Partitions representing change homogeneously - University of
Partitions representing change homogeneously Tim Fernando Trinity College Dublin, Ireland [email protected]
1
Introduction
As leading figures in Amsterdam’s formal semantics scene, Jeroen, Martin and Frank have inspired many semanticists over the years to follow them or in some way respond to their ideas. I am grateful to be somewhere in that non-exclusive disjunction, and wish them the best on their upcoming retirements. The remainder of this note describes some variations on themes familiar from their work, set in the temporal domain. The basic point is to represent change in an interval by partitioning it into subintervals, assuming change is manifested through temporal propositions that are interpreted over intervals. Fix a set of temporal propositions, hereafter called fluents. A -timeline hT, , vi consists of (i) a non-empty set T (of “instants” or “moments”) (ii) a linear order
on T with the set Ivl ( ) = {I ✓ T | (8t, t0 2 I){t00 2 T | t
t00
t0 } ✓ I}
of intervals, and (iii) a relation v ✓ ⇥Ivl ( ) consisting of pairs (', I) of fluents ' 2 that “v-satisfy” '.
and intervals I 2 Ivl ( )
The equivalence ⇡v' on the set Ivl ( ) of intervals I, I 0 given by I ⇡v' I 0 () (v(', I) () v(', I 0 )) induces the partition on Ivl ( ) that we might, following [GS84], interpret as the question ?'. There is a coarseness about ⇡v' , however, in equating two intervals I1 and I2 , neither of which v-satisfies ', even if no subinterval of I1 v-satisfies ' whereas some subinterval of I2 does. In this case, I1 is (', v)-homogeneous, while I2 (which buries ') is not. More precisely, an interval is (', v)-homogeneous if it is ⇡v' -equivalent to each of its subintervals I is (', v)-homogeneous () (8J v I) I ⇡v' J where the subinterval relation v is ✓ restricted to Ivl ( ). Writing I tI 0 for the smallest interval containing I [ I 0 , let us refine ⇡v' to the relation =v' given by I =v' I 0 () I t I 0 is (', v)-homogeneous. Clearly, =v' is contained in ⇡v' , symmetric, and reflexive on (', v)-homogeneous intervals. To conclude that =v' is an equivalence relation on (', v)-homogeneous intervals, it remains to establish transitivity I =v' I 0 and I 0 =v' I 00 =) I =v' I 00 for which it is useful to assume
Partitions representing change
(A1)
T. Fernando
v(', I [ I 0 ) () v(', I) and v(', I 0 )
whenever I, I 0 , I [ I 0 2 Ivl ( ).
One half of (A1), =), ensures ' has the so-called subinterval property v(', I) =) v(', J)
whenever J v I
commonly assumed of fluents ' representing statives since [BP72], as is the second half of (A1) v(', I) and v(', I 0 ) =) v(', I [ I 0 )
whenever I [ I 0 2 Ivl ( ).
Note that if ' satisfies (A1), then so does ¬' with v(¬', I) () (8J v I) not v(', J) for all I 2 Ivl ( ).1 For (', v)-homogeneous I, v(¬', I) reduces to not v(', I). Returning to =v' , we have Proposition 1. Assuming (A1), =v' is an equivalence relation on (', v)-homogeneous intervals. In section 2, we extend =v' conservatively from (', v)-homogeneous intervals to all of Ivl ( ), including those left out by =v' , introducing a second assumption that bounds variation. Section 3 shows how to step from a single fluent ' satisfying (A1) to a set X of such, with an interval defined to be (X, v)-homogeneous if it is (', v)-homogeneous, for every ' 2 X. Finally, section 4 restricts X to be finite, passing over to strings. We assume throughout that (A1) holds for every fluent '.
2
Partitions based on homogeneous intervals
For any set J ✓ Ivl ( ) of intervals, let fJ : Ivl ( ) ! 2J be the function mapping an interval to the set of intervals in J that intersects with it fJ (I) = {J 2 J | J \ I 6= ;} . Equating intervals I and I 0 that intersect the same intervals in J, we get an equivalence relation ⇡J on Ivl ( ) from the clause I ⇡J I 0 () fJ (I) = fJ (I 0 ). In practice, we choose J to be a partition of T . But to relate ⇡J to ⇡v' , we need to constrain J further. Proposition 2. If J ✓ Ivl ( ) is a partition of T , then ⇡J refines ⇡v' () each J 2 J is (', v)-homogeneous. Proof: For =), observe that if J 2 J were not (', v)-homogeneous, then J would have two subintervals that are in ⇡J but not in ⇡v' . To prove the converse, note that for every interval I, there are at most two intervals J 2 fJ (I) such that not J ✓ I (as a third interval would puncture a hole through I). That is, [ fJ (I) = I [ J1 [ J2 for some J1 , J2 2 fJ (I). 1 In
[AF94], ¬ is called strong negation (page 540), while in [Ham71], it is predicate negation (page 131).
Partitions representing change
T. Fernando
S Thus, if v(', I), then v(', fJ (I)), as ' satisfies (A1) and each J 2 J is (', v)-homogeneous. And again, v(', I 0 ) for any I 0 ⇡J I. 2
Clearly, the bigger the (', v)-homogeneous intervals in J, the coarser ⇡J is. The remainder of this section considers the special case where J is finite and consists of the largest (', v)homogeneous intervals. We lift to intervals I, I 0 universally for whole precedence I
I 0 () (8t 2 I)(8t0 2 I 0 ) t
t0 ,
and define a (', v)-alternation to be a finite sequence J1 J2 . . . Jn of intervals Ji 2 Ivl ( ) such that whenever 1 i < n, Ji Ji+1 and v(', Ji+i ) () not v(', Ji ). Sn A (', v)-scale is a (', v)-alternation J1 . . . Jn such that i=1 Ji = T and each Ji is (', v)homogeneous. An obvious property of (', v)-scales is uniqueness. Proposition 3. For all ' and v (satisfying (A1)), there is at most one (', v)-scale.
The next question is existence. Let us agree to call a (', v)-alternation J1 . . . Jn long if its length n is greater than or equal to the length of every (', v)-alternation. Clearly, a (', v)-scale is long. Conversely, we can turn any long (', v)-alternation J = J1 . . . Jn into a (', v)-scale s(J) = J10 . . . Jn0 as follows. For n = 1, J10 = T . Otherwise, n > 1 and J10 = {t 2 T | {t}
Jn0 = {t 2 T | Jn
J2 and {t} ⇡v' J1 } 1
{t} and {t} ⇡v' Jn }
and for 1 < i < n, Ji0 = {t 2 T | Ji
1
{t}
Ji+1 and {t} ⇡v' Ji }.
Before leaping to the conclusion that s(J) is a (', v)-scale, we pause for an instructive example. Let T be the set N = {0, 1, 2, . . .} of non-negative integers (under the usual ordering) and 'ˆ pick out intervals I 2 Ivl (N) that are bounded v(', ˆ I) () (9n 2 N)(8t 2 I) t < n . A long (', ˆ v)-alternation has length 2, but its last subinterval cannot be (', ˆ v)-homogeneous. To sidestep pesky fluents such as ', ˆ let us strengthen (A1) and define ' to be v-pointwise if for every interval I, v(', I) () (8t 2 I) v(', {t}). Proposition 4. If ' is v-pointwise, and a (', v)-alternation J is long, then s(J) is a (', v)-scale. The existence of a long (', v)-alternation (or equivalently, of a (', v)-scale) can be put as (A2)
there is an integer k such that every (', v)-alternation has length < k.
In e↵ect, (A2) says that as an instrument for observing T (or better, ), ' has a limited shelf life and resolution. To overcome these limitations, we might work with not just one fluent ' but many. Of course, many fluents ' can be useful, whether or not they satisfy (A2). That said, we might get around a fluent violating (A2) by breaking it down into infinitely many fluents, each satisfying (A2).
Partitions representing change
3
T. Fernando
From one fluent to many
Recall that the (dependent) product of a family {Ax }x2X of sets Ax indexed by X is the set Y [ Ax = {f : X ! Ax | (8x 2 X) f (x) 2 Ax } x2X
x2X
of functions f with domain X such that f (x) 2 Ax for every x 2 X. Proposition 5. Let X be a set of fluents ' satisfying (A1) and {J(')} T'2X be a family of partitions J(') of T into (', v)-homogeneous intervals. The intersection '2X ⇡J(') is ⇡J(X) , where J(X) is a partition of T into (X, v)-homogeneous intervals given by \ Y J(X) = { f (') | f 2 J(')} {;}. '2X
'2X
From finite partitions J(') (e.g. (', v)-scales, assuming (A2)), Proposition 5 may produce an infinite partition J(X), provided X is infinite. But even if X is finite, the construction holds some interest.
4
From partitions to strings
When X is finite, and each J(') in Proposition 5 is finite, J(X) reduces to a finite set {J1 , . . . , Jn } of (X, v)-homogeneous intervals J1 · · · Jn , with I ⇡{J1 ...Jn } I 0 () (8i 2 [1, n]) (Ji \ I 6= ; () Ji \ I 0 6= ;). (where [j, k] is the set of integers vˆ(', [j, k]) () v(',
j and k). The X-timeline h[1, n],
1
Introduction
As leading figures in Amsterdam’s formal semantics scene, Jeroen, Martin and Frank have inspired many semanticists over the years to follow them or in some way respond to their ideas. I am grateful to be somewhere in that non-exclusive disjunction, and wish them the best on their upcoming retirements. The remainder of this note describes some variations on themes familiar from their work, set in the temporal domain. The basic point is to represent change in an interval by partitioning it into subintervals, assuming change is manifested through temporal propositions that are interpreted over intervals. Fix a set of temporal propositions, hereafter called fluents. A -timeline hT, , vi consists of (i) a non-empty set T (of “instants” or “moments”) (ii) a linear order
on T with the set Ivl ( ) = {I ✓ T | (8t, t0 2 I){t00 2 T | t
t00
t0 } ✓ I}
of intervals, and (iii) a relation v ✓ ⇥Ivl ( ) consisting of pairs (', I) of fluents ' 2 that “v-satisfy” '.
and intervals I 2 Ivl ( )
The equivalence ⇡v' on the set Ivl ( ) of intervals I, I 0 given by I ⇡v' I 0 () (v(', I) () v(', I 0 )) induces the partition on Ivl ( ) that we might, following [GS84], interpret as the question ?'. There is a coarseness about ⇡v' , however, in equating two intervals I1 and I2 , neither of which v-satisfies ', even if no subinterval of I1 v-satisfies ' whereas some subinterval of I2 does. In this case, I1 is (', v)-homogeneous, while I2 (which buries ') is not. More precisely, an interval is (', v)-homogeneous if it is ⇡v' -equivalent to each of its subintervals I is (', v)-homogeneous () (8J v I) I ⇡v' J where the subinterval relation v is ✓ restricted to Ivl ( ). Writing I tI 0 for the smallest interval containing I [ I 0 , let us refine ⇡v' to the relation =v' given by I =v' I 0 () I t I 0 is (', v)-homogeneous. Clearly, =v' is contained in ⇡v' , symmetric, and reflexive on (', v)-homogeneous intervals. To conclude that =v' is an equivalence relation on (', v)-homogeneous intervals, it remains to establish transitivity I =v' I 0 and I 0 =v' I 00 =) I =v' I 00 for which it is useful to assume
Partitions representing change
(A1)
T. Fernando
v(', I [ I 0 ) () v(', I) and v(', I 0 )
whenever I, I 0 , I [ I 0 2 Ivl ( ).
One half of (A1), =), ensures ' has the so-called subinterval property v(', I) =) v(', J)
whenever J v I
commonly assumed of fluents ' representing statives since [BP72], as is the second half of (A1) v(', I) and v(', I 0 ) =) v(', I [ I 0 )
whenever I [ I 0 2 Ivl ( ).
Note that if ' satisfies (A1), then so does ¬' with v(¬', I) () (8J v I) not v(', J) for all I 2 Ivl ( ).1 For (', v)-homogeneous I, v(¬', I) reduces to not v(', I). Returning to =v' , we have Proposition 1. Assuming (A1), =v' is an equivalence relation on (', v)-homogeneous intervals. In section 2, we extend =v' conservatively from (', v)-homogeneous intervals to all of Ivl ( ), including those left out by =v' , introducing a second assumption that bounds variation. Section 3 shows how to step from a single fluent ' satisfying (A1) to a set X of such, with an interval defined to be (X, v)-homogeneous if it is (', v)-homogeneous, for every ' 2 X. Finally, section 4 restricts X to be finite, passing over to strings. We assume throughout that (A1) holds for every fluent '.
2
Partitions based on homogeneous intervals
For any set J ✓ Ivl ( ) of intervals, let fJ : Ivl ( ) ! 2J be the function mapping an interval to the set of intervals in J that intersects with it fJ (I) = {J 2 J | J \ I 6= ;} . Equating intervals I and I 0 that intersect the same intervals in J, we get an equivalence relation ⇡J on Ivl ( ) from the clause I ⇡J I 0 () fJ (I) = fJ (I 0 ). In practice, we choose J to be a partition of T . But to relate ⇡J to ⇡v' , we need to constrain J further. Proposition 2. If J ✓ Ivl ( ) is a partition of T , then ⇡J refines ⇡v' () each J 2 J is (', v)-homogeneous. Proof: For =), observe that if J 2 J were not (', v)-homogeneous, then J would have two subintervals that are in ⇡J but not in ⇡v' . To prove the converse, note that for every interval I, there are at most two intervals J 2 fJ (I) such that not J ✓ I (as a third interval would puncture a hole through I). That is, [ fJ (I) = I [ J1 [ J2 for some J1 , J2 2 fJ (I). 1 In
[AF94], ¬ is called strong negation (page 540), while in [Ham71], it is predicate negation (page 131).
Partitions representing change
T. Fernando
S Thus, if v(', I), then v(', fJ (I)), as ' satisfies (A1) and each J 2 J is (', v)-homogeneous. And again, v(', I 0 ) for any I 0 ⇡J I. 2
Clearly, the bigger the (', v)-homogeneous intervals in J, the coarser ⇡J is. The remainder of this section considers the special case where J is finite and consists of the largest (', v)homogeneous intervals. We lift to intervals I, I 0 universally for whole precedence I
I 0 () (8t 2 I)(8t0 2 I 0 ) t
t0 ,
and define a (', v)-alternation to be a finite sequence J1 J2 . . . Jn of intervals Ji 2 Ivl ( ) such that whenever 1 i < n, Ji Ji+1 and v(', Ji+i ) () not v(', Ji ). Sn A (', v)-scale is a (', v)-alternation J1 . . . Jn such that i=1 Ji = T and each Ji is (', v)homogeneous. An obvious property of (', v)-scales is uniqueness. Proposition 3. For all ' and v (satisfying (A1)), there is at most one (', v)-scale.
The next question is existence. Let us agree to call a (', v)-alternation J1 . . . Jn long if its length n is greater than or equal to the length of every (', v)-alternation. Clearly, a (', v)-scale is long. Conversely, we can turn any long (', v)-alternation J = J1 . . . Jn into a (', v)-scale s(J) = J10 . . . Jn0 as follows. For n = 1, J10 = T . Otherwise, n > 1 and J10 = {t 2 T | {t}
Jn0 = {t 2 T | Jn
J2 and {t} ⇡v' J1 } 1
{t} and {t} ⇡v' Jn }
and for 1 < i < n, Ji0 = {t 2 T | Ji
1
{t}
Ji+1 and {t} ⇡v' Ji }.
Before leaping to the conclusion that s(J) is a (', v)-scale, we pause for an instructive example. Let T be the set N = {0, 1, 2, . . .} of non-negative integers (under the usual ordering) and 'ˆ pick out intervals I 2 Ivl (N) that are bounded v(', ˆ I) () (9n 2 N)(8t 2 I) t < n . A long (', ˆ v)-alternation has length 2, but its last subinterval cannot be (', ˆ v)-homogeneous. To sidestep pesky fluents such as ', ˆ let us strengthen (A1) and define ' to be v-pointwise if for every interval I, v(', I) () (8t 2 I) v(', {t}). Proposition 4. If ' is v-pointwise, and a (', v)-alternation J is long, then s(J) is a (', v)-scale. The existence of a long (', v)-alternation (or equivalently, of a (', v)-scale) can be put as (A2)
there is an integer k such that every (', v)-alternation has length < k.
In e↵ect, (A2) says that as an instrument for observing T (or better, ), ' has a limited shelf life and resolution. To overcome these limitations, we might work with not just one fluent ' but many. Of course, many fluents ' can be useful, whether or not they satisfy (A2). That said, we might get around a fluent violating (A2) by breaking it down into infinitely many fluents, each satisfying (A2).
Partitions representing change
3
T. Fernando
From one fluent to many
Recall that the (dependent) product of a family {Ax }x2X of sets Ax indexed by X is the set Y [ Ax = {f : X ! Ax | (8x 2 X) f (x) 2 Ax } x2X
x2X
of functions f with domain X such that f (x) 2 Ax for every x 2 X. Proposition 5. Let X be a set of fluents ' satisfying (A1) and {J(')} T'2X be a family of partitions J(') of T into (', v)-homogeneous intervals. The intersection '2X ⇡J(') is ⇡J(X) , where J(X) is a partition of T into (X, v)-homogeneous intervals given by \ Y J(X) = { f (') | f 2 J(')} {;}. '2X
'2X
From finite partitions J(') (e.g. (', v)-scales, assuming (A2)), Proposition 5 may produce an infinite partition J(X), provided X is infinite. But even if X is finite, the construction holds some interest.
4
From partitions to strings
When X is finite, and each J(') in Proposition 5 is finite, J(X) reduces to a finite set {J1 , . . . , Jn } of (X, v)-homogeneous intervals J1 · · · Jn , with I ⇡{J1 ...Jn } I 0 () (8i 2 [1, n]) (Ji \ I 6= ; () Ji \ I 0 6= ;). (where [j, k] is the set of integers vˆ(', [j, k]) () v(',
j and k). The X-timeline h[1, n],
