Temporal Query Languages

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Temporal Query Languages

• a Survey
• by Jan Chomicki, January 24, 1995
• Computing and Information Sciences
• Kansas State University
• Presented by Barry Klein, USC, October 3, 2000

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Contents

1. Introduction to temporal databases
2. Temporal databases overview
3. Properties of query languages
4. Abstract query languages
5. Concrete query languages
6. Incomplete temporal information
7. Related work in artificial intelligence

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Introduction to temporal databases

• Key concepts Temporal Domain, Abstract and
  Concrete representations/Query langs, Incomplete
  Temporal Information. Interpreted db domain.
• Examples financial/personnel/medical/legal
  records network monitoring, process control
• Framework integrate temporal research with
  research in db theory, logic and AI.
• Eschew Temporal DB Glossary of Jensen, et al, in
  Tansel book, to comply with accepted db terms.

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Intro to temporal databases (contd)

• ANSI/SPARC architecture 3 levels
• Physical External Conceptual abstract vs.
  concrete
• Abstract formal meaning representation-independ
  t
• Concrete specific, finite rep of a certain data
  model
• Abstract languages
• 1st-order and temporal logic, relational algebra,
  deductive languages
• Concrete languages
• TSQL2 others in 107, 108, 110

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2. Temporal databases

• Major issues
• Choice of temporal domains (only flat types
  considered in this survey)
• Points vs. intervals
• Linear vs. branching
• Dense vs. discrete
• Bounded vs. unbounded time
• Query Language issues
• formal semantics, expressiveness, implementation

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2.1 Temporal domains

• Temporal ontology 2 distinctions from AI and
  logic
• Points, or instants (at particular times)
• Intervals (during ranges of time)
• Point view dominant in database work intervals
  defined as pairs of endpoints, making it easy to
  move between the 2 views in first-order case

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2.1 Temporal domains (contd)

• Mathematical structure on points
• Partial order
• Total (linear) order
• Ex cyclic time modeled with linear transitive
  order, reflexive symmetric, or with ultimately
  periodic sets
• Branching time modeled with partial order
  satisfying left-linearity (no branch to left)

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2.1 Temporal domains (contd)

• Temporal domain first-order structure with a
  given Signature (set of Constant, Function and
  Relation symbols)
• Typical elements of signatures
• lt binary-order relation
• 0 origin or std ref pt of a temporal domain
• s denotes succession of time points
• , - relative distance of time points
• ?k periodicity congruence modulo k

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2.1 Temporal domains (contd)

• Standard temporal domains (in this 1st-order
  structure
• N natural numbers
• Z integers
• Q rationals
• R real numbers
• Equality not necessarily available in domains
  like TSQL2
• Temp domains may have finite universes, or
  bounded subsets of standard domains

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2.1 Temporal domains (contd)

• Common assumption Time is discrete and
  isomorphic to natural numbers vs. AI view that
  time is usually dense.
• Continuous time is becoming valuable in math,
  physics and hybrid systems.
• Constraint formula allows finite representation
  of dense sets in computer storage
• Higher level, or multiple-time granularities
  (hours vs. weeks), require multiple, interrelated
  temporal domains (not included in this survey)

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2.2 Abstract temporal databases

• Model-theoretic view is most basic view
• Treats ATD as a 1st-order structure
• Snapshot view treats ATD as a function mapping
  each instant as a tuple
• Timestamp view maps a set of instants with each
  tuple

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2.2 Abstract temporal dbs (contd)

• Assumptions
• Single temporal dimension and domain T
• Single data domain U containing standard db
  constants (these two will be expanded)
• Context relational data model, then generalized
  to other 1st-order data models
• Fixed db schema with a fixed set of relations

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2.2 Abstract temporal dbs (contd)

• Model-theoretic view
• Abstract temporal relation
• (a1,-,an,t) ? P iff P(a1,-,an) holds at t where
  (a1,-,an) ? U
• where P is a relation of arity n, P of arity
  n1, U is a single data domain, and t is a
  particular instant.
• Formally, an ATD (a finite temporal structure D)
  (U,T,P1,,Pk) for the 2-sorted 1st-order
  language LD containing a new relation symbol for
  each A.T. relation Pi and constant symbol for all
  ? U, and also 0 ? T.
• The final element of Pi is temporal the others
  are data.
• No assumptions are made about temporal domain T
• D is finite if it consists of finite relations

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2.2 Abstract temporal dbs (contd)
Model-Theoretic View
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2.2 Abstract temporal dbs (contd)

• Snapshot view a set of functions s/t
• f Pi(t) (a1,,an) Pi(a1,,an) holds at t
• where each such relation is binary in which the
  values of the 2nd attribute are sets of tuples
  i.e., non-1NF.
• Timestamp view a set of functions s/t
• f Pi( a1,,an) t Pi(a1,,an) holds at t
• Here the db consists of a timestamp relation for
  each relation symbol, and each relation (non-1NF)
  data attributes corresponding to the relation
  symbol, a timestamp attribute, which is a set
  of instants.

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2.2 Abstract temporal dbs (contd)
Snapshot View
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2.2 Abstract temporal dbs (contd)
Timestamp View
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2.2 Abstract temporal dbs (contd)

• Multiple temporal dimensions
• Necessary to model intervals (pairs of pts)
• Multiple kinds of time
• Valid vs. transaction ref. time vs. event time
• Assumption single temporal domain
• Interpretations of MDTD captured by adding axioms
  as integrity contstraints
• An intervals start must precede its end.
• Adding dimensions increases complexity

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2.2 Abstract temporal dbs (contd)

• Example properties of ATDs
• In Valid-time TDs, a model is point-based if
  facts are associated with single instants
• Interval-based if events are associated with
  intervals (represented as pairs of instants)
• The semantics of many query languages and
  integrity constraints can be defined directly,
  regardless of representation method.

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2.3 Concrete temporal databases

• Any model-specific db just a rep of a ATD
• 2 CTDs are equiv if they rep the same ATD
• An ATD may be infinite, but only finite objects
  can be explicitly represented in storage.
• Many TDB models incompatible unless specific
  representations of ATDs.

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2.3 Concrete temporal dbs (contd)

• Two important properties of CTD classes
• Data expressiveness
• How many ATDs can be represented within it (gives
  a metric to expressiveness)
• Succinctness how much space is needed to
  express a given ATD (also good metric).

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2.3 Concrete temporal dbs (contd)

• Concrete Timestamp Databases
• Timestamp view ? most useful for CTD
• Infinite set implicitly represented with
  timestamp formulae 1st-order formulae with one
  free variable in the language of the temporal
  domain.
• Example 0 lt t lt 5 V t gt 10
• See next slide for an example CTS DB

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2.3 Concrete temporal dbs (contd)
Timestamp View with Timestamp Formulae
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2.3 Concrete temporal dbs (contd)

• Timestamp formulae
• For temporal domain (N, lt) timestamps must be
  finite or co-finite subsets of N.
• Presurger arithmetic (N, 0, , lt) timestamps all
  ultimately periodic subsets of N.
• Ex natural numbers beginning with 0, period 7
• ?y, t y y y y y y y
• Equivalently, as congruence formula t ? 7 0
• where ? k means congruent modulo k.

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2.3 Concrete temporal dbs (contd)

• These as timestamps allow infinite ultimately
  periodic ATD to be represented finitely.
• Ultimately periodic means that, if some natural
  number is added to the time coordinate, a given
  set of relationships in the ATD still holds true.
• Calendars can be defined with inf. periodic sets.
• Finite periodic sets may be represented well as
  infinite sets constraints for finiteness.
• Ex all Sundays in a year

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2.3 Concrete temporal dbs (contd)

• Quantifier Elimination main tool in theory of
  timestamp dbs.
• A theory admits Q.E. if every formula in the
  theorys language can ? an equiv formula free of
  quantifiers.
• These must satisfy tests, for example, that a
  specific instant belongs to a timestamp.

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2.3 Concrete temporal dbs (contd)

• Features of timestamp formulae
• Constraints are atomic TSFs
• A TSF is termed separable if it is a conjunction
  of the forms t c, t lt c or t gt c, and c is in T
• To admit gt 1 dimension or rep an interval
  requires timestamp formula to have at least 2
  free variables
• Timestamps may be associated either with tuples
  or with attribute values
• Assumption timestamps are finite or bounded
  sets, unless TSF used to rep inf sets implicitly

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2.3 Concrete temporal dbs (contd)

• Finite Temporal DBs
• For snapshot dbs or temporal structures to be
  used as CDBs requires they be finite and describe
  a finite subset of time domain
• These 2 forms usually waste too much space

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2.3 Concrete temporal dbs (contd)

• Features of Logic Programs
• To represent an (infinite) ATD as finite, LPs
  consist of deductive (Horn) rules a finite db
• The ATD corresponding to such a program is called
  its least Herbrand model
• Ex to rep Sundays sunday(0),
• sunday(s7(T)) - sunday(T).
• Notation (N, 0, s) used for LP syntax where s is
  a unary successor, in only 1 argument of a
  relation

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2.4 Interoperability

• If two temporal data models, ?1 and ?2, use the
  same temporal domains, then the meaning of ?1
  (resp ?2) is defined as a total mapping h1 (resp
  h2) from CTDs defd under ?1 (resp ?2) to ATDs.
• The inverse mappings h-11 and h-12 may be only
  partial (some ATDs may not be repable in the
  given data model)
• Let rep function composition then d1 h-11
  h2(d2) represents the CDB under ?1 corresp to a
  CDB d2 under ?2. Then d1 can be queried the same
  as ?1.
• This provides the access for d2.

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3 Properties of Query Languages

• A semantics is declarative if it assigns meaning
  to a query without ref to evaluation method
• Query evaluation can be in closed form if the
  query result can be expressed in the dbs
  language
• Repl independence a query answer s/b the same
  for any 2 CDBs representing the same ATD
• Query expressiveness 2 queries equivalent if
  they return the same answer for every db.
• Data complexity the computational complexity of
  the set of finite dbs where a fixed query true

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4 Abstract Query Languages

• 4.1 Relational Calculus
• Domain relational calculus the 1st-ord logic of
  an ATD (model theoretic view) can be used as a
  QL.
• Semantics Tarskian, which is declarative the
  answer to a 1st-order query is valuations that
  make the query formula true in the given db
• 1st-order logic can be used as a concrete query
  language

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4.1 Relational Calculus (continued)

• Example query list all countries that lost and
  regained independence. (the example was done in
  2nd-order relational calculus)
• 2 ways to implement query evaluation in CTS DBs
• Translate the query to relational algebra, use
  generalized versions of RA operations.
• Directly eval the query in closed form (resulting
  in a timestamp relation) using quantifier-eliminat
  ion procedures for the temporal and data domains.

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4.2 Relational Algebra

• RA semantics are defined as set theory for
  possibly infinite relations, so it fits the
  model-theoretic view of ATDs
• In snapshot view, RA operations can be used only
  on snapshots corresponding to the same time
  instants in different relations (pointwise).

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4.3 Temporal Logic

• Use a temporal extension of 1st-order LD tl(LD)
• Contains the binary connectives since until
• ?A sometime in the past true since A
• ?A (sometime in the future A) true until A
• These connectives may become invalid when the
  dimension is gt 1

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4.4 Inductive Query Languages

• Natural queries may be inductive or 2nd-order
• Ex find all who are at risk (infected
  previously or been in contact with an at-risk
  person)
• Logic programming languages extend Datalog
  (language of function-free logic programs)
• Datalogltz with integer order constraints
• DatalogltQ with rational order constraints
• Datalog1S a unary successor symbol in 1 arg
• unary successor symbol linear arith constraints

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5 Concrete Query Languages

• 5.1 TQuel
• Supports single temporal domain discrete,
  finite, multi-level (can refer to specific day or
  hour)
• 2 temporal dimensions valid and transaction time
• Data model a timestamp is an interval
• Intervals must be maximal so the timestamps of
  identical facts coalesce for overlapped intervals
• An insertion may trigger a coalesce operation

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5.1 TQuel (continued)

• TQuel can simulate temporal logic queries
• 1st-order language cant express inductive
  temporal queries
• Data complexity polynomial
• Timestamp max req ? a given abstract temporal rel
  is uniquely repd as a TQuel rel
• No support for gt 2 temporal dimensions

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5.2 TSQL2

• SQL2 with time component
• Linearly ordered
• No test for equality of time constraints (can
  give diff answers for discrete dense temp
  domains)
• Point-, not interval-based
• Facts are timestamped with finite unions of maxl
  intervals (one timestamp/fact)
• No semantics or formal properties established ?
  use of interval rels insufficient for unions of
  intervals

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5.3 Hist. Relational Data Model

• HRDM supports single, discrete and infinite
  temporal domain single time dimension
• Relations are finite
• Non-uniform rel attributes some are parts of key
• Non-key attributes can be functions? limited
  non-1NF relations
• No specs for which subsets of the temporal
  domain can serve as function domains difficulty
  assessing expressiveness of the model or comp
  complexity of queries.

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5.3 HRDM (continued)

• Redefines most operators of RA
• Semantics defined with set theory
• Cant express queries relating snapshots at
  different instants
• 1st-order algebra cant express inductive
  temporal queries
• SQL extension not representation-independent

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5.4 Backlog Relations

• Supports single, discrete, infinite temporal
  domain valid- and transaction-time dimensions
• These 2 dimensions are not independent here
• Backlog rels store not data, but change requests
• Only transaction-time instants are stored
• Valid time requires a scan that a tuple was
  inserted but not later deleted
• BRs rep only finite abstract temporal relations

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5.4 Backlog Relations (continued)

• Example - 5 attributes consec of updates
  op-name transaction time country capital.

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6 Incomplete Temporal info

• Example partial ordering of events
• Various solutions proposed
• Non-linear, based on events
• Generalize the temporal db paradigm
• Marked nulls stand for some value in domain
• Same null value may be in diff columns or rows
• Each row has quantifier-free local condition with
  some nulls of this row

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6 Incomplete Temporal info (contd)

• Entire table has a quantifier-free global
  condition relating nulls in different rows
• Indefinite timestamp formulae define indefinite
  timestamps which are sets of timestamps
• An indefinite timestamp table is a finite set of
  tuples with indef ts formulae a global
  condition
• Semantics of an indef temporal table sem(T)
• rep(T) is ts rels ? substing domain vals for
  nulls satisfying the global condition of T

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6 Incomplete Temporal info (contd)

• In van der Meydens framework, only single vals,
  which may be null, can constitute timestamps
• Only finite ATDs can then be represented
• Nulls can be related via global conjunctive cond
• May be arbitrary of temporal dims, but the
  complexity of evaluating queries relies on the
  num.
• Data complexity for certain answers to FO
  queries
• In PTIME for one-dimensional time
• co-NP-complete for greater of dimensions

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7 Related Work in AI

• Most AI approaches restricted to the
  propositional (non-temporal) case
• Often take an interval view of time originating
  in the work of J.F. Allen, where each prop is
  assoc with an interval where it holds true.
• 2 intervals could relate via a rels
  representing a disjunction? allows rep of much
  disjunctive info
• 1st-order queries re quantifiers are not
  supported
• Queries evald via constraint-satisfactn
  algorithms