Prior Probabilities of Allen Interval Relations over Finite Orders

Tim Fernando and Carl Vogel

ADAPT Centre, Computer Science Department, Trinity College Dublin, Ireland

Keywords:

Allen Interval Relations, Probabilities, Events.

Abstract:

The probability that intervals are related by a particular Allen relation is calculated relative to sample spaces

Ω

given by the number n of, in one case, points, and, in another, interval names. In both cases, worlds in

the sample space are assumed equiprobable, and Allen relations are classiﬁed as short, medium and long,

according to the number of shared borders.

1 INTRODUCTION

A useful basis for relating intervals are the 13 rela-

tions described in (Allen, 1983) and widely applied

to temporal relations in text and beyond (Liu et al.,

2018; Verhagen et al., 2009; Allen and Ferguson,

1994; Kamp and Reyle, 1993, among many others).

The present work proceeds from the following ques-

tion.

(Q) Given an Allen relation R, what is the probability

that R relates intervals a and a

, aRa

Let us understand (Q) as saying nothing about a and

, not even that they are distinct (equality being an

Allen relation). As there are 13 Allen relations,

a plausible answer to (Q), under the principle of in-

difference (commonly ascribed to Laplace). But are

Allen relations a matter of indifference when, for ex-

ample, some Allen relations occur more often than

others in the transitivity table of (Allen, 1983)? That

table is a central tool in interval networks formed

from nodes representing intervals, and arcs labelled

by Allen relations that may hold between the inter-

vals. We will return to the transitivity table below.

For now, sufﬁce it to observe that some care is in or-

der when proposing a sample space of equiprobable

outcomes (hereafter, worlds) against which to answer

(Q).

It is natural to interpret (Q) as presupposing a lin-

ear order relative to which a and a

are intervals. To

accommodate all Allen relations, let us assume there

are at least 4 points in that linear order, and for sim-

plicity, let us suppose it is ﬁnite — say, the usual order

on the set

[n] := {i ∈ Z | 1 ≤ i ≤ n}

of integers between 1 and n (inclusive). A pair (l,r)

from the linear order

:= {(l,r) ∈ [n] × [n] | l < r}

on [n] deﬁnes the <

-interval

(l,r] := {i ∈ [n] | l < i ≤ r}

(with left border l and right border r, allowing = with

r but not l). Now, over the linear order <

, the proba-

bility that aRa

becomes the probability that

(l,r] R (l

]

for (l, r) and (l

) drawn from <

. Note that 1 is

excluded from (l,r] for all l,r ∈ [n]. To lift this re-

striction, it sufﬁces to work with copies in <

n+1

given

by mapping i ∈ [n] to i+1 ∈ [n + 1]. Similarly, the re-

quirement that a <

-interval be strictly bounded to the

right can be imposed by passing to <

n−1

with i > 1

mapped to i − 1. Without loss of generality, we iden-

tify <

-intervals with (l,r] for l <

Over a sample space Ω

given by a linear or-

der on n points, probabilities for each Allen relation

R are calculated in section 2, under the assumption

that worlds in Ω

are equiprobable. The probabili-

ties queried by (Q) vary with n and depend on the

extent to which the intervals share borders, given R.

As n approaches inﬁnity, 7 of the 13 Allen relations

have vanishing probabilities, leaving each of the other

6 probability

But should Allen probabilities be assessed around

the number n of points in the linear order? The guid-

ing perspective behind (Allen, 1983) (and many other

As for why an interval should be half open and half

closed, some motivation from Leibniz’s law is presented in

section 4 below.

952

Fernando, T. and Vogel, C.

Prior Probabilities of Allen Interval Relations over Finite Orders.

DOI: 10.5220/0007699609520961

In Proceedings of the 11th International Conference on Agents and Artiﬁcial Intelligence (ICAART 2019), pages 952-961

ISBN: 978-989-758-350-6

works such as (Hamblin, 1971)) is that intervals, not

points, are basic, suggesting that n pertain to inter-

vals, not points. We take up this suggestion in section

3, working with interval names (also known as events

under, for example, the Russell-Wiener construction

of temporal instants described in (Kamp and Reyle,

1993, page 667)). Calculating probabilities becomes

more complex without, as far as we can tell, straying

from the asymptotic behavior determined in section

2: at the limit n → ∞, 7 of the 13 Allen relations have

probability 0, while 6 have

each.

So what? The main thrust of this work is not so

much to calculate numbers but to uncover structure

lurking behind Allen relations. Concrete examples of

structure in natural language semantics are described

in the passage below from (Kamp, 2013, page 11)

when we interpret a piece of discourse — or

a single sentence in the context in which it

is being used — we build something like a

model of the episode or situation described;

and an important part of that model are its

event structure, and the time structure that can

be derived from that event structure by means

of Russell’s construction.

The event structure Kamp has in mind is “made up

by those comparatively few events that ﬁgure in this

discourse” (page 9). The aforementioned Russell con-

struction turns the ﬁnitely many events mentioned in a

(ﬁnite) discourse into a ﬁnite linear order of temporal

instants (each instant being a certain set of events).

This contrasts sharply with the continuum R with

which “real” time is commonly identiﬁed (Kamp and

Reyle, 1993, for example) or, for that matter, any un-

bounded linear order for the time periods of (Allen

and Ferguson, 1994). Indeed, if an event is equipped

with its past and future — or, in the terminology of

(Freksa, 1992), an interval is represented by its semi-

intervals — then the resulting time structure amounts

to ordering the left and right borders l and r of events

(Fernando, 2016, page 3635). The case of two events

yields the Allen relations, which can be formulated

naturally in terms of strings (Durand and Schwer,

2008). That formulation is recounted in Table 1 in

section 2 below.

The appeal to left and right borders runs counter

to the use of the transitivity table in (Allen, 1983),

where borders are buried out of sight. That said, both

sections 2 and 3 end with links to the transitivity table.

A more serious issue is the assumption of equiprob-

able worlds, which we reconsider in section 4, after

the nature of the sample spaces becomes clearer. That

space is formed in section 3 out of strings that go well

beyond pictures of Allen relations between two inter-

vals. Throughout this paper, however, our focus is on

answering the question (Q) against a ﬁnite temporal

structure (given by a ﬁnite discourse).

2 PROBABILITIES OVER n

ORDERED POINTS

Let AR be the set of 13 names

b, bi, d, di, o, oi, m, mi, s, si, f, ﬁ, e

of Allen relations. For each R ∈ AR , Table 1 pictures

(l,r] R (l

] as a string s

of boxes arranged from left

to right so that all borders in the same box are equal

and are < borders in boxes to the right (Durand and

Schwer, 2008).

Table 1: Allen relations in strings, following Figure 4 of

(Durand and Schwer, 2008).

(l,r] R (l

] s

−1

(l,r] b (l

] l r l

bi l

l r

(l,r] d (l

] l

l r r

di l l

(l,r] o (l

] l l

r r

oi l

l r

(l,r] m (l

] l r,l

mi l

,l r

(l,r] s (l

] l,l

r r

si l,l

(l,r] f (l

] l

l r,r

ﬁ l l

r, r

(l,r] e (l

] l,l

r, r

For example, l r l

depicts the ordering

l < r < l

< r

characteristic of (l,r] b (l

]

while l, l

r, r

depicts the ordering

l = l

< r = r

characteristic of (l,r] e (l

Each R ∈ AR can be classiﬁed as either long

{R ∈ A R | length(s

) = 4} = {b,d,o,bi,di,oi}

or medium

{R ∈ AR | length(s

) = 3} = {m,s,f,mi,si,ﬁ}

or short

{R ∈ AR | length(s

) = 2} = {e}

according to the length of s

, which also happens

to be the cardinality of the set {l,l

,r, r

} when

(l,r] R (l

]. The probabilities assigned in this paper

to each R ∈ AR will turn out to depend on whether R

is long, medium or short.

More precisely, given an integer n ≥ 4, let us agree

an n-world is a function

f : {x,y,x

} → [n]

Prior Probabilities of Allen Interval Relations over Finite Orders

953

assigning four distinct variables x,y, x

integers in

[n] such that

f (x) < f (y) and f (x

) < f (y

For each R ∈ AR , we say an n-world f satisﬁes R if

( f (x), f (y)] R ( f (x

), f (y

)].

Now comes a key observation.

Lemma 1. Given an integer n ≥ 4,

(i) the number of n-worlds satisfying e (equal) is





n(n −1)

(ii) for each medium R ∈ AR , the number of n-worlds

satisfying R is









n −2

(iii) for each long R ∈ A R , the number of n-worlds

satisfying R is









n −3

Proof. Let im be the map from an n-world f to its

image

im( f ) = { f (x), f (y), f (x

), f (y

)} ⊆ [n].

For each R ∈ AR , let im

be the restriction of im to

n-worlds satisfying R. It sufﬁces to observe that im

is a bijection to subsets of [n] of cardinality







4 if R is long

3 if R is medium

2 if R is e.



Let Ω

be the set of n-worlds, and for each R ∈

AR , let p

(R) be the fraction of Ω

satisfying R

(R) =

cardinality({ f ∈ Ω

| f satisﬁes R})

cardinality(Ω

)

Representing the medium relations by meet, m, and

long relations by before, b, we have from Lemma 1,

(m)

(e)

n −2

and

(b)

(m)

n −3

which with

1 =

∑

R∈AR

(R) = p

(e) +6p

(m) +6p

(b)

allows us to solve for p

(e). A simpler alternative sug-

gested by a referee is to use

cardinality(Ω

) =









(as Ω

consists of all choices of pairs l,r and l

from [n]). Either way, we obtain

Theorem 2. For n ≥ 4 and R,R

∈ A R ,

(R) = p

) if length(s

) = length(s

)

where the short relation e (equal) has probability

(e) =

n(n −1)

while medium relations have probabilities

(m) =

2(n −2)

3n(n −1)

and long relations have probabilities

(b) =

(n −3)(n −2)

6n(n −1)

Corollary 3. For R ∈ AR ,

lim

n→∞

(R) =



0 if R is short or medium

otherwise.

To put Corollary 3 in context, the probabilities at the

start are strikingly different, with e the most probable

at n = 4, m catching up at n = 5, and b at n = 6 (and

the most probable from n ≥ 8).

Table 2: Some probabilities from Theorem 2.

n p

(e) p

(m) p

(b)

4 1/6 1/9 1/36

5 1/10 1/10 1/20

6 1/15 4/45 1/15

8 1/28 1/14 5/56

Recall from the Introduction that 1 should be added to

or subtracted from n to lift or impose bounds. At any

rate, there is an arbitrariness in any choice of n that

calls out for attention. Letting n approach +∞ (as in

Corollary 3) is an admittedly crude way to attend to

this. A more sophisticated approach would build on a

probability distribution on the lengths n — a direction

not pursued below.

What is pursued is the short-medium-long classi-

ﬁcation of Allen relations, which we pause now to

note is implicit in the transitivity table at the center

of (Allen, 1983). That table maps a pair (R

) of

Allen relations to the set t(R

) of Allen relations

R such that there are intervals i, j and k for which

j and jR

k and iRk.

Let us deﬁne the t-number of an Allen relation R to

be the sum

#(R) :=

∑

∈AR

cardinality(t(R,R

))

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954

of the numbers of entries in the row for R, including

the Allen relation of equality, e, omitted from the tran-

sitivity table in (Allen, 1983), which we incorporate

into t as expected

t(R,e) = t(e,R) = {R} for each R ∈ AR .

Proposition 4. For R ∈ AR ,

#(R) =







41 if R is long

25 if R is medium

13 if R is e.

Proposition 4 characterizes short, medium and long

Allen relations in terms of a notion #(R) that does not

explicitly mention interval borders. The same sum

#(R) arises down the column of the transitivity table

#(R) =

∑

∈AR

cardinality(t(R

,R))

and is the cardinality of the set

{(R

) ∈ AR × AR | R

∈ t(R,R

)}.

In the next section, t-numbers #(R) are built into the

probabilites assigned to Allen relations R when three

or more intervals are considered.

3 PROBABILITIES OVER n

INTERVAL NAMES

The sample space Ω

in section 2 ﬁxes the number

n of linearly ordered points. An alternative is to let

n ≥ 2 be the number of intervals under consideration,

construing each element i of [n] not as a point but as an

interval. Following (Allen, 1983), we might redeﬁne

an n-world to be a function

ω : ([n] × [n]) → AR

that labels every pair (i, j) from [n]×[n] with an Allen

relation ω(i, j) ∈ AR in a consistent manner.

Con-

sistency of ω here can be understood as the existence

of functions

α : [n] → [2n]

and

β : [n] → [2n]

such that for all i ∈ [n],

α(i) < β(i) (1)

Subsets of AR assigned to edges between intervals in

(Allen, 1983) are reduced to singletons to keep worlds dis-

joint, and avoid double counting when basing probabilities

on world counts.

and for all j ∈ [n],

ω(i, j) is the Allen relation R such that

(α(i),β(i)] R (α( j),β( j)]. (2)

Together, (1) and (2) turn i, j ∈ [n] into <

-intervals

(α(i),β(i)] and (α( j),β( j)] that satisfy the speciﬁca-

tion encoded by ω. The functions α and β above need

not be unique, as [2n] may offer plenty of room to

satisfy (1) and (2). An extreme example is where all

intervals in [n] are equal

ω(i, j) = e for all i, j ∈ [n] (3)

in which case there are





pairs

α,β : [n] → [2n]

that work. At the other extreme, exactly one such pair

satisﬁes ω if each interval i < n is before i + 1

ω(i,i +1) = b for i ∈ [n − 1]. (4)

These two extreme examples make clear that n is the

number of interval names, as opposed to intervals. In

the former case, (3), there is just one interval; in the

latter, (4), there are un-named intervals between those

named in [n]. Should we not insist that n count in-

tervals and not just some names? But what, in the

ﬁnite case, are intervals other than pairs of endpoints?

Counting these pairs would lead us back to section 2,

with





many intervals from k points (give or take 1,

for bounds explained in the Introduction). Moreover,

it bears noting that interval names are events, which

are important ingredients in not only philosophical

reconstructions of time but also natural language se-

mantics (Kamp and Reyle, 1993; Kamp, 2013).

For a handle on consistent labellings ω : [n] ×

[n] → AR , we turn to strings of sets. Recall from

Table 1, the strings s

for Allen relations R, such as

the string

= l r,l

of length 3, the middle symbol of which is the set with

r and l

as its elements. It will be crucial below not to

conﬂate the notions l,l

,r, r

even when, as with r and

in the middle box of s

, they name the same point.

Reconstrual of l, l

,r, r

in Table 1. The letters l, l

and r

appearing in the strings s

in Table 1 are un-

interpreted terms (e.g., variables), each distinct from

the other (whether or not they co-occur in a box of a

string).

We draw boxes instead of curly braces {·} so as not

to confuse string symbols with sets such as

| R ∈ AR }

Prior Probabilities of Allen Interval Relations over Finite Orders

955

which we can form from l r and l

through a

certain ternary relation & on strings s of sets

&( l r , l

, s) ⇐⇒ s ∈ {s

| R ∈ AR }. (5)

(5) is a consequence of deﬁning & by induction ac-

cording to

(i0)

&(ε,ε,ε)

(i1)

&(s,s

)

&(sa,s

(a ∪a

))

(i2)

&(s,s

)

&(sa,s

(i3)

&(s,s

)

&(s,s

)

where ε is the empty string, and a,a

are sets, qua

string symbols (Fernando, 2018).

The base case (i0)

puts (ε,ε,ε) into &, which is closed under rules (i1)

for superposition, and (i2), (i3) for shufﬂing. For ex-

ample,

&( l r , l

, s

)

follows from (i0), (i2), (i1) and (i3)

(i0)

(ε,ε,ε)

(i2)

( l , ε, l )

(i1)

( l r , l

, l r,l

)

(i3)

( l r , l

, l r,l

Collecting strings into sets (i.e., languages), we

can express & as a binary operation on languages

L,L

, deﬁning

L&L

:= {s

| (∃s ∈ L)(∃s

∈ L

) &(s, s

)} .

We apply & repeateadly to form languages L

encod-

ing consistent labellings ω : [n] × [n] → AR . Let

:= 1 1

(following the custom of conﬂating a string s with the

singleton language {s}) and

n+1

:= L

& n +1 n +1 for n ≥ 1.

To see how L

encodes consistent labellings, a few

deﬁnitions are in order. Given a set X and a string

s = a

···a

of sets,

(i) the X -reduct ρ

(s) of s is its componentwise in-

tersection with X

···a

) := (a

∩ X)···(a

∩ X)

(ii) the X-projection π

(s) of s is the result of deleting

all occurrences of the empty box  in ρ

(s)

A special case, mix, of the join operation in (Durand

and Schwer, 2008) sufﬁces for an unmarked version of (5).

The calculation of probabilities below is, however, based on

(i0)–(i3).

(Durand and Schwer, 2008). For example,

{2,3}

( 1,2,4 1 2,3 3 4 ) = 2 2,3 3

{2,3}

( 1,2,4 1 2,3 3 4 ) = 2 2,3 3

and for any string s, 3 occurs exactly twice in s if

{3}

(s) = 3 3 . Clearly, L

is the set

{s ∈ (2

[n]

− {})

| (∀i ∈ [n]) π

{i}

(s) = i i }

of strings of non-empty subsets of [n] where each i ∈

[n] occurs exactly twice. Next, for distinct i, j ∈ [n]

and R ∈ AR , we let s

R/i, j

be the string s

(from Table

1) with l,r replaced by i, and l

replaced by j. For

example,

m/2,3

= 2 2,3 3 and s

e/1,2

= 1,2 1,2

and

= {s

R/1,2

| R ∈ AR }.

For i 6= j, we can always invert s

7→ s

R/i, j

because

i and j each occur exactly twice in s

R/i, j

. If s =

···a

∈ L

, and i, j ∈ [n], then

{i, j}

(s) = s

R/i, j

⇐⇒ (l,r] R (l

]

where l,r are positions in s marked by i

l := (least p ∈ [k]) i ∈ a

r := (greatest p ∈ [k]) i ∈ a

and similarly for l

and j

:= (least p ∈ [k]) j ∈ a

:= (greatest p ∈ [k]) j ∈ a

Accordingly, let us agree s satisﬁes iR j if its {i, j}-

projection is s

R/i, j

s |= iR j ⇐⇒ π

{i, j}

(s) = s

R/i, j

Proposition 5. Let n ≥ 2.

(i) For all s ∈ L

and (i, j) ∈ [n]×[n], there is a unique

R ∈ AR such that s |= iR j.

(ii) For all s ∈ L

, let ω

: [n]×[n] → AR be the func-

tion that sends (i, j) to the unique R ∈ AR such

that s |= iR j (given by part (i)). The map s 7→ ω

is a bijection from L

onto the set of consistent

labellings from [n] × [n] to AR .

Proposition 5 follows by induction on n. Henceforth,

we adopt L

as our ofﬁcial sample space, equating the

probability of R (for each R ∈ AR ) with the propor-

tion of L

in which interval 1 is R-related to interval

(R) :=

cardinality(L

(R))

cardinality(L

)

(6)

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956

where L

(R) is the subset

(R) := {s ∈ L

| s |= 1R2}

of L

satisfying R. The languages L

(R) vary with

R ∈ AR , but have a common part (in a sense to be

made precise presently), the language L

3:n

, deﬁned as

follows

3:2

:= ε

3:n+1

:= L

3:n

& n +1 n +1 for n ≥ 2.

Note that ε is the identity of the binary operation &,

which is associative and commutative.

Proposition 6. For n ≥ 2, and R ∈ AR ,

(R) = s

R/1,2

& L

3:n

Behind Proposition 6 is a relationship between & and

that can be explained with a couple more deﬁni-

tions. An X-component of a string s of sets is a string

of subsets of X such that

&(s

,s) for some string s

subsets disjoint from X.

We say s is an S-word (Durand and Schwer, 2008) if

 does not occur as a symbol in s — i.e.,

s = π

voc(s)

(s)

where the vocabulary voc(s) of s is the least set X

such that s ∈ (2

)

∗

voc(a

···a

) =

[

i=1

Lemma 7. For all strings s of sets, and disjoint sets

X and Y ,

&(π

(s),π

X∪Y

(s)) (when X ∩Y =

and if s is an S-word, then π

(s) is the unique S-word

that is an X-component of s.

X-components of S-words need not be S-words (e.g.,

1 is a {1}-component of 1 2 ) but they are

unique after deleting .

Proposition 8. Let n ≥ 2 and s be a string of length

k > 1 with n 6∈ voc(s). The set

s& n n

consists of strings of length k, k + 1, and k + 2, of

which there are exactly

(k) :=

k(k − 1)

strings of length k,

(k) := k(k + 1) strings of length k + 1, and

(k) :=

(k + 1)(k + 2)

strings of length k + 2.

A string in s& n n of length k chooses 2 positions

from s in which to put n, whence

(k) =





while length k + 1 chooses a position from s and one

of k + 1 positions not in s

(k) = k(k +1)

and length k + 2 chooses 2 positions outside s, which

may be different or the same

(k) =



k + 1



+ k +1 =

(k + 1)(k + 2)

Returning now to the probabilities deﬁned by line (6)

above, let c

(R) be the number

(R) := cardinality(L

(R))

of strings in L

satisfying R. It is instructive to ob-

serve that c

(R) is just the t-number #(R) deﬁned at

the end of section 2 as the sum of the transitivity table

row for R

(R) =

∑

∈AR

cardinality(t(R,R

)).

For all n ≥ 2, we can calculate the quantities c

(R) in

terms of

(R;k) := cardinality({s ∈ L

(R) | length(s) = k})

for which we have the recurrence

(R;k) =



1 if length(s

) = k

0 otherwise

(7)

n+1

(R;k) = c

(R;k)d

(k) + c

(R;k − 1)d

(k − 1)

+ c

(R;k − 2)d

(k − 2)

= d

(k)(c

(R;k) + 2c

(R;k − 1)

+ c

(R;k − 2)) (8)

from Proposition 8, with Lemma 7 ruling out the pos-

sibility that (8) double counts. Propositions 6 and 8

reduce the variation in p

(R) to the length of s

(R) = c

) if length(s

) = length(s

)

for all R,R

∈ A R and n ≥ 2. For the record,

Prior Probabilities of Allen Interval Relations over Finite Orders

957

Table 3: Some probabilities of e, m, b.

n p

(e) p

(m) p

(b) γ

1 −6p

(b)

1 1

≈ 0.538461538

3 0.031784841 0.061124694 0.100244499 2 2 0.398533007

10 0.002527761 0.021841026 0.144404347 9 7 0.133573915

100 0.000023782 0.002283051 0.164379652 96 72 0.013722086

500 0.000000959 0.000460405 0.166206102 480 361 0.002763387

1000 0.000000240 0.000230840 0.166435786 961 721 0.001385281

1500 0.000000107 0.000153893 0.166512755 1442 1082 0.000923468

Theorem 9. For n ≥ 2 and R ∈ AR , the probabili-

ties p

(R) = c

(R)/c

can be calculated as follows

(e) =

2n−2

∑

k=2

(e;k) (9)

(R) =

2n−1

∑

k=3

(R;k) for medium R

(R) =

∑

k=4

(R;k) for long R

where c

(R;k) is given by lines (7) and (8) above, and

= c

(e) +6(c

(m) +c

(b)) (10)

(representing medium relations by meet, m, and long

relations by before, b).

The summation index k in Theorem 9 ranges over

the possible lengths of strings in L

(R), according to

whether R is short, medium or long. One can map the

language L

to L

n+1

(e) by a bijection that renames

interval i to i + 1 and inserts 1e2, establishing

= c

n+1

(e).

Hence, as an alternative to (9), we can specify c

(e)

by the recurrence

(e) = 1 (= c

(m) = c

(b))

n+1

(e) = c

(e) +6(c

(m) +c

(b)) for n ≥ 2.

It is (9) and c

(e;k), however, that appear in Sloane’s

On-line Encyclopedia of Integer Sequences for the

“number of different relations between n intervals on

a line”

a(n) =

∑

i=2

λ(i,n) where λ(i,n) = c

(e;i)

(according to (7), (8) above)

in https://oeis.org/A055203.

It is conjectured there that a(n) = 1 mod 12, which is

equivalent to the claim that c

(m) + c

(b) is even, by (10)

in Theorem 9.

Some values of p

(R) are listed in Table 3, along-

side integers γ

and γ

that compare p

(m) to p

(e)



(m)

(e)





(m)

(e)



and p

(b) to p

(m),



(b)

(m)





(b)

(m)



respectively. The inequalities

(m)

(e)

n+1

(m)

n+1

(e)

and

(b)

(m)

n+1

(b)

n+1

(m)

have been veriﬁed computationally for 2 ≤ n ≤ 1500,

providing evidence but not a proof that the asymp-

totic probabilities described in Corollary 3 carry over

to L

. The case n = 2 reproduces our ﬁrst answer to

the question (Q) in the Introduction above

(R) =

while the transitivity table numbers #(R) are the basis

for n = 3

(R) =

#(R)

∑

∈AR

#(R

)

which varies according to whether R is short, medium

or long.

4 DISCUSSION

The study of probabilities above has led us to partition

Allen relations between the short, medium and long,

which is far less common than that between overlap

 =

{d,di,o,oi,s,si,f,ﬁ,e},

precedence

≺ =

{m,b},

NLPinAI 2019 - Special Session on Natural Language Processing in Artiﬁcial Intelligence

958

and its converse

 =

{mi,bi}

(Kamp and Reyle, 1993; Durand and Schwer, 2008,

among others). Using section 2, the asymptotic prob-

abilities

p() = lim

n→∞

(d) + p

(di) + p

(o) + p

(oi)+

(s) + p

(si) + p

(f) + p

(ﬁ) + p

(e)

p(≺) = lim

n→∞

(m) + p

(b) =

p() = lim

n→∞

(mi) + p

(bi) =

do not differ vastly from the numbers

9/13, 2/13, 2/13

obtained by replacing the probabilities p

(R) of an

Allen relation R uniformly with 1/13, the probability

(R) where section 3 starts (at n = 2). While varia-

tions in n are of limited consequence for , ≺ and ,

it is a another matter once , ≺ and  are reﬁned to

Allen relations. But why invite such complications?

An important reason to be interested in n is granu-

larity, which takes on particular signiﬁcance when it is

varied. One way to see this is through Leibniz’s law,

indiscernibility as identity. The requirement that any

difference x 6= y is discernible via some property P can

be expressed in monadic second-order logic (Libkin,

2010, for example) as

x 6= y ⊃ (∃P)¬(P(x) ≡ P(y)). (LL)

If we replace 6= by adjacency S and restrict P to be

given by some ﬁnite set X, (LL) becomes “time steps

require change

”

xSy ⊃ x 6≡

y (LL

S,X

)

where x 6≡

y means: x and y differ over some predi-

cate from X

x 6≡

y :=

i∈X

¬(P

(x) ≡ P

(y)).

For each i ∈ X , let us mark P

’s left and right borders

with subscripts l(i) and r(i) for predicates P

l(i)

saying:

is false but S-after true

l(i)

(x) ≡ ¬P

(x) ∧(∃y)(xSy ∧ P

(y)) (11)

and P

r(i)

saying: P

is true but not S-after

r(i)

(x) ≡ P

(x) ∧¬(∃y)(xSy ∧ P

(y)). (12)

Formulating x 6≡

y as

i∈X

((¬P

(x) ∧P

(y)) ∨(P

(x) ∧¬P

(y))

brings us, under xSy, to

i∈X

l(i)

(x) ∨P

r(i)

(x)))

xSy ⊃ (x 6≡

y ≡

i∈X

l(i)

(x) ∨P

r(i)

(x)))

assuming (11), (12) and S is deterministic

(∀z)(xSy ∧ xSz ⊃ y = z). (13)

That is, under (11)–(13), (LL

S,X

) says:

(∃y)(xSy) ⊃

i∈X

l(i)

(x) ∨P

r(i)

(x)). (14)

To enforce (14), we let X

•

be the set

•

:= {l(i) | i ∈ X} ∪ {r(i) | i ∈ X}

of borders in X, and deﬁne a translation

β : (2

)

∗

→ (2

•

)

∗

with for example,

β( i, i

) = l(i),l(i

) r(i) r(i

)

mapping, in general, a string a

···a

of subsets of X

to the string b

···b

of subsets of X

•

according to (11)

and (12)

:= {l(i) | i ∈ a

x+1

− a

} ∪

{r(i) | i ∈ a

− a

x+1

} for x < k (15)

:= {r(i) | i ∈ a

}

(Fernando, 2018). While (13) is built into every

string, (14) is not. For a non-ﬁnal position x, (15)

says

6=  ⇐⇒ (a

x+1

− a

) ∪(a

− a

x+1

) 6= 

⇐⇒ a

x+1

6= a

That is, for b

···b

= β(a

···a

···b

k−1

is an S-word ⇐⇒ a

···a

has no stutter

where a stutter of a

···a

is a non-ﬁnal position x ∈

[k − 1] such that

= a

x+1

An S-word β(s) satisﬁes (14) and a bit more

(∀x)

i∈X

l(i)

(x) ∨P

r(i)

(x))

without the precondition

(∃y)(xSy)

that x is not S-ﬁnal.

For each Allen relation R, we can picture 1R2 not

only as the S-word s

R/1,2

from Table 1, but also as a

stutterless string s

◦

in Table 4 (Fernando, 2016, page

3635), stepping outside S-words for

◦

= 1 2

and

◦

= 2 1 .

Prior Probabilities of Allen Interval Relations over Finite Orders

959

Table 4: Allen relations via stutterless strings.

R s

◦

−1

◦

−1

b 1 2 bi 2 1

o 1 1, 2 2 oi 2 1,2 1

m 1 2 mi 2 1

d 2 1, 2 2 di 1 1,2 1

s 1, 2 2 si 1,2 1

f 2 1,2 ﬁ 1 1,2

e 1,2

From Table 4, Table 1 is a small step away

≈ β(s

◦

) for l ≈ l(1), r ≈ r(1),

≈ l(2), r

≈ r(2).

For example, R = m gives

β( 1 2 ) = l(1) r(1),l(2) r(2)

≈ l r,l

Stutterless strings arise from de-stuttering

saas

sas

(16)

just as S-words arise from -removal

ss

. (17)

(17) implements the Aristotelian slogan

no time without change

under the assumption that

(†) all predicates in a string symbol a express change.

By contrast, (16) reﬂects the assumption that strings

are built from cumulative predicates, where by deﬁni-

tion, a predicate P on intervals is cumulative if when-

ever an interval i meets an interval i

for the combined

interval i t i

P(i) and P(i

) =⇒ P(i t i

The converse

P(i ti

) =⇒ P(i) and P(i

)

(for i meets i

) is what it means for P to be divisive. P

is cumulative and divisive precisely if it satisﬁes the

condition (H) for homogeneity

(H) for all intervals i and i

whose union i ∪ i

is an

interval,

P(i ∪i

) ⇐⇒ P(i) and P(i

A bias towards stutterless strings (as opposed to S-

words) is in line with the well-known aspect hypoth-

esis from (Dowty, 1979) claiming

the different aspectual properties of the vari-

ous kinds of verbs can be explained by pos-

tulating a single homogeneous class of predi-

cates — stative predicates — plus three or four

sentential operators or connectives. (page 71)

That said, it is no accident that non-stative borders are

strung together in Table 1 for use in both sections 2

and 3, whereas their stative interiors are relegated (for

present purposes) to Table 4. Our analysis of Allen

relations above focuses not on the static condition of

interiors (described by (H)), but on the change marked

by borders (in accordance with (†)).

There are reasons to shift the aforementioned fo-

cus towards a more even balance in future work.

Statives and non-statives are boxed together in dis-

course representation structures (Kamp and Reyle,

1993), which can be put one after another in strings

to describe regularities (such as the preconditions and

effects of actions) beyond chance. Chance is as-

sessed above relative to sample spaces Ω

consist-

ing of worlds linked to model-theoretic interpreta-

tions of discourse representation structures. These

model-theoretic interpretations can be recast in ordi-

nary predicate logic, on which probabilities can be

deﬁned. An equation assigning probabilities p(x) to

worlds x that has received considerable attention in

recent years is

p(x) =

exp(

∑

i∈I

(x)) (18)

(Domingos and Lowd, 2009) given some ﬁnite set I

of ﬁrst-order formulas i and weights w

∈ R that shape

the probability of x according to the number n

(x) of

groundings in x that satisfy i. (18) is applied to in-

terval networks for event recognition in (Morariu and

Davis, 2011), one of a number of works with data-

driven assignments of probabilities to Allen relations

(Zhang et al., 2013; Liu et al., 2018, among others).

The contribution of I to (18) is neutralized if every

weight w

is 0 (or equivalently, I =

0), resulting in

equiprobable worlds (with Z in (18) equal to the num-

ber of such worlds). It is this null, data-free case on

which we focus when raising in the Introduction the

question (Q) of the probability of aRa

, for arbitrary

intervals a, a

. Our answers, Theorem 2 in section 2

and Theorem 9 in section 3, are based on ﬁnite sam-

ple spaces Ω

of temporal entities that divide Allen

relations into the short, medium and long. No previ-

ous attention has, as far as we know, been paid to this

division. Does the division fade into insigniﬁcance

once an account of actions is introduced through a

non-empty set I of formulas and non-zero weights in

(18)? That would depend on I, which we have put

aside in answering (Q).

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960

5 CONCLUSION

The probability an Allen relation holds between two

arbitrary intervals is speciﬁed in Theorems 2 and 9

under the assumption that intervals are drawn from

a ﬁnite model by a fair method (in accordance with

the principle of indifference). The ﬁnite model as-

sumed depends on the particular application at hand.

(For example, the passage above from (Kamp, 2013)

describes a range of applications where that model

is based on the events mentioned in a discourse.)

Whether or not the notion of a fair coin can or should

extend to the choice of intervals from any such model

is a natural question that, in our view, merits study.

ACKNOWLEDGEMENTS

We are grateful to three anonymous referees for their

helpful comments.

This research is supported by Science Foun-

dation Ireland (SFI) through the CNGL Pro-

gramme (Grant 12/CE/I2267) in the ADAPT Cen-

tre (https://www.adaptcentre.ie) at Trinity College

Dublin. The ADAPT Centre for Digital Content

Technology is funded under the SFI Research Centres

Programme (Grant 13/RC/2106) and is co-funded un-

der the European Regional Development Fund.

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