Temporal Representation

1

• Temporal Representation Reasoning
• Quantitative Temporal Representation Reasoning

2
Representation and Reasoning with Time

• Many choices have to be made when modeling time
• Principle Different tasks require different
  models.
• Absolute or relative descriptions of time (date)
• Discrete or continuous time
• Different sizes of (smallest) time unit
  (granularity)
• Quantitative or qualitative approach
• Linear, parallel or branching time
• Time points or intervals
• Events, processes or states

3
Absolute vs. Relative

• compare John will arrive Wednesday, 21st.
• John will arrive tomorrow.

Wed 21/03/04
John arrives
Tomorrow
Today
John arrives
4
Discrete or Continuous Time

• Discrete time
• time steps, next/previous time point
• like integers
• t1 t2 t3 t4
• Continuous time
• always time in between two time points
• like real numbers
• t0 t0.5 t1

5
Size of Smallest Unit of Time

• Relevant for discrete time (mainly).
• year, month, day, hour, minute, second,...
• Choice affects
• fidelity and precision
• costs (storage, efficiency)
• Sometimes, a hierarchical approach, with
  multiple resolutions used together, works best.

6
Quantitative vs. Qualitative

• For both discrete and continuous time.
• Qualitative
• ordering between time points Tk gt T0
• relative properties Tk - T0 lt Tx - Ty
• Quantitative
• absolute size Tk 1.5, T0 0
• absolute properties Tk - T0 1.5

7
Points or Intervals

• Time points
• event E occurs at time Ti.
• property P holds at time Ti.
• Time intervals
• event E takes place during Ti,Tj .
• property P holds during Ti,Tj .
• Distinction points/intervals not the same as
  distinction discrete/continuous.

8
Quantitative Temporal Representation Reasoning

• Often we want and can talk about precise times
• ILP we have precise time points
• scheduling we want to commit to exact time
  points
• timetabling we have the exact times and want to
  state them
• How do we represent such information?
• time points actions and events are
  instantaneous, or we consider their
  beginning and ending
• time intervals all actions and events have
  duration

9
Representation of Quantitative (Numeric) Temporal
Information

• The Temporal Constraint Satisfaction (TCSP)
  framework of Dechter, Meiri, and Pearl is the
  most widely used framework for the representation
  of quantitative temporal information.
• In the TCSP
• Temporal events are represented by intervals
• Each interval has a starting and ending time
  point
• Constraints capture distances between time
  points
• Constraints can be disjunctive (to represent
  imprecise information)
• The restriction of TCSP where disjunctive
  constraints are not allowed is called a Simple
  Temporal Problem (STP)

10
The TCSP Framework

• Variables represent time points - starting time
  point and ending time point of intervals
• Numeric temporal information is represented by a
  set of unary and binary constraints, each
  specifying a set of permitted intervals.
• Propositions stand for events, and each
  proposition Pi is associated with an interval Ii
  ai,bi. Information about the timing of events
  can be presented by means of constraints on the
  intervals
• I.e. their associated beginning and ending
  points.
• Time points are considered as variables with
  continuous domains, where a time point may be a
  beginning or an ending point of some event.

11
The TCSP Framework

• A constraint is a distance between two time
  points. It is of the form Xj - Xi lt C , if Xi
  and Xj are two time points. Namely, constraints
  are a set of linear inequalities on the Xis
• Disjunctive sets of constraints are allowed
• A special variable representing the start of
  events is introduced
• to capture absolute times
• We are interested in the following reasoning
  tasks
• finding all feasible time that a given event can
  occur
• finding all possible relationships between two
  given events
• generating one or more consistent scenario

12
Example

• John and Fred work for a company that has local
  and main offices in Los Angeles. They usually
  work at the local office, in which case it takes
  John less than 20 minutes and Fred 1520 minutes
  to get to work. Twice a week John works at the
  main office, in which case his commute to work
  takes at least 60 minutes. Today John left home
  between 705710 a.m., and Fred arrived at work
  between 750755 a.m. We also know that Fred and
  John met at a traffic light on their way to work.

13
The TCSP Framework (example)

• J is a proposition standing for event "John going
  to work" and X1, X2 are the starting time and
  ending times of J, respectively. F is a
  proposition standing for event "Fred going to
  work" and X3, X4 are starting time and ending
  time of F, respectively. X0 is the beginning of
  the events.
• We can represent with inequalities information of
  the form
• John goes to work either by car (30-40 minutes),
  or by bus (at least 60 minutes).
• 30 lt X2 - X1 lt 40 or X2-X1 gt60
• Fred goes to work either by car (20-30 minutes),
  or on foot (40-50 minutes).         20 lt X4 -
  X3 lt 30 or 40 lt  X4 - X3 lt 50

14
The TCSP Framework (example)

• Today John left home between 710 and 720
• X0 7.00
• 10 lt X1 X0 lt20
• Fred arrived at work between 800 and 810
• 60 lt X4 X0 lt70
• John arrived at work about 10-20 minutes after
  Fred left home.
• 10 lt X2 X3 lt20

15
The TCSP

• A TCSP involves a set of variables, X1,,Xn ,
  having continuous domains each variable
  represents a time point. Each constraint is
  represented by a set of intervals. An interval
  (Ii)  can be represented by two time points,
  ai,bi, where ai is the starting point and bi is
  the ending point of Ii . 
• Therefore , set of intervals I1,,
  Ina1,b1,,an,bn
• A unary constraint, Ti, restricts the domain of
  variable Xi, to the given set of intervals
  namely, it represents the disjunction
• (a1 ltXi ltb1) V . . . V (an ltXi ltbn)  
• A binary constraint, Tij, constraints the
  permissible values for the distance Xj Xi it
  represent the disjunction
• (a1 ltXj - Xi ltb1) V . . . V (an ltXj - Xi
  ltbn)
• Constraints are always given in a canonical form
  where all intervals are pairwise disjoint.

16
The TCSP

• A network of binary TCSP can be represented by a
  directed constraint graph. A network of a binary
  TCSP consists of
• Nodes that represent variables (X1, . . . , Xn),
  
• Edges that represent unary and binary
  constraints, which are labeled by the interval
  set.
• Each constraint, Tij, implies an equivalent
  constraint Tji however only one of them will
  usually be shown in the constraint graph.
• A special time point, X0, is introduced to
  represent the "beginning of the world". So a
  unary constraint is represented in an equivalent
  binary constraint with X0
• A solution is a tuple X x1,. . ., xn, if
  the assignment X1 x1, . . . , Xn xn satisfies
  all the constraints. V is a feasible value if
  there is a solution including V. The minimal
  domain is the set of all feasible values of a
  variable.

17
The TCSP

• There are three operations on binary constraints
  Union, Intersection and Composition. Definitions
  of the three binary operations on TCSP are
• Let TI1, . . Il and S J1, . . Jm be
  constraints
• T ? S I1, . . Il , J1, . . Jm
• T  ? S K1, . . Kn where Kk Ii ? Jj for
  some I and j (admits only values that are allowed
  by both of them) Note that n lt l m
• T ? S K1, . . Kn where Kk ac,bd for
  some II a,b and Jj c,d Note that n lt l
  m

18
The TCSP

• For example let C1 1,4), (6,8) and C2
  (0,1, (3,5), 6,7
• C1 ? C2 1, (3,4), (6,7
• Let C3 1,2, (6,8) and C4 0,3), (12,15
• C3 ? C4 1,5), (6,11), (13,17, (18,23)

19
The TCSP

• A binary constraint T is tighter than S, denoted
  by T ? S, if every pair of values allowed by T is
  also allowed by S
• The Universal Constraint is the most relaxed
  constraint 
• Network T is tighter than S if T and S have the
  same set of variables and the partial order ? is
  satisfied for all the corresponding constraints
  namely, for all pairs i, j, Tij ? Sij
• Two networks are equivalent if they represent the
  same solution set. A network can have many
  equivalent representations. However, there is one
  minimal equivalent network with respect to ?.

20
The STP

• A TCSP in which all constraints specify a single
  interval is called a simple temporal problem
  (STP). A STP can be represented by a distance
  graph. In such a distance graph, each arc(i,j),
  is labeled by an interval, a, b, representing
  the constraint in inequalities form.
• An example a distance graph has arc(i,j) which
  is labeled by a,b
• represents linear inequalities
• alt Xj - Xi ltb Xj - Xilt b Xi Xjlt -a

21
Solving an STP

• An STP is consistent if its distance graph has no
  negative cycle. Finding a minimal graph of a
  consistent STP can give a solution of the STP.
• A Minimal graph can be constructed from the
  directed-graph (d-graph). A consistent STP can be
  effectively specified by a complete directed
  graph, called d-graph, where each edge, i-gtj, is
  labeled by the shortest path length. The d-graph
  represents a tighter of the original STP. A
  d-graph of a consistent STP is the minimal graph
  representation of the STP.
• The d-graph of a STP can be constructed by
  applying Floyd-Warshalls all-pairs-shortest-paths
  algorithm to the distance graph. The algorithm
  runs in time O (n3), and negative cycles are
  detected simply by examining the sign of the
  diagonal elements dij.

22
Solving an STP

• All-pairs-shortest parts algorithm 1. for i1
  to n do dii ?0 2. for i,j1 to n do dij ? aij
  3. for k1 to n do    4. for i,j 1 to n do
            dij ? mindij,dikdkj 
• Floyd-Warshalls algorithm
• Constructing a distance graph runs in time O
  (n3). It constitutes a polynomial time algorithm
  for determining the consistency of an STP, and
  for computing both the minimal domains and the
  minimal network. Once the d-graph is available,
  assembling a solution requires only 0 (n2) time
  because each successive assignment needs to be
  checked against previous assignments and is
  guaranteed to remain unaltered. Thus, finding a
  solution from minimal graph can be achieved in O
  (n3)

23
Solving a TCSP

• Solving a General TCSP by STPs
• Deciding consistency and solving for a TCSP is
  NP-hard.
• A TCSP can be solved by decomposing the TCSP
  into several STPs and solving each one of them,
  and then combining the results. The complexity of
  solving a general TCSP by generating all the
  labeling and solving them independently is
  O(n3Ke), where K is the maximum number of
  intervals labeling an edge, and e is the number
  of edges.
• Alternatively we can use CSP search algorithms
  that branch on single intervals

24
Combining Quantitative and Qualitative Information

• Meiri proposed a new network-based computational
  model for temporal reasoning that is capable of
  handling both qualitative and quantitative
  information.
• In Meiris model, a temporal object (Oi) can be a
  point or an interval. Intervals correspond to
  time periods during which events occur or
  propositions hold, and points represent the
  beginning and ending points of some events, as
  well as neutral points of time.

25
Qualitative Constraints

• A qualitative constraint between two objects Oi
  and Oj is a disjunction of the form
• (Oi r1 Oj) V . . . V (Oi rk Oj)
• where each of the ris is a basic relation that
  may exist between the two objects. There are
  three type of basic relations.
• Basic Interval-Interval (II) relations that can
  hold between a pair of intervals (Allens
  relations) denoted by the set b,m,s,d,f,o,bi,mi,s
  i,di,fi,oi,
• Basic Point-Point (PP) relation relations that
  can hold between a pair of points, denoted by the
  set lt, , gt
• Basic Point-Interval (PI) relation relations
  that can hold between a point and an interval,
  and basic Interval-Point (IP) relations that can
  hold between an interval and a point.

26
Qualitative Constraints
The basic relations between a point, p, and an
interval, I I- ,I
27
Qualitative Constraints

• Qualitative constraints of Meiris algebra (QA)
  are all legal constraints
• 213 II relations, 23 PP relations, 25 PI
  relations and 25 IP relations. Two binary
  operations are defined on these elements
  intersection and composition
• R' ? R'' the intersection of two qualitative
  constraints is the set-theoretic intersection R'
  ? R''
• R' ? R'' the composition of two qualitative
  constraint is defined by the following
  transitivity tables

28
Qualitative Constraints
A full transitivity table
Composition of PP and PI relations
Composition of PI and IP relations
29
Qualitative Constraints
Composition of IP and PI relations
30
Quantitative Constraints

• A quantitative constraint represents an absolute
  location or the distance between points. There
  are two types of quantitative constraints.
• A unary constraint, on points Pi, restricts the
  location of Pi to a given set of intervals
• A binary constraint, between points Pi and Pj,
  constrains the permissible values for the
  distance Pj-Pi
• Two binary operations are defined on these
  elements intersection and composition
• C' Å C'' xxÎ I , xÎ I
• C' Ä C'' z x Î I , yÎ I ,xyz

31
Relationship between Qualitative and Quantitative
Constraints

• The existence of a constraint of one type
  sometimes implies the existence of a constraint
  of the other type.
• Case 1 If a quantitative constraint, C, between
  Pi and Pj is given then the implied qualitative
  constraint, QUAL(C) is defined as follows
• If 0 Î I1, . . ., Ik, then ""Î QUAL(C).
• If there exist a value v gt 0 such that v Î I1,
  . . ., Ik, then "lt"Î QUAL(C).
• If there exist a value v lt 0 such that v Î I1,
  . . ., Ik, then "gt"Î QUAL(C).

32
Relationship between Qualitative and Quantitative
Constraints

• Case 2 If a qualitative constraint, C, between
  Pi and Pj is given then the implied quantitative
  constraint, QUAN(C) is defined as follows
• If "lt" Î R, then 0, Î QUAN(C).
• If "" Î R, then 0Î QUAN(C).
• If "gt" Î R, then - ,0Î QUAN(C).

The QUAN translation
33
Relationship between Qualitative and Quantitative
Constraints

• The intersection and composition operations are
  extended when the constraints are of different
  types. If C is a quantitative constraint and C
  is qualitative, then intersection is defined as
  quantitative intersection
• C' Å C'' C'Å QUAN(C'' )
• Composition depends on the type of C'' . If C''
  is a PP relation, then composition is
  quantitative
• C' Ä C'' C'Ä QUAN(C'' )
• If C'' is a PI relation, then composition is
  qualitative.
• C' Ä C'' QUAN(C' ) Ä C''

34
Example

• For example let C1 (0,3) be a quantitative
  constraint and
• C2 lt , be a PP relation, and C3 b,d be
  a PI relation
• C1 ? C2 (0,3) ? 0,?) (0,?)
• C1 ? C3 lt ? b,d b,s,d

35
General Temporal Constraint Networks

• A general temporal constraint network involves a
  set of variables X1, . . ., Xn, each
  representing a temporal object ( a point or an
  interval) and a set of unary and binary
  constraints.
• When a variable represents a time point, its
  domain is the set of real numbers.
• When a variable represents a temporal interval,
  its domain is the set of ordered pairs of real
  number, namely, (a,b)a,b ÎÂ , a lt b.
• Constraints may be quantitative or qualitative.
• A constraint network is represented by a
  directed constraint graph, where nodes represent
  variables and an arc (i, j) indicates that a
  constraint Cij between variables Xi and Xj is
  specified. The arc is labeled by an interval set
  or by a QA element.

36
Example

• Given JP1, P2 and FP3, P4 are intervals of
  the events that John and Fred go to work. P0 is
  the beginning of the world. The constraint graph
  of the proposition is

37
Augmented Qualitative Networks

• An augmented qualitative network is a qualitative
  network a CPA network or a PA network
  augmented by unary constraints on its domain.
• The simplest way of combining qualitative and
  quantitative information

A CPA network over multiple-interval domains
38
Augmented Qualitative Networks
Complexity of deciding consistency in augmented
qualitative networks
39
Disjunctive Temporal Reasoning

• Not all kinds of quantitative temporal reasoning
  are expressible using TCSPs
• Consider the following example from job-shop
  scheduling
• Let I and J be intervals corresponding to
  operations Oi and Oj. Oi and Oj will be executed
  on a machine that can handle only one operation
  at a time and has a set-up time of 2 minutes. The
  following is a constraint on the scheduling of
  operations Oi and Oj
• I - J- ? -2 ? J - I- ? -2
• This constraint cannot be expressed in the TCSP
  framework

40
Disjunctive Temporal Reasoning

• The Disjunctive Temporal Framework subsumes the
  TCSP and can be used to model a variety of
  temporal reasoning problems from scheduling,
  planning, temporal databases, etc.
• In this framework a temporal constraint is a
  disjunction of the form
• X1 X2 ? A1 ? X3 X4 ? A2 ?
• where each variable Xi ranges over the reals and
  each Ai is a real constant
• Various algorithms to solve disjunctive temporal
  problems have been developed
• CSP-based, SAT-based