Ontology Of Time

The Ontology of Time

Formal Ontology for Knowledge Representation and Natural LanguageProcessing – ICT School Doctorate Course

May 3rd, 2005

2qMay 3rd, 2005

Introduction

3qMay 3rd, 2005

Time

An important philosophical domain of inquiry

A foundational ontology

An essential domain in Physics

An essential domain in Knowledge Representation andReasoning

A basic ingredient of linguistic statements

4qMay 3rd, 2005

Which time?

Absolute or Relative?I Absolute: There is a purely temporal space in which

objects are temporally located.I Standard physicsI DOLCE’s Time Quality Space

I Relative: Time is an implicit structure induced by temporalrelations

I Leibniz - Newton controversyI commonsense and psychological evidence

Which structure?

I Which temporal relationsI Which axioms

5qMay 3rd, 2005

Which primitive temporal entities?

Absolute case

I “instants”, “moments”: pointsI “intervals”, “periods”: regions

Relative case

I All concrete entitiesI Only “perdurants”, “events”, “processes”, “eventualities”. . .

The endurant / perdurant discussion applies in both cases

I Three- vs. four-dimensionalismI do objects have temporal parts?

I Co-localizationI does a given 4-D “worm” identify a single entity?

6qMay 3rd, 2005

Which temporal location relation?

I Temporal locationI Binary relation relating a perdurant and a time: Occurs(e,t)

(cf. Casati & Varzi’s L for spatial location and its relation tomereology)

I DOLCE, time is a quality: Inheres + Has-qualeI Matching dimensions

I extended events: intervals are best suitedI both instantaneous and extended events: intervals and

instants, or simply instants

7qMay 3rd, 2005

Temporal location and temporal logics

I “Fluents”, “properties” and types of eventualitiesI Reified temporal logics [Allen, 1984; Shoham 1986]

Holds(p,t) iff proposition p is true at/during tI Distinctions according to dissectivity, cumulativity, etc.I Eventuality types, not tokens

I Reasoning about time vs. reasoning about propositions intime

I Modal temporal logics [Prior, 1967]I Implicit timeI Past

I It has always been the case that p (Hp)I It has been the case that p (Pp)

I FutureI It will always be the case that p (Gp)I It will be the case that p (Fp)

8qMay 3rd, 2005

Structures of time

9qMay 3rd, 2005

Instant structures

I Primitive relationI precedence: strict order

I Which order?I total (linear) or partial (branching)

I parallel times: alternative worldsI linear to the left: branching future

I unbounded (?)I dense or discrete (atomic)

I commonsenseI Achille and the turtleI computers are discrete

10qMay 3rd, 2005

Interval structures - 1

I Convex “intervals”I Allen’s relations [Allen 83, 84; Allen & Hayes 85, 89]I 13 possible relations between any ordered pair of intervals

Before x

yMeets x

y

Overlaps x

yStarts x

y

During x

yFinishes x

y

Equals x

y

I Inverse relations: After, Met-by, Overlapped-by, Started-by,Contains, Finished-byAfter(x, y) ↔ Before(y, x)

11qMay 3rd, 2005


I Allen & Hayes’s theory [85, 89]I based on a unique primitive: Meets, noted ‖I Before(x, y) , ∃z(x‖z∧ z‖y)I Equals(x, y) , ∃z, t(z‖x∧ x‖t ∧ z‖y∧ y‖t)I unbounded, “continuous” and linear time

I Alternative axiomatizations (Ladkin, Galton, Hajnicz )UNI ∀x, y, z, v ((x‖z∧ x‖v∧ y‖z) → y‖v)UEND ∀x, y, z, v ((x‖z‖y∧ x‖v‖y) → z = v)LIN ∀x, y, z, v ((x‖y∧ z‖v) → (x‖v∇ ∃s(x‖s‖v) ∇ ∃s(z‖s‖y)))UNB ∀x ∃y, z (y‖x‖z)SUM ∀x, y (x‖y→ ∃z, v, s(z‖x‖y‖v∧ z‖s‖v))DENSM ∀x, y, z, u (P<(x, y, z, u) →

∃v, w (P<(x, y, v, w) ∧ P<(v, w, z, u)))

with P<(x, y, z, u) , x‖y∧ z‖u∧ ∃w(x‖w‖u)

12qMay 3rd, 2005


I TheoremsIRREF ∀x (¬(x‖x))ASYM ∀x, y (x‖y→ ¬(y‖x))ANTRA ∀x, y (x‖y→ ¬∃z (x‖z∧ z‖y))

13qMay 3rd, 2005

From points to intervals and vice-versa

I Let 〈Q, <〉 be a linear, unbounded, dense strict orderstructure, then

I 〈I , ‖〉 s.t.I I = {(q1, q2) | q1, q2 ∈ Q and q1 < q2}

where (q1, q2) = {q ∈ Q | q1 < q < q2}I (q1, q2)‖(q3, q4) iff q2 = q3

is a model of interval theory (an interval structure)I Interval theory is countably categorical, i.e., all models of

cardinality ω0 are isomorphic to 〈I , ‖〉I Let 〈I , ‖〉 be an interval structure, then

I 〈P, <〉 s.t.I P = {[x, y] | x, y ∈ I and x‖y}

where [x, y] = {(z, v) | z, v ∈ I and x‖v and z‖y}I [x, y] < [z, v] iff ∃w (x‖w‖v).

is a linear, unbounded, dense strict order structure

14qMay 3rd, 2005

Period structures [van Benthem, 83]

I Two order relations: a precedence ≺ and a mereology vI ≺ is an unbounded strict order, “discrete”:

I NEIGH ∀x, y (x≺ y→ (∃z1 (x≺ z1 ∧ ¬∃u (x≺ u≺ z1))∧∃z2 (z2 ≺ y∧ ¬∃u (z2 ≺ u≺ y))))

I v is reflexive, transitive, antisymmetric, andI supplementation, called FREE by van Benthem∀x, y (∀z (zv x→ zOy) → xv y)where xOy , ∃z (zv x∧ zv y)

I existence of the product, called CONJ here∀x, y (xOy→ ∃z(zv x∧ zv y∧ ∀u((uv x∧ uv y) → uv z)))z is noted xu y

I existence of the “convex sum”, called DISJ∀x, y (xUy→ ∃z (xv z∧ yv z∧∀u((xv u∧ yv u) → zv u)))where xUy , ∃z (xv z∧ yv z); z is noted xt y

I DIR ∀x, y ∃z (xv z∧ yv z)I DENS ∀x∃y, z(y≺ z∧ x = y + z)

where x = y + z iff x = yt z∧ ∀w (wv x→ (wOy∨ wOz))

15qMay 3rd, 2005

Period Structures – Links between ≺ and v

I MON ∀x, y (x≺ y→ ∀z((zv x→ z≺ y) ∧ (zv y→ x≺ z)))I MOND ∀x, y (x≺ y→ (∀z (z≺ y→ (ut z) ≺ y)∧∀z (y≺ z→ y≺ (ut z))))

I CONV∀x, y, z ((x≺ y∧ y≺ z) → ∀u ((xv u∧ zv u) → yv u))

I LIN ∀x, y(x≺ y∨ y≺ x∨ ∃z(zv x∧ zv y))I ORI ∀x, y (xOy→ (x = y ∨

(xv y∧ (∃z (x≺ z∧ y = x + z) ∨ ∃z (z≺ x∧ y = x + z)∨∃z1, z2 (z1 ≺ x≺ z2 ∧ y = (z1 + x) + z2)))

(yv x∧ (∃z (y≺ z∧ x = y + z) ∨ ∃z (z≺ y∧ x = y + z)∨∃z1, z2 (z1 ≺ y≺ z2 ∧ x = (z1 + y) + z2)))∨

∃z1, z2 (z1 ≺ z2 ∧ x = z1 + z2 ∧ z1 ≺ y∧ z2 v y∧∃z3(y = z2 + z3 ∧ x≺ z3)))∨

∃z1, z2 (z1 ≺ z2 ∧ y = z1 + z2 ∧ z1 ≺ y∧ z2 v x∧∃z3 (x = z2 + z3 ∧ y≺ z3)))))

16qMay 3rd, 2005

From points to periods and vice-versa

I Let 〈Q, <〉 be a linear, unbounded, dense strict orderstructure, then

I 〈I ,≺,v〉 s.t.I I = {(q1, q2) | q1, q2 ∈ Q and q1 < q2}

where (q1, q2) = {q ∈ Q | q1 < q < q2}I (q1, q2) ≺ (q3, q4) iff q2 ≤ q3

I (q1, q2) v (q3, q4) iff q3 6 q1 < q2 6 q4

is a model of the period theory (a period structure)I Period theory is countably categorical, i.e., all models of

cardinality ω0 are isomorphic to 〈I ,≺,v〉I Let 〈I ,≺,v〉 be a period structure, then

I 〈P, <〉 s.t.I P = {F ⊆ I |(∀x ∈ F)(∀s∈ I)(xv s→ s∈ F) and

(∀x, y ∈ F)(xu y ∈ F)};I F1 < F2 iff (∃x ∈ F1)(∃y ∈ F2)(x≺ y).

is a linear, unbounded, dense strict order structure

17qMay 3rd, 2005

Reasoning with time representations

18qMay 3rd, 2005

Approaches to reasoning

I Transitive closureI given R1(x, y) and R2(y, z) what is the relationship between x

and z?I Constraint satisfaction problem

I given a network of relations over a set of variables, is therea consistent variable assignment?

I Reasoning over a set of jointly exhaustive pairwise disjointbase relations, and “generalized relations”, i.e., anydisjunction of base relations

I Allen’s 13 relations form a JEPD set of relations overintervals

I <, =, > is a JEPD set of relations over instants

I Transitivity tables

19qMay 3rd, 2005

Relation algebras

I A JEDP set of relations can be taken as the set of atoms ofa “relation algebra” [Tarski]

I 〈A,+, ·,−, 0, 1, ◦,̆ , 1′〉 or sometimes simply 〈A, ◦,̆ , 1′〉I 〈A,+, ·,−, 0, 1〉 is a Boolean algebra. Elements of A are any

disjunction of base relations. 0 is the empty relation, and 1is the universal relation (no information).

I An additional binary operator, the composition law ◦:R1 ◦ R2 = {〈x, y〉 : ∃z R1(x, z) and R2(z, y)}, s.t.:

I ◦ is associative and distributive over +I 1′ is an identity (neutral element) for ◦I An “inverse” operator˘that is idempotentI ˘distributes over +I (x ◦ y)̆ = y̆ ◦ x̆I (x̆ ◦ −(x ◦ y)) ≤ −y

20qMay 3rd, 2005

Composition tables

I The behavior of the ◦ operator can be characterized by a“composition table”.

I If a Boolean algebra, with distinguished element 1′ andwith two operators ◦ and˘distributive over + and s.t.

I 0̆ = 0 and (0 ◦ x) = (x ◦ 0) = 0I for any base relations (atoms) x, y, z,

I x̆ is an atom too,I x ◦ (y ◦ z) ≤ (x ◦ y) ≤ zI x ◦ 1′ = xI x≤ y ◦ z implies x̆≤ z̆◦ y̆ and y≤ x ◦ z̆

then it is a relation algebra

21qMay 3rd, 2005

Composition Tables - 2

I Point algebraI Points are for instants in time, but actually, valid for any

dense linear strict order

◦ < = >

< < < {<,=, >}= < = >

> {<,=, >} > >

I Similarly, the set of 13 Allen’s relations form a relationalgebra, the interval algebra, characterized by itscomposition table (cf. hand out).

22qMay 3rd, 2005

Calculability

I Complexity of transitive closure (constraint propagation) inthe point algebra is polynomial

I In the interval algebra, it is NP-hardI The solution: finding sub-algebras of acceptable

complexity.

23qMay 3rd, 2005

Conceptual neighborhoods

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24qMay 3rd, 2005

Convex relations

I Commonsense relevanceI Out of 213 relations in the interval algebra, only 82 are

convex.I Convex relations are closed under intersection and

compositionI Consistency check in the sub-algebra of convex relations is

polynomial

25qMay 3rd, 2005

Events

26qMay 3rd, 2005

The need for events

I Psychological evidence: infants discriminate and countevents

I Causal reasoning and planning are based on events, esp.actions

I Linguistic evidence: event names, event anaphora...

27qMay 3rd, 2005

Davidson’s case for the existence of events

I The problemI Jones buttered the toast

Butter(Jones, toast)I Jones buttered the toast with the knife

Butter(Jones, toast, knife)I Jones buttered the toast with the knife in the bathroom

Butter(Jones, toast, knife, bathroom)I . . .

I How many arguments?I Many different predicates require many different meaning

postulates:I Butter2(x, y, z) → Butter1(x, y)I Butter3(x, y, z, t) → Butter2(x, y, z) . . .

28qMay 3rd, 2005






I How many arguments?

I Many different predicates require many different meaningpostulates:

I Butter2(x, y, z) → Butter1(x, y)I Butter3(x, y, z, t) → Butter2(x, y, z) . . .

29qMay 3rd, 2005






I How many arguments?I Many different predicates require many different meaning

postulates:I Butter2(x, y, z) → Butter1(x, y)I Butter3(x, y, z, t) → Butter2(x, y, z) . . .

30qMay 3rd, 2005


I The solutionI Reify events, and introduce them as implicit arguments of

verbs

I Distinguish necessary arguments from optional parameters(1) ∃e Butter(e, Jones, toast)(2) ∃e (Butter(e, Jones, toast) ∧With(e, knife))

I (2) logically entails (1)I Other advantages

I Event anaphora: It happened at midnightevent nominalization: The buttering was slow

I Quantification: In every burning, oxygen is consumed andAnn burned the wood, therefore Oxygen was consumed.

I Predication over events: I enjoyed reading the book, I sawyou enter, I heard the explosion

I Eliminativists exist

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verbsI Distinguish necessary arguments from optional parameters

(1) ∃e Butter(e, Jones, toast)(2) ∃e (Butter(e, Jones, toast) ∧With(e, knife))






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I (2) logically entails (1)

I Other advantagesI Event anaphora: It happened at midnight

event nominalization: The buttering was slowI Quantification: In every burning, oxygen is consumed and

Ann burned the wood, therefore Oxygen was consumed.I Predication over events: I enjoyed reading the book, I saw

you enter, I heard the explosion


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35qMay 3rd, 2005










36qMay 3rd, 2005

What events are not

I Events vs. objectsI Endurant / perdurant discussion

I Four-dimensionalism: stages of objectsI Objects occupy space, events can be co-located:

the ball / the piece of metal vs.the spinning of the ball / the warming up of the ball

I Objects can move, events cannotI What relationship? Dependence, participation

37qMay 3rd, 2005

What events are not

I Events vs. objectsI Endurant / perdurant discussionI Four-dimensionalism: stages of objects

I Objects occupy space, events can be co-located:the ball / the piece of metal vs.the spinning of the ball / the warming up of the ball


38qMay 3rd, 2005

What events are not

I Events vs. objectsI Endurant / perdurant discussionI Four-dimensionalism: stages of objectsI Objects occupy space, events can be co-located:



39qMay 3rd, 2005

What events are not



I Objects can move, events cannot

I What relationship? Dependence, participation

40qMay 3rd, 2005

What events are not




41qMay 3rd, 2005

What events are not

I Events vs. facts, propositions and states of affairs

I Caesar’s death / that Caesar died, my standing here / that Iam standing here

I Events are concrete (= situated in space-time), facts andsoa are abstract

I Events occur once, propositions and soas can repeatedlybe the case / obtain

I Caesar’s death = Caesar’s violent death, that Caesar died6= that Caesar died violently

42qMay 3rd, 2005

What events are not

I Events vs. facts, propositions and states of affairsI Caesar’s death / that Caesar died, my standing here / that I

am standing here

I Events are concrete (= situated in space-time), facts andsoa are abstract



43qMay 3rd, 2005

What events are not


am standing hereI Events are concrete (= situated in space-time), facts and

soa are abstract



44qMay 3rd, 2005

What events are not



soa are abstractI Events occur once, propositions and soas can repeatedly

be the case / obtain


45qMay 3rd, 2005

What events are not



soa are abstractI Events occur once, propositions and soas can repeatedly

be the case / obtainI Caesar’s death = Caesar’s violent death, that Caesar died6= that Caesar died violently

46qMay 3rd, 2005

How many events?

I The spinning of the ballThe warming up of the ball

I John’s answering my questionJohn’s shouting

I Brutus’s stabbing CaesarBrutus’s killing CaesarCaesar’s death

I My alerting the burglarMy illuminating the roomMy turning on the lightMy pushing on the buttonMy moving my finger . . .

47qMay 3rd, 2005

Event identity

I “No entity without identity”I Identity criteria

I Co-localization, but four-dimensionalismI Causal equivalence, but temporal shiftsI Logical equivalence, but slingshot argumentI Many different properties: exemplification of properties at a

time

I A general semantic problem? (cf. definite descriptions)I “Linguistic ontology” and multiplicationism

48qMay 3rd, 2005

Event categoriesI Linguistic tests:

I possibility of the progressiveI am running / ?? I am knowing linguistics, ?? I amrecognizing you

I accepts modification with “for an hour”I ate for an hour / ?? I ate an apple for an hour

I accepts modification with “in an hour”I ate an apple in an hour / ?? I ate in an hour / ?? I ateapples in an hour

I accepts modification with “at noon”I recognized you at noon / ?? I knew linguistics at noon

I accepts modification with “since noon”I know linguistics since noon / ?? I recognize you sincenoon

I Telicity: existence of a culmination pointI run, eat, know, loveI eat an apple, climb a mountain, recognize you, reach the

top

49qMay 3rd, 2005

Event categoriesI Linguistic tests:

I possibility of the progressiveI am running / ?? I am knowing linguistics, ?? I amrecognizing you

I accepts modification with “for an hour”I ate for an hour / ?? I ate an apple for an hour

I accepts modification with “in an hour”I ate an apple in an hour / ?? I ate in an hour / ?? I ateapples in an hour

I accepts modification with “at noon”I recognized you at noon / ?? I knew linguistics at noon

I accepts modification with “since noon”I know linguistics since noon / ?? I recognize you sincenoon

I Telicity: existence of a culmination pointI run, eat, know, loveI eat an apple, climb a mountain, recognize you, reach the

top

50qMay 3rd, 2005

Vendler’s categoriesI Activity: + progressive, + “for an hour” (atelic)

I runI sitI eatI eat apples

I Accomplishment: + progressive, - “for an hour” (telic)I eat an appleI climb a mountainI draw a circle

I Achievement: - progressive, + “at noon” (telic)I recognize someoneI reach the top of a mountainI find something

I State: - progressive, - “at noon” (atelic)I know something / know someone / know that...I be sickI love

51qMay 3rd, 2005

Ontological distinctions

I Linguistic tests are unstable I was reaching the top, when...Difficult to eliminate aspect

I Need to characterize the categories in ontological terms[Dowty]

I cf amounts of matter / objectsI Homogeneity: Statives (states and some activities, sitting)

every part of an event of type V is itself an event of type VI Homogeneity up to a certain limit: Activities (most activities)

every part of an event of type V down to a certain limit insize is itself an event of type V

I Non-homogeneity: “Events” (accomplishments andachievements)no part of an event of type V is itself an event of type V

I Temporally extended: accomplishmentsPunctual: achievements

52qMay 3rd, 2005

Ontological distinctions

I Linguistic tests are unstable I was reaching the top, when...Difficult to eliminate aspect

I Need to characterize the categories in ontological terms[Dowty]

I cf amounts of matter / objectsI Homogeneity: Statives (states and some activities, sitting)

every part of an event of type V is itself an event of type VI Homogeneity up to a certain limit: Activities (most activities)

every part of an event of type V down to a certain limit insize is itself an event of type V

I Non-homogeneity: “Events” (accomplishments andachievements)no part of an event of type V is itself an event of type V

I Temporally extended: accomplishmentsPunctual: achievements

53qMay 3rd, 2005

Event categories in DOLCE

I Use both homogeneity and cumulativity to avoid the “grain”problem

I Use only mereological notions to avoid reference totemporal locations

I States: cumulative and homogeneousI Processes: cumulative and not homogeneousI Achievements: not cumulative and atomicI Accomplishments: not cumulative and not atomic

54qMay 3rd, 2005

Participation and roles

I Relation between endurants and perdurants: participationI Loves(e, x, y) → Participates-in(x, e) ∧ Participates-in(y, e)I Participation is a kind of existential dependence of the

perdurant on the endurantI Specialization of participation: roles

I Loves(e, x, y) → Agent-of(x, e) ∧ Patient-of(y, e)

55qMay 3rd, 2005

Algebra of eventsI Plural events and collective events

I John and Mary went to Rome → John went to Rome andMary went to Rome: Distributive, plural event made up oftwo events

I John and Mary lifted the piano 6→ John lifted the piano andMary lifted the piano: Collective, one event with a pluralendurant as argument

I Three boys invited four girls is ambiguousI Distributive reading: 3 events, 3 groups of 4 girls (and

between 4 to 12 girls in total)I Cumulative reading: between 4 and 12 events

there are three boys that invited a girl and four girls thatwere invited by a boy

I Collective reading: 1 eventthe three boys as a group invited the four girls as a group

56qMay 3rd, 2005

Algebra of eventsI Plural events and collective events

I John and Mary went to Rome → John went to Rome andMary went to Rome: Distributive, plural event made up oftwo events

I John and Mary lifted the piano 6→ John lifted the piano andMary lifted the piano: Collective, one event with a pluralendurant as argument

I Three boys invited four girls is ambiguousI Distributive reading: 3 events, 3 groups of 4 girls (and

between 4 to 12 girls in total)I Cumulative reading: between 4 and 12 events

there are three boys that invited a girl and four girls thatwere invited by a boy

I Collective reading: 1 eventthe three boys as a group invited the four girls as a group

57qMay 3rd, 2005

Algebra of events

I Algebra of plural events (distributive and cumulativecases): similar to the treatment of plural endurants, twomereologies or semi-lattices, with constitution relations

I The “homogeneous” substrate is sometimes considered as“process”

I Many other part-of relations between eventsI Mary puts a brick on another / Mary builds a houseI The actor plays the part of Hamlet / The troupe plays

HamletI the first half of the soccer match / the soccer match

58qMay 3rd, 2005

Algebra of events

I Algebra of plural events (distributive and cumulativecases): similar to the treatment of plural endurants, twomereologies or semi-lattices, with constitution relations

I The “homogeneous” substrate is sometimes considered as“process”

I Many other part-of relations between eventsI Mary puts a brick on another / Mary builds a houseI The actor plays the part of Hamlet / The troupe plays

HamletI the first half of the soccer match / the soccer match

59qMay 3rd, 2005

Temporal relations between events [Kamp]

I Precedence ≺: a strict partial orderI Overlap o: a reflexive and symmetric relationI Mixed axioms

I ∀x, y (x≺ y→ ¬xoy)I ∀x, y, z, t ((x≺ y∧ yoz∧ z≺ t) → x≺ t)I ∀x, y (x≺ y∨ xoy∨ y≺ x)

I Construction of an instant structure from an event structure(Russell-Wiener)

I 〈E,≺, o〉: event structureI instants are maximal sets of two by two overlapping events

I ⊆ 2E s.t. for any i ∈ I ∀x, y ∈ i xoy and∀x ∈ E (x 6∈ i → ∃y (y ∈ i ∧ ¬xoy))

I for all i, j ∈ I , i ≤ j iff ∃x, y (x ∈ i ∧ y ∈ j ∧ x≺ y)I 〈I ,≤: event structure

60qMay 3rd, 2005

Temporal relations between events [Kamp]

I An interval structure can be built on top of the instantstructure

I Not an isomorphism between the original event structureand this interval structure

I No sum and no product existence within events: moreintervals than events

I Atomic events generate instants: degenerate intervals areneeded

I Hypothesis that only “events” (accomplishments andachievements) contribute to time structure

I every state and activity is started and ended by a (changeof state) event

Ontology Of Time

Education

Transcript of Ontology Of Time