has a sp ecied elementofits j i A jAjWe use the notation jjAjj to denote the cardinality of the universe of A universe c j Since we are only interested in nite structures we let STRUC denote the set of nite structures of vo cabulary We dene a complexity theoretic problem to b e anysubset S STRUC for some A problem is the same thing as a b o olean query For anyvo cabulary there is a corresp onding rstorder language L built up from the symb ols of and the numeric relation symb ols and BIT and numeric constants min max using logical connectives variables x y z and quantiers represents a total ordering on the universe whichmaybeidentied with the set f n g The constants min max represent the minimum and maximum elements in this ordering th BITx y means that when x is co ded as a log n bit numb er the y bit of this enco ding is a one In static complexitytheentire input structure A is xed and weareinterested in deciding whether AS forarelevant prop erty S In the dynamic case the structure changes over time The actions wehave in mind are a sequence of insertions and deletions of tuples in the input relations We will usually think of our dynamic structure A hf n gR R c c iashaving a xed size p otential universe jAj f n g r s and a unary relation R sp ecifying the elements in the active domain The initial structure n of size n for this vo cabulary will b e taken to b e A hf n g fg i having R fg indicating that the single element is in the active domain and all other relations are empty FirstOrder Reductions Here we briey describ e rstorder reductions a natural way of reducing one problem to another in the descriptivecontext Firstorder reductions are used in Section to build new reductions that honor dynamic complexityFurthermore reductions are used in Section as a motivation for the denition of Dynamic Complexity More information ab out rstorder reductions can b e found in IL Recall that