IP: An Instant-Period{based Theory of Time

IP :
An Instant-Period{based Theory of Time
Llus Vila
Cam de Sta. Barbara, s/n, 17300 Blanes, Catalonia, Spain
Instants have been criticised as temporal primitive for common-sense reasoning on
ontological {for being just abstract entities not concerned with common-sense{ and
semantical {the Divided Instant Problem (DIP){ arguments. Therefore, period-based
theories have received special attention. In this paper we provide arguments for incorporating instants to the time ontology. We present an axiomatization of time based
on instants and periods (IP ), characterize all the models of the theory, and explore
its relations with other Period{ and Instant-Period{based theories appeared in the
AI community. Finally, we discuss its suitability for supporting temporal knowledge
representation and the Divided Instant Problem is revisited.
1 Introduction
Several theories of time have been porposed aiming at providing a formal basis for commonsense reasoning in Articial Intelligence. Historically, the contenders as ontological primitive
of time has been two: points of time {instants{ and intervals of time {periods. Recalling the
\temporal pioneers" in AI, McDermott [11] proposed a theory based on a left-linearly rightpartially ordered unbounded and dense collection of instants. Allen [1, 2], on the other hand,
claims that the period should be the only temporal primitive since it is the most adequate
concept for representing both states and events. A general criticism against instant-based
theories is their articiality {as long as, according to Allen & Hayes [4], they start with some
mathematical structure which uses to be too \rich" wrt. the common-sense intuitions one
is intended to capture. Allen [1] takes a basic set of 13 primitive binary period relations
that correspond to every possible simple qualitative relationship that may exist between
two periods. Given these relationships, Allen formulates a set of axioms that dene their
behaviour in terms of the following three axiom schemas [2]:
1. given any period there exists another period related to it by each relationship
2. the relationships are mutually exclusive
3. the relationships have a transitive behaviour
We shall below refer to this theory as IA . Allen's theory is formally re-dened in terms
of the single period relation Meets in [3] and formally analyzed by P. Ladkin in [9] from
where we reproduce the axioms {Meets is noted by k and denotes the or-exclusive logical
8p; q; r; s: pkq ^ pks ^ rkq =) rks
8p; q; r; s: pkq ^ rks =) pks 9t: pktks 9t: rktkq
8p: 9q; r: qkpkr
8p; q; r; s: pkqks ^ pkrks =) q = r
Let's call it IAH . Ladkin relates this theory with other period-based theories, completely
characterizes its models and proposes a completion to obtain an axiomatization of the theory
of intervals over rational endpoints which is proved to be countably categorical [13].
Nevertheless, from a common-sense point of view, many situations suggest the need of
including instants in our model of time as an entity dierent from periods: Instantaneous
holding of states {e.g. \the pressure of the valve at the moment the device stopped",
\the patient temperature at 9.00"{, Accomplishment events {e.g. \shoot the gun", \
turn o the light"{, Transitions, the time where state propositions change its truth value
are usually seen as instantaneous {e.g. \when the light is turned o the room changes from
being illuminated to darkness", Continuous change1 .
As from that point one seemingly has two alternatives: (i) starting with a concerted
instant-period ontology, or (ii) dening instants from periods. Some semantic problems
have been put forward against rst choice like the famous Divided Instant Problem (DIP)
[7, 13] which leaded a number of researchers {Allen & Hayes among them{ to follow the
second way. In [14] ontological arguments are furnished against such instant constructions
on period-based theories. To summarize them, Allen & Hayes end-up with a sort of models
which constitute a non-homogeneous picture of time, present some technical glitches and
are not necessary to overcome the DIP. Moreover, as remarked by Allen & Hayes, if either
approach {instant or inteval-based{ is pushed to its limits one may end up at the same place.
It seems to invalidate initial considerations against instants since their undesired richness
turns out to be not exclusive of them. Therefore, we nd no arguments against following the
rst route. Next we introduce an instant-period theory of time, study its models and analize
its relation with other theories. Then we discuss its advantages as a suport for temporal
knowledge representation. Full proofs and details can be found in the IIIA reports [16, 15].
2 An Ontology made of Instants and Periods
In this section we propose a time ontology inspired in the idea of having both instants and
periods (Galton [6], Bochman [5]). We follow a simple and straight forward route which
consists of relating periods to pairs of instants, according to the intuition that a period is
characterized by its pair of endpoints. A period is set out as the portion of time beginning at
a certain instant and nishing at a dierent later one. The problem with such an objection
stems from viewing a period as a collection of instants, position that we explicitly refuse2 .
2.1 The Instant-Period Theory IP
Our language has two sorts of symbols3 , the instants sort (I ) and the periods sort (P ),
which are formed by two innite disjoint sets of symbols, and three primitive binary relation
symbols, : I I and begin; end : I P .
The rst-order logical formulation of our simple theory of time (let's name it IP ) consists
of the following axioms4:
1 A. Galton [6] provides arguments for which the notion of instant appears to be \necessary in order to
accommodate the possibility of representing facts concerned with continuous change".
2 Notice that our approach is conceptually dierent from viewing periods as convex sets of instants despite
that some authors seemingly consider them as being equivalent.
3 Using Shoham terms [12], they are the two time-ontological entities over which we shall interpret
4 We note variables for instants as
n and variables for periods as 1
n . Free variables denote
outermost universally quantied variables. The following priority is assumed for connectors: = : ^ _ )
{though one may nd some redundancies.
i ;:::;i
P ;:::;P
IP5 1
IP5 1
IP7 1
IP7 2
IP8 1
IP8 1
:(i i)
i i0 =) :(i0 i)
i i0 ^ i0 i00 =) i i00
i i0 _ i i 0 _ i = i 0
9i0 : i0 i
9i0 : i i0
begin(i; P ) ^ end(i0 ; P ) =) i i0
9i: begin(i; P )
9i: end(i; P )
begin(i; P ) ^ begin(i0 ; P ) =) i = i0
end(i; P ) ^ end(i0 ; P ) =) i = i0
i i0 =) 9 P: begin(i; P ) ^ end(i0 ; P )
begin(i; P ) ^ end(i0 ; P ) ^ begin(i; P 0 ) ^ end(i0 ; P 0 ) =) P = P 0
IP1 IP4 are the conditions for to be an strict linear order {irreexive, asymmetric,
transitive and linear{ relation over the instants set.
Axioms IP1 ,IP2 and IP4 can be more compactly expressed using the axiom
IP40 i i0 i i0 i = i0
where denotes exclusive disjunction. IP5 imposes unboundness on this ordered set. IP6 is
intended to order the extremes of a period. This axiom rules out durationless periods which
are not necessary since we have instants as a primitive. An instant cannot be identied as a
non durative period. Our theory keeps instants and periods apart {though closely related{
since they exhibit dierences in their fundamental behaviour. The pairs of axioms IP7 and
IP8 formalize the intuition that the beginning and ending instants of a period always exist
and are unique respectively. Conversely, axioms IP9 and IP10 are intended to ensure the
existence and uniqueness of periods to close the connection between instants and periods.
2.2 The Models of the Theory
Denition 1 (IP -structure) An IP -structure is a tuple hI ; P ; < ; begin ; end i where
I and P are two sets of temporal objects, < is a binary relation on I and begin ; end
are binary relations on I ; P .
The basis of our ontological approach for time is considering periods as ordered pairs of
Denition 2 (pairs) Given a set S over which an ordering relation < is dened, we note
by pairs(S ) the set of <-ordered pairs of distinct elements of S : pairs(S ) = f(x; y) j x; y 2
S ^ x < yg. Over a set of pairs we dene the following relations: (i) rst(x; (y; z )) = x = y
and (ii) second(x; (y; z )) def
= x = z.
Notice that < over S {seen as the tuples that satisfy the relation{ and pairs(S ) are by
denition exactly the same. We just introduce pairs for the sake of following the signature
of our language. We show {similarly to Ladkin [9]{ that the elements and the pairs of an
unbounded linear order S form a model for IP .
Theorem 1 (a model) Given an innite set S and an unbounded strict linear order < on
it then the IP -structure hS ; pairs(S ); <; rst; secondi forms a model of IP .
Proof: (sketch) It is easy to see that every axiom of IP is satised if we interpret the
instants on the set S , the periods on the set pairs(S ), the ordering as <, and begin and
end relations as rst and second respectively.
We want to show that these are \the only" models of IP .
Theorem 2 (the models) Any model M = hI ; P ; < ; begin ; end i of IP is isomorphic
to the structure hI ; pairs(I ); < ; rst; secondi where pairs, rst and second are dened
as above.
Proof: The mapping (f ) is dened as:
i 2 I : f (i) = i
p 2 P : f (p) = fhi1 ; i2 i j begin (i1 ; p) and end (i2 ; p)g
The relations begin and end are obviously preserved. It is easy to see that it is an
isomorphism: Sup. p; p0 such that f (p) = (a; b) = f (p0 ). Then we have begin (a; p),
begin (a; p0 ), end (b; p) and end (b; p0 ). By axiom IP10 (period uniqueness) we have p = p0 .
Corollary 1 Every model of IP is characterized by an innite set S and an unbounded
strict linear order < on it.
2.3 Allen & Hayes's Interval-based theory
Instant-Period Theory IPd
and the Dense
Although Allen & Hayes theory (IAH ) is based on time intervals and its axioms have a
rather dierent appearance from those in IP , both theories are rather close. Its language is
exclusively based on time intervals and the meets relation dened among them. Ladkin [9]
characterizes the models of IAH as the intervals from an arbitrary unbounded strict linear
Theorem 3 (IAH models) Given an arbitrary unbounded strict linear order < on a set
S , the intervals of S (dened as ordered pairs of elements of S ) form a model of IAH and
these are the only models up to isomorphism.
Therefore, the period parts of our models are the same.
Furthermore, our instant part can be associated with the classes coming out from the
relation \having the same meeting point"5 which can be dened over the set of periods.
Denition 3 (\having the same meeting point" I ) Given the intervals p; q; r; s,
[p; q] I [r; s] , pkq ^ rks ^ pks.
We can introduce the instants obtained from such a construction to extend the universe
of elements of the universe IAH . Let's call it IAHI . We obtain a theory which class of
models is the same than our instant-period axiomatization, i.e. the theories are equivalent.
Theorem 4 IP a` IAHI .
Ladkin also proposes an extension of IAH to obtain an axiomatization of the countable
categorical theory Th(INT(Q)) {i.e. the theory of rational intervals{ previously studied by
vanBenthem [13] and Ladkin & Maddux [10]. The extension consists of an additional axiom
expressing the instants density. Since we are explicitly dealing with instants, we can directly
and clearerly dene a similar extension by adding the following axiom
IP11 i i0 =) 9i00: i i00 i0
to our theory. We name the extended axiomatization IPd .
Corollary 2 (dense models) The models of IPd are characterized by the set of elements
and the set of ordered pairs of distinct elements of an unbounded, \dense", strict linearly
ordered set. Hence IPd is yet another axiomatization of Th(INT(Q)).
5 Such a construction indeed corresponds to the denition, due to Bolzano, of a point as the set of pairs
of intervals which end and begin at that point.
2.4 Galton's Instant-Period theory IPG
Anthony Galton [6] is intended to build a theory of instants and periods based on the idea
of \. . . there being an instant at the point where two periods meet. . . . "6 . Galton's theory is
dened as an extension upon Allen's period theory by postulating the existence of instants,
introducing the instant-to-period relations Within and Limits and providing a rather loose
axiomatization in which the period relation In is also involved. The axiomatics (namely
IPG ) is as follows:
I1 8I:9i: Within(i; I )
I2 Within(i; I ) ^ In(I; J ) ) Within(i; J )
I3 Within(i; I ) ^ Within(i; J ) ) 9K:[In(K; I ) ^ In(K; J )]
I4 Within(i; I ) ^ Limits(i; J ) ) 9K:[In(K; I ) ^ In(K; J )]
The actual axiomatization that Galton borrows from Allen is not precisely stated in
Galton's paper: \. . . together with the various relations between intervals catalogued by
Allen . . . ". Since Galton's paper is a revision of the theory expounded in [2] it seems that
the basic theory taken by Galton is IA . We assume as its formalization the following set of
axiom schemas where IR denotes the set of the 13 period relations:
A1 8P 2 P ; R 2 IR: 9P 0: R(P; P 0 )
A2 8P; P 0 2 P ; R 2 IR: 8R0 2 IR ? R: R(P; P 0 ) ) :R0(P; P 0 )
A3 Allen's transitive table (see [1])
We keep calling IPG to the resulting theory. At rst sight one may observe that the
instants in this theory does not correspond at all with the places where periods meet {
which is supposed to be Galton's starting intuition. For example, nothing forces the instant
within a period which existence is determined by axiom I1 to be a place where two periods
meet. Neither do Galton's instants correspond to the idea of period endpoints which is our
approach. The instant-period axiomatization proposed by Galton is just something much
weaker than other approaches to concerted instant-period axiomatics. On the one hand any
model of IPd is also a model of IPG under the denitions of Within, Limits and In given
Theorem 5 IPd ` IPG .
Proof: (sketch) IPG axioms are proven from IP10 , linearity, extremes ordering, existence
of both instants and periods and density.
It seems unlikely that such a weak theory of time has any interest other than providing
a suitable support to the characterization of Galton's states of motion. Its weakness causes
it to accept a number of rather unintuitive models mostly originated by the extremely loose
connection between instants and periods systems. We present some examples:
Example 1 Let's take a basic model M composed of an innite set of periods and Allen's
relations satisfying IA axioms plus an innite set of instants which make M satisfy I1 {for
example INT(Q) as periods and Q as instants:
Example model 1: M plus a single instant i 62 Q which limits a certain period P in M
and only that one. In particular it does not limit any of those periods that meet or are
met by P .
Example model 2: M plus a single instant i 62 Q which limits a certain period P in M
and is not within any period. In particular it is not within any of those periods that
overlap P .
6 Galton aims at dening a theory which embraces an account for continuous time and claims that the
notion of instant is indispensable for such a purpose.
Example model 3: M plus a single instant i 62 Q which limits a certain period P in M
and also is within P .
The case of the theory obtained from incorporating Galton's instants to IAH suers from
the same kind of problems due also to so loosely connected instants.
In summary, though Galton's instant-period theory appears to be strong enough for
Galton's purposes is too weak as a common-sense theory of time: it does accept too many
models. It might be interesting to characterize the subclass of models of IPG that indeed
correspond to the intuition of having instants at the period meeting places. For such a
purpose we proceed as follows: rst, we rescue IA theory through an apparently slight
extension with amazing consequences; then, we incorporate instants by adding an axiom
embodying the intuition just stated above; nally, we discuss the class of models for the
constructed instant-period theory in relation with IP .
A more accurate look to IA reveals that nothing takes account for the intuition that
\periods are all contained in a single time dimension"7. Our renement of IA is based on
adding an axiom schema with such a role:
Denition 4 (IA0 ) IA0 is the axiomatization IA plus the following additional axiom
A0 1 8P; P 0: 9R 2 IR: R(P; P 0)
This renement produces a rather remarkable change in the models that are going to
be accepted. We demonstrate it by analyzing how this theory relates with Allen & Hayes
theory. We use the denitions of period relations in terms of the single relation Meets given
in [3].
Lemma 1 IA0 ` IAH.
Lemma 2 IAH ` A0 1; A2; A3 .
Regarding A1 , the relationships Meets and Before (and their inverses) can be easily
derived from one and two applications respectively of axiom M3 towards the future (towards
the past), but is not so the case for the remaining ones. To derive them we would require
a sort of density axiom like N1 which that Ladkin proposes to complete IAH theory ([9],
theorem 4):
N1 8p; q; r; s: 9x; y: Pointless(p; q; x; y)&Pointless(x; y; r; s)
Lemma 3 IAH; N1 ` A1 .
Theorem 6 IAH; N1 a` IA0 .
Proof: Given lemmas 1, 2 and 3, it suces to prove that IA0 ` N1 that is quite straight
forward by applying A1 with the relationship Started by on the period bounded by the
initial points.
Let us dene a variant of the instant-period theory IPG by incorporating instants into
IA0 . We start with an ontology made of non-empty sets of instants and periods, the primitive
relations Within, Limits, the 13 period-to-period relations and In is a derived relation
dened from Meets as usual. We take the axioms of IA0 plus the following axioms:
IM1 8P; P 0: P kP 0 ) 9i: Limits(i; P ) ^ Limits(i; P 0)
IM2 8i: 9P; P 0 : P kP 0 ^ Limits(i; P ) ^ Limits(i; P 0)
We name the resulting axiomatization IPG0 . It is rather straight forward that its theory
does actually include the theory proposed by Galton:
7 This is an idea which Allen seems to be in sympathy with since he explicitly refuses alternative structures
like McDermott's branching time construction.
Lemma 4 IPG0 ` IPG .
Proof: (sketch)
It's easy to show that IPG0 ` I1 ; I2 ; I3 ; I4 .
We have nally reconstructed Galton's theory by following the lead that Galton suggested
of having and instant at the place where two periods meet {and only there. To characterize
the models of this new theory we state and prove the relation of the resulting theory with
our instant period one.
Theorem 7 IPG0 a` IPd .
Proof: It follows from theorems 4, 6, IM1 and IM2 .
We end up at the same place. IP takes account of Galton's intuition about instants but
also every instant is guaranteed to be a period meeting place.
3 Temporal Representation
Since IP theory is based on both instants and periods we claim that it is better suited than
period-based theories to support both:
a natural modeling of domain phenomena from a common-sense point of view {like
those presented in section 1.
a natural and exible denition of a logic of time and action.
We illustrate it by demonstrating how the DIP scenario is represented. Let us rst
introduce the set of predicates available for talking about the truth value of propositions
over time. Our two-fold ontology of time allows one to talk about propositions holding both
at an instant and throughout a period.
We introduce the predicates Holds and Holds [8, 6]:
Holds (p; P ) p holds throughout the period P
Holds (p; i) p holds at the instant i
They are related by the following axiom
Holds (p; P ) () 8i: Within(i; P ) ) Holds (p; i)
Note that nothing is said about either
Holds (p; i) ^ begin(i; P )
Holds (p; i) ^ end(i; P )
The holding of non-atomic propositions at an instant are dened by
Holds (:p; i) () :Holds (p; i)
Holds (p ^ q; i) () Holds (p; i) ^ Holds (q; i)
The holding of non-atomic propositions on a period are trivially obtained from the denition
of Holds in terms of Holds . The holding of a negated proposition is
Holds (:p; P ) () 8i: Within(i; P ) ) :Holds (p; i)
Holds (p ^ q; P ) () 8i: Within(i; P ) ) Holds (p ^ q; i)
Using this logic of time, the \tossed ball" example would be expressed as follows:
Holds (\velocity is positive"; P1 )
Holds (\velocity is zero"; i)
Holds (\velocity is negative"; P2)
end(i; P1 ) ^ begin(i; P2 ) (thus P1 kP2 by denition)
The DIP is formalized as follows:
Holds (p; P1 )
Holds (:p; P2 )
P1 kP2
Borrowing also the predicate Occurs we could assert that
Occurs (\transition event from p to :p"; end(P1 ))
but nothing can be inferred from this information about
Holds (p; i)
However, the possibility is left open to explicitly assert either it or the opposite. The solution
given to the DIP is based on the idea of simply not considering as directly related a state
holding at an instant and this state holding on a period which the instant is an extreme of.
4 Concluding Remarks
From an analysis on previous works of dening time, we claim that it is more natural and
simple to start with both instants and periods as primitives, instead of starting with a
purely interval-based set of more complicated axioms. Intuitions are more clearly reected
by axioms and models are easily identied. In this paper we furnished a rather trivial
axiomatics for instants and periods and used an interpretation of periods as ordered distinct
pairs of points to characterize all the models of the theory. Besides it, we have shown (i)
that our theory is equivalent to IAH appropriately extended with a denition of instants, (ii)
that it can be extended with denseness to obtain yet another axiomatization of Th(INT(Q))
and (iii) its relations with other instant- period based theories constructed upon Allen &
Hayes's and Allen's period based and Galton's instant-period based axiomatics.
Furthermore, we claim that our instant-period based theory provides desirable properties
of diverse nature: ontological {period decomposability, uniformity of time{, representational
{instantaneous holding of states, distinction between open/closed periods{ and semantical
{the divided instant problem can be adequately accommodated in our instant-period based
theory of time, which is not so the case of the attempts of introducing instants from intervalbased theories [4].
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