A Temporal Logic Based Approach for Querying Lists, Trees, and DAGs in Databases

A Temporal Logic Based Approach for QueryingLists,Trees,and D AG s in DatabasesPeter BeckerWilhelm-Schickard-Institut f¨u r Informatik,Universit¨a t T¨u bingenSand13,D-72076T¨u bingenE-Mail:AbstractAlthough lists,trees,and directed acyclic graphs(dags)are of fundamental interest in computer science,they have scarcely been noticed as data types byexisting database systems and database models.In this article I present query lan-guage concepts for(object-oriented)database systems offering such non-traditionaldata resp.object types.For the main task of such a query language,namely thespecification of predicates on instances of these types,a temporal logic approach isused.For this,an existing timely oriented approach for querying lists is extendedby incorporating concepts from the area of branching-time logics.This articlemainly presents the logic concepts of the query language and an implementationof these concepts in an object base systems for discrete structures.1IntroductionLists,trees,and graphs are some of the most important data structures in computer science.They are not only used as implementation concepts for the efficient realization of several data types,but especially serve for the modeling of hierarchically or complex assembled entities.Typical application areas for databases,where e.g.trees as data type are useful are computer linguistics and text databases,cf.[CACS94,GT87].In such applications, trees can serve as a modeling tool for describing whole entities,e.g.parse trees resulting from a syntactical analysis(cf.[AB94])or trees to represent the S GML structure of a document(cf.[CACS94]).Despite their importance as modeling tools,lists,trees,and graphs have scarcely been noticed as data types by existing database systems.Traditional database systems only offer sets resp.relations for the representation of entity collections and simple data types like integer or string for the representation of entity attributes.Non-traditional data types like lists,trees,and graphs have to be represented by these facilities in a laborious way.Overcoming the problem of representing complex entities was one major reason for the development of object-oriented database systems.But up to now,only simplecollection types like tuples,sets,lists,and bags have been incorporated into commonobject-oriented database systems(see e.g.[Cat94]).Moreover,the query facilities of such database systems are quite constrained with respect to these collection types.Especially query predicates that concern several items in single lists(e.g.“From agiven set of lists,select all lists,where the items are monotonically increasing”)are hardly supported.A user has to write a dedicated method in an associated databaseprogramming language to execute a query like the one given above.Such a proceeding may be quite elaborate.In this article,query language concepts for databases offering lists,trees,anddirected acyclic graphs(dags)as data resp.object type are presented.The query language approach that is presented here was inspired by the work of J.Richardson givenin[Ric92].In this article,a temporal logic approach for the specification of predicateson lists has been presented.This approach seems natural,because“temporal formulae (for discrete,linear time),which usually serve to describe time-ordered sequences ofstates,can equally well describe spatially-ordered sequences of objects”,cf.[Ric92].In this work I extend the approach given in[Ric92]to trees and dags.For this,con-cepts of the area of branching-time logics are incorporated into the former approach.Inparticular concepts of the so-called computational tree logic(C TL*resp.C TL)[CES86] are used.C TL*,which has mainly been used for the specification and verification ofconcurrent systems,extends the concepts of linear time logic,which is the basis of[Ric92],by introducing path quantifiers.Beside the logic concepts,the integration of these concepts in an object base systemfor discrete structures is shown in this article.For this,the object-oriented model and method base system M[Bec94]has been used as an implementation platform.The rest of the paper is structured as follows:In Section2a review of the mostimportant facts of[Ric92]is given.In Section3,I present the extension of this approach to trees by using concepts of C TL*.In Section4,I show the incorporationof the concepts given in the former sections into the object base system M.Section5examines the application of the temporal logic concepts to the problem of querying dags and Section6gives a summary and presents some open problems.2A Timely Approach for Querying ListsIn this section a review of the main concepts of[Ric92]is given.Mainly,the set oftemporal operators that will form the basis for querying lists are introduced.These operators are similar to the usual operators in propositional linear time logic(P LTL),cf.[Eme90].A list is seen as a sequence of objects,.A distinction is madebetween the value at point,denoted by and the-th item of a list,denoted by .To refer to the value at the current position of a list(the value at point“now”) the symbol is used.An atomic fomula is defined as a predicate that can directly be evaluated by lookingat an individual list element.Such a formula does not contain logical operators and returns a boolean result.is an example of an atomic formula.It is true,if the actual list value is less thanfive.The truth of a formula is defined as truth at a point of a list.The relation assigns for each List,for each,and for each list logic formula a truth value according to the following definition.Definition1(Syntax and Semantics of List Logic)is an atomic formula and evaluates to true forq(q)is the next(previous)operator.It has the meaning that a fomula is valid at the next position.E.g.the list logic formula means that the element after the actual element is negative.()is the until(since)operator.The formula()has the meaning that formula is valid at some point in the future(past),i.e. after(before)the actual point,and that from now(from there)until then(since now) the formula ist true.We write as a shorthand for.If we have,we say that the list is a model for the list logic formula.Observe that the operators and q are weak operators,i.e.(q)is always true,independent of,at the last(first)position of a list.Based on these definitions, we derive additional temporal operators in the following definition.Definition2(Additional Operators for List Logic)strong next(previous)q qeventually(past)true truealways(past)The operator q)is the strong counterpart to(q).(Up to now we have only considered predicates in that the state of each individual list value is compared with itself or with global constants.But for a query like,“Is the list monotonically increasing”,we need some facility to compare several list values with each other.For this,the binding operator has to be used(see Definition1).The binding operator introduces the variable whose scope is.denotes the formula derived from by replacing all free occurrences of in with. Example2The list is monotonically increasing:The list contains no duplicates:Observe the functionality of the binding operator in thefirst of the two examples above.The always operator()examines each list item.In these examinations the actual list value is bound to by the binding operator.This bound value is then compared with the value at the next position by using the next()operator.Beside the logical concepts presented above,[Ric92]gives other valuable contri-butions like an extension of an existing query algebra with the concepts shown above. Here,we do not review this further,but in Section4,which describes the incorporation of my approach into an object base sytem,similar query algebra concepts to that in [Ric92]are presented.3A Logic for Querying TreesNow,we consider unordered rooted trees of arbitrary degree.As usual, denotes the node set of the tree and the edge set of.The label function assigns to each node of a value of some arbitrary domain.Analogous to lists,denotes the value at a node of,and denotes the node itself.The approach presented in the last section is mainly an adaption of propositional linear time logic(P LTL).In linear time there is just one possible future,like in the case of lists,where each element has at most one successor.To generalize this approach to trees,I use concepts from the area of branching-time logics.With branching-time there may be several incompatible futures.This is analogous to trees,where each element may have several distinct successors.Here,I use an adaption of the computational tree logic(C TL*resp.C TL).C TL* has been proposed for thefirst time in[CES86]as an extension of C TL.It has the advantage on other branching-time logics that it fully incorporates P LTL.Moreover,it is more expressive than a lot of other branching-time logics proposed,see[Eme90]. Nevertheless,it facilitates easy and intelligible query formulation,as we will show in this and the following section.The main idea of C TL*is the introduction of path quantifiers.Let be a P LTL formula.Then the C TL*formula has the informal meaning that there exists a path begining in the actual node that satisfies.Such paths are simply viewed as lists.The dual operator has the meaning that is valid for all paths starting in the actual node.Because the logic that I am going to present here is closely related to C TL*,I denote it as C TL.As for lists,it is assumed that there exists a set of atomic formulas.In C TL,we have two different types of formulas,namely state formulas and path formulas.State formulas correspond to nodes,i.e.their truth value is determined with respect to a given node,while path formulas correspond to a path of a given tree.First, the syntax of C TL and the restricted language C TL are defined.Definition3(Syntax of C TL and C TL)State Formulas(S1)Each atomic formula is a state formula.(S2)If and are state formulas,so are:and.(S3)If is a path formula then,and are state formulas.Path Formulas(P1)Each state formula also is a path formula.(P2)If and are path formulas,so are:and.(P3)If and are path formulas,so are:,and q.The set of state formulas generated by the above rules forms the language C TL.The set of state formulas generated by(S1–3)and the following rule(P0)forms the language C TL:(P0)If and are state formulas,then,and q are path formulas.Observe,that the restricted logic C TL is obtained by disallowing boolean combina-tions and nestings of list logic operators.We will need C TL especially for dags,cf. Section5.Before the semantics of C TL is presented,another definition has to be given. Definition4Let be a node of a tree.A path of is called a maximal path for,if and only if is a leaf.Definition5(Semantics of C TL and C TL)State Formulas:The relation assigns for each tree,for each node,and for each state formula a truth value in the following way:(S1)evaluates to true for(S2)(S3)maximal pathmaximal pathsfor the pathfrom the root toPath Formulas:The relation assigns for each tree,for each maximal path,for each,and for each path formula a truth value: (P1)(P2)(P3)qAdditional logical operators for path formulas can be defined analogously to Defi-nition2,e.g.object base system for discrete structures(e.g.graphs and trees).One of its main features is the easy and efficient integration of data resp.object types for such discrete structures as well as of dedicated software packages for the solution of discrete problems.M is based on the object-oriented paradigm including concepts for the support of persistence and transactions,it uses a client server architecture.Operations concerning objects of the model resp.object base are defined by the means of a specialized model base language called MeBaL.They are issued from a client as requests to the server, which does all administration and manipulation of the object base.In the prototype implementation the methods resp.for verifying,whether a list logic resp.C TL formula is valid for a given list resp.tree,are offered via M. Let denote a C TL formula.Then the MeBaL statementverifies,whether the tree is a model for.As you can see,parameters are named in M.is the parameter name,while is the parameter value.To test whether a list has only positive elements(the list logic formulation is ),the following statement can be used:Formulas are represented by a so-called lambda expressions,which give the definition of a function,but are simple atomic data types like integer or real.They are syntactically represented by an opening,arbitrary MeBaL statements,and a closing parantheses.In the example above we have a nested lambda expression.The inner one()tests whether the actual list element is positive.The variable corresponds to the symbol of list logic.The outer expression()applies the operator to.In this process,the variable is bound to the list for which the method has been applied.Each operator of list logic has a corresponding M method.Here,the method is used to apply the operator.M offers some abbreviation concepts to facilitate easier query formulation.The character can be used instead of the identifier.Moreover,the M facility for the definition of operators has been used to represent each“time-oriented”logic operator by a dedicated and new defined MeBaL operator.Such an operator definition states, how an operator application is transformed into an appropiate method ing these techniques the query above can be expressed by:Here,the operators and substitute the corresponding methods.Observe that and are parts of the operator names.Variable bindings can be simulated by M’s assignment operator().The following query tests whether a tree has the heap property,i.e.whether each node is greater than all of its successors.The corresponding query in C TL is:Beside model checking for single lists and trees,list logic resp.C TL formulas can be used as selection predicates in object base queries.Assume that is a classof trees.Then the following MeBaL statement selects all instances of that have the heap property:The result of such a query would be a set of object identifiers.Arbitrary MeBaL statements can be used as selection predicates or atomic formulas.There is,however, the single constraint that these statements have to return a boolean value.Formulas can also be used to select single items of a list or a tree.It is also possible to nest the application of the resp.operator.The following statement checks whether each of the sublists of contains the value zero:Observe that the operator(always)is applied to the whole(outer)list,while the operator(eventually)is applied to each single(inner)list.The method facilitates the application of generic list methods to paths of trees.The following query yields all trees of that contain a path,where the overall sum is greater than ten:Beside the variable the following variables can be used in lambda expressions representing list logic or C TL formulas:denotes the actual position resp.the actual node number,denotes the actual path in a C TL path formula(this path is given as a list of node numbers),and denotes the actual position in a path while checking a C TL path formula.I close this section with a more complex query example.The request“Give the height and the number of nodes for all trees,that contain a path of length greater than or equal tofive with only non-negative elements”is expressed in M by:The variable and the predefined list method are used to verify the path length.The operator transforms the items of the result set into the de-sired format.For this,the transformation rule uses the tuple constructor.represents the unique object identificator,which is bound to.and are predefined methods for trees.5A Query Approach for DagsDue to the fact that C TL offers path quantification,it is on principial possible to also use C TL for dags.E.g.the C TL formula becomes true for a node of a dag ,if all paths from to some node that has no successor satisfy.Because dags are acyclic,it is guaranteed that there are afinite number of maximal paths that have tobe considered.Only the past oriented operator has to be adopted to apply C TL ingeneral to dags.Unfortunately,this approach leads to a severe problem.In contrast to trees,forwhich the number of maximal paths from a node to some leaf is linearly bound in thenumber of nodes of a tree,a dag may have an exponential number of such paths.For this reason,the query logic for dags is restricted to a variant of C TL.Observe thatC TL formulas are always state formulas,i.e.they become true for and are checked against some node.Therefore,constructing maximal paths and checking path formulascan be avoided,if such a restricted logic is used,as it will be shown below.Due to space limitations only a quite informal definition of the logic for dags, denoted by C TL,is given.Definition7(Syntax and Semantics of C TL)The syntax of C TL is defined analogously to C TL.C TL uses the future-oriented C TL operators and their past-oriented counterparts.The seman-tics of the future-oriented operators is given as in Definition6and Definition7.Thepast-oriented operators are defined analogously to their future-oriented counterparts.Now,the problem of efficient model checking for C TL formulas is considered.Thefixpoint characterizations of the logic operators can be used for it.E.g.we havethe followingfixpoint characterization for the C TL operator.I.e.is true at some node of a dag,if is true at and is true for eachsuccessor of.Such afixpoint characterization exists for each C TL operator withthe exception of the next()and previous()operators,cf.[Eme90]. The usage of thesefixpoint characterizations prevents the construction of maximal paths and model checking becomes more efficient than checking the formulas according to their semantic definition.Lemma1Let be a dag with nodes.Model checking for a C TL formula,where is a C TL operator,can be done by testing for at most nodes. Proof:The lemma is obviously true for the next and previous operators.For all other operators model checking is done due to theirfixpoint characterization.Here,the operator is used as an example.Assume,that the formula has to be checked for some node of.To achieve the result stated above,we simply proceed in a depth-first traversal and mark all nodes that have already been visited.For this,we add to each node of a marker,before the model checking is started.Moreover,we will store for each checked node the result of the check.Then we start the model checking and proceed recursively due to thefixpoint characterization of.Now,assume we have to check for a node.If has already been checked for,we instantly return the truth value,which is stored in the node.Otherwise we check for.If is not valid for,we mark as checked and return false.Otherwise we recursively checkfor each successor of.After that,we mark and return true,if all checks have beensuccessful and false otherwise.Observe that due to the marking the truth value foris computed at most once for each node,i.e.we have at most tests for.6SummaryIn this article query language concepts for the retrieval of lists,trees,and dags have been presented.The proposed concepts are based on a temporal logic approach.An already based on propositional linear time logic presented language for the retrieval of lists has been extended to unordered rooted trees by incorporating concepts from computational tree logic.The resulting language C TL is powerful and facilitates intelligible and easy query formulation.Moreover,it has been shown,how the temporal logic based languages for lists and trees have been integrated in M,an object base systems for discrete structures.Furthermore,the logic C TL for the retrieval of directed acyclic graphs has been presented and a technique to improve the efficiency of model checking for this logic has been proposed.General optimization issues for the retrieval of lists,trees,and dags may be the most important ones among the open problems.Up to now,selection of trees is implemented in the prototype system by means of verifying the C TL formula for each single tree within the base set.This may be rather time consuming.[AB94]gives an optimization technique for tree matching based retrieval of trees.This techniques could be reused,if C TL is extended by tree matching operators.Queries that use such operators could then be optimized.References[AB94]H.Argenton and P.Becker.Efficient retrieval of labeled binary trees.In S.Uemura and M.Yoshikawa,editors,Proceedings of the International Symposium on AdvancedDatabase Technologies and Their Integration,pages176–182,1994.[Bec94]P.Becker.M—an object-oriented model and method base system for discrete optimization problems.In D.Patel,Y.Sun,and S.Patel,editors,OOIS’94—Proceedings of the1994International Conference on Object Oriented InformationSystems,pages74–87.Springer,1994.[CACS94]V.Christophides,S.Abiteboul,S.Cluet,and M.Scholl.From structured documents to novel query facilities.In Proceedings of the ACM SIGMOD Conference,pages313–324,1994.[Cat94]R.G.G.Cattell,editor.The Object Database Standard:ODMG-93.Morgan Kaufmann,1994.[CES86] E.M.Clarke,E.A.Emerson,and A.P.Sistla.Automatic verification offinite-state concurrent systems using temporal logic spcifications.ACM TOPLAS,8(2):244–263,1986.[Eme90] E.A.Emerson.Temporal and modal logic.In J.van Leeuwen,editor,Handbook of Theoretical Computer Science,chapter16,pages995–1072.Elsevier,1990.[GT87]G.H.Gonnet and F.W.Tompa.Mind your grammer:a new approach to modelling text.In Proceedings of the13th VLDB Conference,pages339–346,1987.[Ric92]J.Richardson.Supporting lists in a data model(a timely approach).In Proceedings of the18th VLDB Conference,pages127–138,1992.。