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ACL-OCL / Base_JSON /prefixW /json /W19 /W19-0401.json
Benjamin Aw
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 {
"paper_id": "W19-0401",
"header": {
"generated_with": "S2ORC 1.0.0",
"date_generated": "2023-01-19T06:29:06.086669Z"
},
"title": "Projecting Temporal Properties, Events and Actions",
"authors": [
{
"first": "Tim",
"middle": [],
"last": "Fernando",
"suffix": "",
"affiliation": {
"laboratory": "",
"institution": "Trinity College Dublin",
"location": {}
},
"email": "[email protected]"
}
],
"year": "",
"venue": null,
"identifiers": {},
"abstract": "Temporal notions based on a finite set A of properties are represented in strings, on which projections are defined that vary the granularity A. The structure of properties in A is elaborated to describe statives, events and actions, subject to a distinction in meaning (advocated by Levin and Rappaport Hovav) between what the lexicon prescribes and what a context of use supplies. The projections proposed are deployed as labels for records and record types amenable to finite-state methods.",
"pdf_parse": {
"paper_id": "W19-0401",
"_pdf_hash": "",
"abstract": [
{
"text": "Temporal notions based on a finite set A of properties are represented in strings, on which projections are defined that vary the granularity A. The structure of properties in A is elaborated to describe statives, events and actions, subject to a distinction in meaning (advocated by Levin and Rappaport Hovav) between what the lexicon prescribes and what a context of use supplies. The projections proposed are deployed as labels for records and record types amenable to finite-state methods.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Abstract",
"sec_num": null
}
],
"body_text": [
{
"text": "Reflecting on years of work on discourse semantics, Hans Kamp writes 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 (Kamp, 2013, page 13) .",
"cite_spans": [
{
"start": 408,
"end": 429,
"text": "(Kamp, 2013, page 13)",
"ref_id": null
}
],
"ref_spans": [],
"eq_spans": [],
"section": "Introduction",
"sec_num": "1"
},
{
"text": "The event structure Kamp has in mind is \"made up of those comparatively few events that figure in this discourse\" (page 9). Let us put aside for the moment how to extract from a discourse D the set E D of events that figure in D, and observe that if the set E D is finite (as typically happens in practice), so is the linear order returned by the Russell construction for time (details in section 2 below). This is in sharp contrast to the continuum R, with which \"real\" time is commonly identified (Kamp and Reyle, 1993) , or to any unbounded linear order supporting the temporal interval structure defined in Allen and Ferguson (1994) , where a different perspective on events is adopted.",
"cite_spans": [
{
"start": 499,
"end": 521,
"text": "(Kamp and Reyle, 1993)",
"ref_id": "BIBREF18"
},
{
"start": 611,
"end": 636,
"text": "Allen and Ferguson (1994)",
"ref_id": "BIBREF1"
}
],
"ref_spans": [],
"eq_spans": [],
"section": "Introduction",
"sec_num": "1"
},
{
"text": "We take the position that events are primarily linguistic or cognitive in nature. That is, the world does not really contain events. Rather, events are the way by which agents classify certain useful and relevant patterns of change (Allen and Ferguson, 1994, page 533 ).",
"cite_spans": [
{
"start": 232,
"end": 267,
"text": "(Allen and Ferguson, 1994, page 533",
"ref_id": null
}
],
"ref_spans": [],
"eq_spans": [],
"section": "Introduction",
"sec_num": "1"
},
{
"text": "Allen and Ferguson specify temporal structure before introducing events (or, for that matter, properties and actions), reversing the conceptual priority events enjoy over time in the Russell construction mentioned by Kamp. Without embracing this reversal, the present paper builds on elements of Allen and Ferguson (1994) and other works to construct time from not only events, but also properties and actions. The aim is to find a temporal ontology of finite strings that is not too big (which R or any infinite linear order arguably is) and not too small (which the linear order from Russell's construction can be, depending on the event structure it is fed as input). Insisting on temporal structure that is just right is reminiscent of Goldilocks, and perhaps more germanely, the Goldilocks effect observed in Kidd et al. (2012) as the tendency of human infants to look away from events that are overly simple or overly complex. Whether or not any useful link can be forged between that work and the present paper, I am not able to say. But I do claim that the notions of projections brought out below provide helpful handles on granularity, especially when granularity is varied.",
"cite_spans": [
{
"start": 217,
"end": 222,
"text": "Kamp.",
"ref_id": null
},
{
"start": 296,
"end": 321,
"text": "Allen and Ferguson (1994)",
"ref_id": "BIBREF1"
},
{
"start": 814,
"end": 832,
"text": "Kidd et al. (2012)",
"ref_id": "BIBREF19"
}
],
"ref_spans": [],
"eq_spans": [],
"section": "Introduction",
"sec_num": "1"
},
{
"text": "That granularity is given, in the simplest case, by a finite set A of properties, expressing in section 2 events, as conceived in the Russell construction. More sophisticated pictures of events are considered and \"relevant patterns of change\" captured through an explicit account of action and incremental change in section 3. Strings and languages are presented in section 4 as records and record types labeled with projections, bringing out certain affinities with Type Theory with Records (Cooper and Ginzburg, 2015 ).",
"cite_spans": [
{
"start": 492,
"end": 518,
"text": "(Cooper and Ginzburg, 2015",
"ref_id": "BIBREF3"
}
],
"ref_spans": [],
"eq_spans": [],
"section": "Introduction",
"sec_num": "1"
},
{
"text": "Leibniz's law, decreeing that any difference x = y be discernible via some property, can be expressed in monadic second-order logic (MSO, e.g. Libkin (2010)) as the implication",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Introduction",
"sec_num": "1"
},
{
"text": "x = y \u2283 (\u2203P )\u00ac(P (x) \u2261 P (y)) (LL)",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Introduction",
"sec_num": "1"
},
{
"text": "with \u00ac(P (x) \u2261 P (y)) asserting P separates x from y. A special case of inequality = is the successor relation S that specifies a notion of step. We link that step to a set {P a } a\u2208A of properties P a named with a finite set A (conflating the property P a with its name a \u2208 A when convenient), and adopt the abbreviation x \u2261 A y for the conjunction expressing the inseparabilty in A of x and y",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Introduction",
"sec_num": "1"
},
{
"text": "x \u2261 A y := a\u2208A (P a (x) \u2261 P a (y)).",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Introduction",
"sec_num": "1"
},
{
"text": "Two substitutions in (LL), S for =, and the negation of x \u2261 A y for its consequent, turn (LL) into",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Introduction",
"sec_num": "1"
},
{
"text": "xSy \u2283 x \u2261 A y (LL A )",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Introduction",
"sec_num": "1"
},
{
"text": "(pronounced \"S-steps require change A \"). If we represent x by its A-profile",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Introduction",
"sec_num": "1"
},
{
"text": "A[x] := {a \u2208 A | P a (x)}",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Introduction",
"sec_num": "1"
},
{
"text": "specifying the properties in A that hold of x, we can can study S-chains ",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Introduction",
"sec_num": "1"
},
{
"text": "x 1 Sx 2 and x 2 Sx 3 and \u2022 \u2022 \u2022 and x n\u22121 Sx n through strings A[x 1 ]A[x 2 ] \u2022 \u2022 \u2022 A[x n ] of",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Introduction",
"sec_num": "1"
},
{
"text": "S n := {(i, i + 1) | i \u2208 [n \u2212 1]} +1 on [n]",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Introduction",
"sec_num": "1"
},
{
"text": ", and P a as the set",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Introduction",
"sec_num": "1"
},
{
"text": "[[P a ]] \u03b1 1 \u2022\u2022\u2022\u03b1n := {i \u2208 [n] | a \u2208 \u03b1 i }",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Introduction",
"sec_num": "1"
},
{
"text": "of positions in s where a occurs (for each a \u2208 A). For example, the string a a, a a of length 5 (with string symbols drawn as boxes) corresponds to the model with universe [5] = {1, 2, 3, 4, 5}, interpreting P a as {2, 3} and P a as {3, 4}. (Note is the empty set \u2205 qua string of length 1, not to be confused with the null string of length 0 or the empty language.) The vocabulary of s, voc(s), is the smallest set A such that s is a string of subsets of A",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Introduction",
"sec_num": "1"
},
{
"text": "voc(\u03b1 1 \u2022 \u2022 \u2022 \u03b1 n ) = n i=1 \u03b1 i",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Introduction",
"sec_num": "1"
},
{
"text": "(making, for example, {a, a } the vocabulary of a a, a a ).",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Introduction",
"sec_num": "1"
},
{
"text": "Rather than fixing A once and for all, we let A vary, keeping it finite for bounded granularity (restricting our attention to finite strings of finite sets). If A = \u2205, then x \u2261 A y, which is to say, the strings that satisfy (LL \u2205 ) are exactly those of length 1 (or 0, if we allow a model with empty universe). Evidently, the space of models of (LL A ) increases as we enlarge A. Given a string s of sets that may or not be subsets of A, we define the A-reduct of s to be the string obtained by intersecting s componentwise with A (Fernando, 2016) . For instance, the {a}-reduct of the string a a, a a is \u03c1 {a} ( a a, a a ) = a a .",
"cite_spans": [
{
"start": 531,
"end": 547,
"text": "(Fernando, 2016)",
"ref_id": "BIBREF8"
}
],
"ref_spans": [],
"eq_spans": [],
"section": "Introduction",
"sec_num": "1"
},
{
"text": "\u03c1 A (\u03b1 1 \u2022 \u2022 \u2022 \u03b1 n ) := (\u03b1 1 \u2229 A) \u2022 \u2022 \u2022 (\u03b1 n \u2229 A)",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Introduction",
"sec_num": "1"
},
{
"text": "Whereas a a, a a satisfies (LL {a,a } ), its {a}-reduct satisfies neither (LL {a,a } ) nor (LL {a} ). The problem is that a a stutters. In general, a stutter of a string",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Introduction",
"sec_num": "1"
},
{
"text": "\u03b1 1 \u2022 \u2022 \u2022 \u03b1 n is a position i \u2208 [n \u2212 1] such that \u03b1 i = \u03b1 i+1 . a a",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Introduction",
"sec_num": "1"
},
{
"text": "has two stutters, 2 and 4. It is easy to see that a string s is stutterless iff it satisfies (LL voc(s) ). The consequent x \u2261 A y of (LL A ) is equivalent to the disjunction",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Introduction",
"sec_num": "1"
},
{
"text": "a\u2208A ((\u00acP a (x) \u2227 P a (y)) \u2228 (P a (x) \u2227 \u00acP a (y))",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Introduction",
"sec_num": "1"
},
{
"text": "where each a \u2208 A can separate x from y in one of two ways, corresponding to a's left and right borders, l(a) and r(a), respectively. We introduce predicates P l(a) saying: P a is false but S-after true",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Introduction",
"sec_num": "1"
},
{
"text": "EQUATION",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [
{
"start": 0,
"end": 8,
"text": "EQUATION",
"ref_id": "EQREF",
"raw_str": "P l(a) (x) \u2261 \u00acP a (x) \u2227 (\u2203y)(xSy \u2227 P a (y))",
"eq_num": "(1)"
}
],
"section": "Introduction",
"sec_num": "1"
},
{
"text": "and P r(a) saying: P a is true but not S-after",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Introduction",
"sec_num": "1"
},
{
"text": "EQUATION",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [
{
"start": 0,
"end": 8,
"text": "EQUATION",
"ref_id": "EQREF",
"raw_str": "P r(a) (x) \u2261 P a (x) \u2227 \u00ac(\u2203y)(xSy \u2227 P a (y)).",
"eq_num": "(2)"
}
],
"section": "Introduction",
"sec_num": "1"
},
{
"text": "Then",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Introduction",
"sec_num": "1"
},
{
"text": "x \u2261 A y is equivalent under xSy to a\u2208A (P l(a) (x) \u2228 P r(a) (x))) xSy \u2283 (x \u2261 A y \u2261 a\u2208A (P l(a) (x) \u2228 P r(a) (x))).",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Introduction",
"sec_num": "1"
},
{
"text": "Hence, (LL A ) is equivalent to",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Introduction",
"sec_num": "1"
},
{
"text": "EQUATION",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [
{
"start": 0,
"end": 8,
"text": "EQUATION",
"ref_id": "EQREF",
"raw_str": "(\u2203y)(xSy) \u2283 a\u2208A (P l(a) (x) \u2228 P r(a) (x))",
"eq_num": "(3)"
}
],
"section": "Introduction",
"sec_num": "1"
},
{
"text": "assuming (1), (2), and ",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Introduction",
"sec_num": "1"
},
{
"text": "EQUATION",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [
{
"start": 0,
"end": 8,
"text": "EQUATION",
"ref_id": "EQREF",
"raw_str": "xSy \u2227 xSz \u2283 y = z.",
"eq_num": "(4)"
}
],
"section": "Introduction",
"sec_num": "1"
},
{
"text": "\u03b2 i := {l(a) | a \u2208 \u03b1 i+1 \u2212 \u03b1 i } \u222a {r(a) | a \u2208 \u03b1 i \u2212 \u03b1 i+1 } for i < n (5) \u03b2 n := {r(a) | a \u2208 \u03b1 n } (Fernando, 2018). For example, b( a, a a ) = l(a), l(a ) r(a) r(a ) .",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Introduction",
"sec_num": "1"
},
{
"text": "In general, (5) says that for a non-final position i,",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Introduction",
"sec_num": "1"
},
{
"text": "\u03b2 i = \u21d0\u21d2 (\u03b1 i+1 \u2212 \u03b1 i ) \u222a (\u03b1 i \u2212 \u03b1 i+1 ) = \u21d0\u21d2 \u03b1 i+1 = \u03b1 i .",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Introduction",
"sec_num": "1"
},
{
"text": "That is, s is stutterless iff b(s) is -lite, where by definition, a string",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Introduction",
"sec_num": "1"
},
{
"text": "\u03b1 1 \u2022 \u2022 \u2022 \u03b1 n is -lite if for each i \u2208 [n \u2212 1], \u03b1 i is not .",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Introduction",
"sec_num": "1"
},
{
"text": "For the record, we have Proposition 1. For any sets A and X, and for any string s \u2208 (2 X ) * , the following are equivalent.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Introduction",
"sec_num": "1"
},
{
"text": "(i) Mod(s) satisfies (LL A ) (ii) \u03c1 A (s) is stutterless (iii) Mod(b(\u03c1 A (s))) satisfies (3) (iv) the A \u2022 -reduct of b(s) is -lite.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Introduction",
"sec_num": "1"
},
{
"text": "Implicit in Proposition 1 are two notions of string compression, s\u03b1\u03b1s s\u03b1s (6) for strings over the alphabet 2 A to satisfy (LL A ), and",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Introduction",
"sec_num": "1"
},
{
"text": "EQUATION",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [
{
"start": 0,
"end": 8,
"text": "EQUATION",
"ref_id": "EQREF",
"raw_str": "s s ss for s =",
"eq_num": "(7)"
}
],
"section": "Introduction",
"sec_num": "1"
},
{
"text": "for strings over the alphabet 2 A\u2022 to satisfy the border translation (3) of (LL A ). Destuttering (6) is implemented fully by block compression bc",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Introduction",
"sec_num": "1"
},
{
"text": "bc \u22121 \u03b1 1 \u2022 \u2022 \u2022 \u03b1 n = \u03b1 + 1 \u2022 \u2022 \u2022 \u03b1 + n for stutterless \u03b1 1 \u2022 \u2022 \u2022 \u03b1 n while -removal d implements (7) without the proviso s = d \u22121 \u03b1 1 \u2022 \u2022 \u2022 \u03b1 n = * \u03b1 1 * \u2022 \u2022 \u2022 * \u03b1 n * for -less \u03b1 1 \u2022 \u2022 \u2022 \u03b1 n",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Introduction",
"sec_num": "1"
},
{
"text": "where a -less string is a string of non-empty sets. In Durand and Schwer (2008) , -less strings are called S-words (\"S for Set\"), and the S-projection over A of s defined to be d (\u03c1 A (s)). To make room for bc and link up with Allen and Ferguson (1994) and the Russell construction mentioned in the Introduction, let us agree that, given strings s and s of sets,",
"cite_spans": [
{
"start": 55,
"end": 79,
"text": "Durand and Schwer (2008)",
"ref_id": "BIBREF7"
},
{
"start": 227,
"end": 252,
"text": "Allen and Ferguson (1994)",
"ref_id": "BIBREF1"
}
],
"ref_spans": [],
"eq_spans": [],
"section": "Introduction",
"sec_num": "1"
},
{
"text": "(i) s bc-projects to s if the voc(s )-reduct of s without stutters is s bc(\u03c1 voc(s ) (s)) = s (ii) s -projects to s if the voc(s )-reduct of s without any is s d (\u03c1 voc(s ) (s)) = s (iii) an s-period is an a \u2208 voc(s) such that s bc-projects to a .",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Introduction",
"sec_num": "1"
},
{
"text": "The occurrences of to the left and right in a represent the left and right bounds on a period in Allen and Ferguson (1994) . As with intervals, periods a and a can be related by exactly one element of the set",
"cite_spans": [
{
"start": 97,
"end": 122,
"text": "Allen and Ferguson (1994)",
"ref_id": "BIBREF1"
}
],
"ref_spans": [],
"eq_spans": [],
"section": "Introduction",
"sec_num": "1"
},
{
"text": "AR := {b, bi, o, oi, m, mi, d, di, s, si, f, fi, e}",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Introduction",
"sec_num": "1"
},
{
"text": "of Allen relations (Allen, 1983) . Each R \u2208 AR is pictured as a stutterless string s aRa in Table 1 so that for any string s of sets, and all distinct a, a , a and a are both s-periods \u21d0\u21d2 (\u2203R \u2208 AR) s bc-projects to s aRa . Let us call a string s an A-timeline if s is stutterless and every a \u2208 A is an s-period. For a = a , the {a, a }-timelines are exactly the strings s aRa , for R \u2208 AR. How do these {a, a }-timelines compare to the linear orders obtained by the Russell construction on event structures over {a, a }? Without entering into all the details of the event structure A, \u227a, on which the Russell construction is applied, suffice it to say we can derive s a m a from a \u227a a , s a mi a from a \u227a a, and s a e a from a a , while every other string s aRa is ruled out by the following fact about a linear order < obtained via Russell ( \u2020) the instants related by < are certain subsets of A, no two of which are related by \u2286.",
"cite_spans": [
{
"start": 19,
"end": 32,
"text": "(Allen, 1983)",
"ref_id": "BIBREF0"
}
],
"ref_spans": [
{
"start": 92,
"end": 99,
"text": "Table 1",
"ref_id": "TABREF1"
}
],
"eq_spans": [],
"section": "Introduction",
"sec_num": "1"
},
{
"text": "For example, for A = {a, a }, < cannot describe s a o a = a a, a a since a \u2286 a, a . But can we not get around the antichain condition ( \u2020) by fleshing s a o a out as a, pre(a ) a, a post(a), a and similarly for all other strings s aRa ? In general, the idea would be for any set A and string s of sets, to form the A-closure of s, cl A (s), by setting",
"cite_spans": [],
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"text": "cl A (\u03b1 1 \u2022 \u2022 \u2022 \u03b1 n ) to \u03b2 1 \u2022 \u2022 \u2022 \u03b2 n where \u03b2 i := \u03b1 i \u222a {pre(a) | a \u2208 (A \u2212 \u03b1 i ) \u2229 n k=i+1 \u03b1 k } \u222a {post(a) | a \u2208 (A \u2212 \u03b1 i ) \u2229 i\u22121 k=1 \u03b1 k }",
"cite_spans": [],
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"text": "adding two negations, pre(a) and post(a), for every a \u2208 A (familiar in the A-series of McTaggart (1908) as past and future). The difficulty with cl A (s) is that if a is an s-period, then neither pre(a) nor post(a) can be a cl A (s)-period, as",
"cite_spans": [
{
"start": 87,
"end": 103,
"text": "McTaggart (1908)",
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"text": "bc(\u03c1 {pre(a)} (cl A (s)) = pre(a) and bc(\u03c1 {post(a)} (cl A (s)) = post(a) .",
"cite_spans": [],
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"text": "To cover pre(a) and post(a), infinitely many periods are assumed in Allen and Ferguson (1994) , each bounded to the left and right. An alternative is to drop the bounds on periods, and work with semi-intervals (Freksa, 1992) . Or rather than introducing pre(a) and post(a) through the A-closure cl A (s), we might apply the border translation b(s) for left and right borders l(a) and r(a) that capture moments of change (as opposed to instants, under the Russell construction, of pairwise overlapping events). Table 2 Table 2 with l(a) and r(a) replaced both by a, and l(a ) and r(a ) replaced both by a leads to Figure 4 in (Durand and Schwer, 2008, page 3288) . These replacements simplify, for example, b( s a b a ) to a a a a with the first occurrence of a understood as a's left border, and the second as a's right. Insofar as these simplifications suffice to represent Allen relations in strings, MSO is overkill. The \"relevant patterns of change\" associated with events in Allen and Ferguson (1994) are, however, another matter, or so the next section argues, pointing to action and activity left out of l(a) and r(a).",
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"start": 68,
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"text": "Allen and Ferguson (1994)",
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"start": 210,
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"text": "(Freksa, 1992)",
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"start": 625,
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"text": "(Durand and Schwer, 2008, page 3288)",
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"start": 510,
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"start": 613,
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"text": "R b( s aRa ) R \u22121 b( s aR \u22121 a ) b l(a) r(a) l(a ) r(a ) bi l(a ) r(a ) l(a) r(a) d l(a ) l(a) r(a) r(a ) di l(a) l(a ) r(a ) r(a) o l(a) l(a ) r(a) r(a ) oi l(a ) l(a) r(a ) r(a) m l(a) r(a), l(a ) r(a ) mi l(a ) r(a ), l(a) r(a) s l(a), l(a ) r(a) r(a ) si l(a), l(a ) r(a ) r(a) f l(a ) l(a) r(a), r(a ) fi l(a) l(a ) r(a), r(a ) e l(a), l(a ) r(a), r(a ) e",
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"text": "3 Border action and activity While (B) holds for the strings in Table 2 , it fails for those in Table 1 , the approriate compression in which is destuttering bc, or be cumulative. By definition, a predicate P on intervals is cumulative if whenever an interval i meets (abuts) an interval i for the combined interval i i ,",
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"start": 64,
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"start": 96,
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"text": "P (i) and P (i ) =\u21d2 P (i i ).",
"cite_spans": [],
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"text": "The converse",
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"section": "Introduction",
"sec_num": "1"
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"text": "P (i i ) =\u21d2 P (i) and P (i ) whenever i meets i",
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"sec_num": "1"
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"text": "is the defining condition for divisive predicates P . Cumulativity and divisiveness combine for the condition (H) for homogeneity (H) for all intervals i and i whose union i \u222a i is an interval, Developing Dowty's aspect hypothesis in terms of strings arguably runs counter to assumption (B) above. Many non-statives are given by result verbs that center around some prescribed post-state, as opposed to some manner of change (Levin and Rappaport Hovav, 2013, for example). It is natural to identify that post-state with the consequent state in Moens and Steedman (1988) , where the Aristotle-Ryle-Kenny-Vendler verb classification (Dowty, 1979) is reworked according to Table 3 . Table 3 formulates the culimination resulting in consequent state a as the string pre(a) a , which is associated with the left border l(a) by the border translation b and closure cl A from the previous section. Line (1) in that section implies",
"cite_spans": [
{
"start": 544,
"end": 569,
"text": "Moens and Steedman (1988)",
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{
"start": 631,
"end": 644,
"text": "(Dowty, 1979)",
"ref_id": "BIBREF5"
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"ref_spans": [
{
"start": 670,
"end": 677,
"text": "Table 3",
"ref_id": "TABREF3"
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"start": 680,
"end": 687,
"text": "Table 3",
"ref_id": "TABREF3"
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"section": "Introduction",
"sec_num": "1"
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"text": "P (i \u222a i ) \u21d0\u21d2 P (i) and P (i ).",
"cite_spans": [],
"ref_spans": [],
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"section": "Introduction",
"sec_num": "1"
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{
"text": "EQUATION",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [
{
"start": 0,
"end": 8,
"text": "EQUATION",
"ref_id": "EQREF",
"raw_str": "\u00acP a (x) \u2227 (\u2203y)(xSy \u2227 P a (y)) \u2283 P l(a) (x)",
"eq_num": "(8)"
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],
"section": "Introduction",
"sec_num": "1"
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{
"text": "which can be read as a law of inertia (Dowty, 1986) saying pre(a) persists (forward) unless a force is applied, l(a). If we associate a result verb with a force, it is not surprising that a force f should represent a manner verb lacking a lexically prescribed post-state (Levin and Rappaport Hovav, 2013) , marked \u2212conseq in Table 3 (with f below). The point (semelfactive) string ap(f ) ef(f ) is built from two properties, ap(f ) saying f is applied, and ef(f ) representing a contextually supplied effect of that application. We are borrowing here a basic distinction drawn in Levin and Rappaport Hovav (2013) between the meaning of a verb that is lexically specified (before the verb is used) and the meaning inferred from a specific context of use. When ef(f ) is a, it is tempting to reduce ap(f ) to l(a), except that the lexical/contextual distinction tells us to resist that reduction. Whereas the contextually supplied effect of a manner verb may vary with the use of the verb, the lexically prescribed post-state of a result verb does not. Moreover, while a point (semelfactive) can apply successively (for a process/activity), the implication P l(a) (x) \u2283 \u00acP a (x) (saying l(a) cannot co-exist with a in the same box) blocks successive culminations. How is it possible that ap(f ) and ef(f ) can be boxed together, as in the rightmost column in Table 3 (when pre(a) and a cannot)? An instructive example, given by incremental change tracked by a scale \u227a on a set D of degrees, is a force \u2191 D for a \u227a-increase, with the effect at y",
"cite_spans": [
{
"start": 38,
"end": 51,
"text": "(Dowty, 1986)",
"ref_id": "BIBREF6"
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{
"start": 271,
"end": 304,
"text": "(Levin and Rappaport Hovav, 2013)",
"ref_id": "BIBREF20"
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{
"start": 580,
"end": 612,
"text": "Levin and Rappaport Hovav (2013)",
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{
"start": 325,
"end": 332,
"text": "Table 3",
"ref_id": "TABREF3"
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"start": 1357,
"end": 1364,
"text": "Table 3",
"ref_id": "TABREF3"
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"section": "Introduction",
"sec_num": "1"
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{
"text": "EQUATION",
"cite_spans": [],
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"eq_spans": [
{
"start": 0,
"end": 8,
"text": "EQUATION",
"ref_id": "EQREF",
"raw_str": "P ef(\u2191D) (y) \u2248 (\u2203d \u2208 D) P d (y) \u2227 (\u2203xSy)(\u2203d \u227a d)P d (x) .",
"eq_num": "(9)"
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],
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"sec_num": "1"
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"text": "Unfortunately, the right side of \u2248 in (9) quantifies over d and d , which appear as subscripts in P d (y) and P d (x), not as arguments y and x. Working instead with any finite subset D \u2022 of D (which may well be infinite), we turn (9) into the MSO formula ",
"cite_spans": [],
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"eq_spans": [],
"section": "Introduction",
"sec_num": "1"
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{
"text": "EQUATION",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [
{
"start": 0,
"end": 8,
"text": "EQUATION",
"ref_id": "EQREF",
"raw_str": "P ef(\u2191D) (y) \u2261 d\u2208D\u2022 P d (y) \u2227 (\u2203x)(xSy \u2227 P \u2248d (x))",
"eq_num": "(10)"
}
],
"section": "Introduction",
"sec_num": "1"
},
{
"text": "EQUATION",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [
{
"start": 0,
"end": 8,
"text": "EQUATION",
"ref_id": "EQREF",
"raw_str": "P ap(\u2191D) (x) \u2227 xSy \u2227 \u00acP ap(\u2193D) (x) \u2283 P ef(\u2191D) (y).",
"eq_num": "(11)"
}
],
"section": "Introduction",
"sec_num": "1"
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{
"text": "If we unwind the disjunction characterizing ef(\u2191 D) in (10), (11) gives",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Introduction",
"sec_num": "1"
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{
"text": "EQUATION",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [
{
"start": 0,
"end": 8,
"text": "EQUATION",
"ref_id": "EQREF",
"raw_str": "P \u2248d (x) \u2227 P ap(\u2191D) (x) \u2227 xSy \u2227 \u00acP ap(\u2193D) (x) \u2283 P d (y) (d \u2208 D 0 ).",
"eq_num": "(12)"
}
],
"section": "Introduction",
"sec_num": "1"
},
{
"text": "To allow P d (x) in place of P \u2248d (x) in (12), we modify (10) slightly to",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Introduction",
"sec_num": "1"
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{
"text": "EQUATION",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [
{
"start": 0,
"end": 8,
"text": "EQUATION",
"ref_id": "EQREF",
"raw_str": "P ef(\u2191D) (y) \u2261 d\u2208D\u2022 P d (y) \u2227 (\u2203x)(xSy \u2227 (P d (x) \u2228 P \u2248d (x)))",
"eq_num": "(13)"
}
],
"section": "Introduction",
"sec_num": "1"
},
{
"text": "which means \u2191 D may have the effect not of change but rather preservation (of P d ). Pressure to change P d comes from \u2193 D, for which we have \u2193-counterparts to (11)",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Introduction",
"sec_num": "1"
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{
"text": "EQUATION",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [
{
"start": 0,
"end": 8,
"text": "EQUATION",
"ref_id": "EQREF",
"raw_str": "P ap(\u2193D) (x) \u2227 xSy \u2227 \u00acP ap(\u2191D) (x) \u2283 P ef(\u2193D) (y)",
"eq_num": "(14)"
}
],
"section": "Introduction",
"sec_num": "1"
},
{
"text": "and to (13)",
"cite_spans": [],
"ref_spans": [],
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"section": "Introduction",
"sec_num": "1"
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"text": "P ef(\u2193D) (y) \u2261 d\u2208D\u2022 P \u227ad (y) \u2227 (\u2203x)(xSy \u2227 (P \u227ad (x) \u2228 P \u2248d (x))).",
"cite_spans": [],
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"section": "Introduction",
"sec_num": "1"
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"text": "The implications (11) and (14) reveal shortcomings that the borders l(a) and r(a) have as pictures of transitions associated with events. The account of inertia from the half of line (1) expressed by (8) is unproblematic enough: change requires force. But the other half of (1), the converse of (8), misrepresents how complicated determining the effects of forces can be. Incrementality (the possibility of more than two degrees) opens the door to competition, necessitating the \"no-intervention\" provisos, \u00acP ap(\u2193D) (x) and \u00acP ap(\u2191D) (x), in the antecedents of (11) and of (14). In Allen and Ferguson (1994) , thwarted forces lead to a predicate Try(f, t) that takes an action (or force) term f and time period t, corresponding above to P ap(f ) (t). 1 Whether we refer to f as a force or an action, what are we to make of the property ap(f )? As far as the point (semelfactive) entry ap(f ) ef(f ) in Table 3 is concerned, ap(f ) is clearly non-stativei.e., subject to -removal, as opposed to destuttering d . But turning to a force f given by incremental change, our revision (13) of (10) has the effect beyond (12) of adding (via (11)) the implications",
"cite_spans": [
{
"start": 583,
"end": 608,
"text": "Allen and Ferguson (1994)",
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"start": 752,
"end": 753,
"text": "1",
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{
"start": 903,
"end": 910,
"text": "Table 3",
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"sec_num": "1"
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"text": "P d (x) \u2227 P ap(\u2191D) (x) \u2227 xSy \u2227 \u00acP ap(\u2193D) (x) \u2283 P d (y) (d \u2208 D 0 ).",
"cite_spans": [],
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"section": "Introduction",
"sec_num": "1"
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"text": "Conservative forces that guard against change are left out of l(a), 2 along with incrementality and competition. If \u2191 D can have the effect of not changing P d , what becomes of the assumption (B) above behind box-removal d ? In Moens and Steedman (1988) , the difference between a state and a process (activity)",
"cite_spans": [
{
"start": 229,
"end": 254,
"text": "Moens and Steedman (1988)",
"ref_id": "BIBREF23"
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"eq_spans": [],
"section": "Introduction",
"sec_num": "1"
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{
"text": "EQUATION",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [
{
"start": 0,
"end": 8,
"text": "EQUATION",
"ref_id": "EQREF",
"raw_str": "ap(f ) ap(f ), ef(f ) ef(f )",
"eq_num": "(15)"
}
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"section": "Introduction",
"sec_num": "1"
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{
"text": "is blurred by a progressive state. Arguably, that progressive state pertains to the second box ap(f ), ef(f ) in (15), perhaps with ap(f ) replaced by a stative variant, ap s (f ). Aspectual type shifts are commonly associated with reconstruals, and rather than attempt to resolve the aspectual character of ap(f ) definitively, suffice it to repeat Levin and Rappaport Hovav (2013)'s claim that context is required to spell out the effect ef(f ) of a manner verb f . That wrinkle is a sign of, in Robin Cooper's words, \"semantics in flux,\" challenging a legacy from Montague (1974) the impression of natural languages as being regimented with meanings determined once and for all by an interpretation (Cooper, 2012, page 271) .",
"cite_spans": [
{
"start": 567,
"end": 582,
"text": "Montague (1974)",
"ref_id": "BIBREF24"
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{
"start": 702,
"end": 726,
"text": "(Cooper, 2012, page 271)",
"ref_id": null
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"section": "Introduction",
"sec_num": "1"
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{
"text": "This impression is congenial with Allen and Ferguson (1994) 's avowed position that temporal structure is prior to properties, events and actions -a position open to dispute (harking back to Russell).",
"cite_spans": [
{
"start": 34,
"end": 59,
"text": "Allen and Ferguson (1994)",
"ref_id": "BIBREF1"
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"ref_spans": [],
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"section": "Introduction",
"sec_num": "1"
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"text": "Semantic flux is an important motivation for Type Theory with Records (TTR), against which it is instructive to understand the present paper's",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Projections within records and record types",
"sec_num": "4"
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"text": "Main Claim Temporal notions such as those in Allen and Ferguson (1994) and Moens and Steedman (1988) can be represented in strings structured by MSO and finitary projections, on which we can reason through finite-state methods.",
"cite_spans": [
{
"start": 45,
"end": 70,
"text": "Allen and Ferguson (1994)",
"ref_id": "BIBREF1"
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{
"start": 75,
"end": 100,
"text": "Moens and Steedman (1988)",
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"sec_num": "4"
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"text": "The promise of finite-state methods (mentioned in the Main Claim) rests on (i) a classic theorem due to B\u00fcchi, Elgot and Trakhtenbrot (Libkin, 2010) mapping MSO-sentences to finite automata checking satisfaction (and back), and (ii) the computability by finite-state transducers of the projections proposed. These projections operate between finite sets A and A , composing f \u2208 {bc, d , id} (where id is the identity function) with \u03c1 A for the function f A,A = \u03c1 A ; f : (2 A ) * \u2192 (2 A ) * mapping a string s of subsets of A to the string f (\u03c1 A (s)) of subsets of A that f returns when fed the A-reduct \u03c1 A (s) of s.",
"cite_spans": [
{
"start": 134,
"end": 148,
"text": "(Libkin, 2010)",
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"sec_num": "4"
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"text": "Proposition 2. Given any set \u0398, let Fin(\u0398) be the set of finite subsets of \u0398. For f \u2208 {bc, d , id}, the family {f A,A : (2 A ) * \u2192 (2 A ) * } A,A \u2208F in(\u0398) is a projective system -i.e., f A,A is the identity on (2 A ) * and f A,A is the composition f A ,A ; f A;A for all A \u2286 A \u2286 A \u2208 Fin(\u0398).",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Projections within records and record types",
"sec_num": "4"
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"text": "Recall from section 2 the introduction of strings and the projections \u03c1 A , bc and d through a bounded form of Leibniz's law in MSO (linking stutterless and -less strings according to Proposition 1). MSO properties are restricted to unary predicates over string positions, compelling us in section 3 to sidestep the formula",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
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"sec_num": "4"
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{
"text": "EQUATION",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [
{
"start": 0,
"end": 8,
"text": "EQUATION",
"ref_id": "EQREF",
"raw_str": "(\u2203d \u2208 D) P d (y) \u2227 (\u2203xSy)(\u2203d \u227a d)P d (x)",
"eq_num": "(16)"
}
],
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"sec_num": "4"
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"text": "(in (9)) saying the D-degree at y is greater than at its predecessor. Logical hygiene around P a (x) dictates separating the temporal entities over which the property-argument x ranges from the bits incorporated into the property-index a. Among the latter bits are degrees d in P d and P \u2248d , as well as actions/forces f in P ap(f ) and P ef(f ) . That said, any finite \u227a-chain",
"cite_spans": [],
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"sec_num": "4"
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"text": "d 1 \u227a d 2 \u227a \u2022 \u2022 \u2022 \u227a d n in D",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Projections within records and record types",
"sec_num": "4"
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"text": "yields an approximation of (16) as the finite disjunction",
"cite_spans": [],
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"sec_num": "4"
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{
"text": "EQUATION",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [
{
"start": 0,
"end": 8,
"text": "EQUATION",
"ref_id": "EQREF",
"raw_str": "n i=1 P >i (y) \u2227 (\u2203x)(xSy \u2227 P i (x))",
"eq_num": "(17)"
}
],
"section": "Projections within records and record types",
"sec_num": "4"
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"text": "much as time is sampled in section 2 by a string s, with string positions populating the MSO-model Mod(s). 3 A basic flaw, however, in (17) is that the indices > i and i (appearing as subscripts in P >i and P i ) leave out the attribute that is being graded. That is, the degree d in P d ought properly to be fleshed out as an attribute-value pair ( , v) with a grade or value v that a force \u2191 D can raise (and \u2193 D lower). The letter for attribute can also be understood as a abel in a record",
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"text": "{ i , v i } i\u2208[k] or record-type { i , T i } i\u2208[k]",
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"text": ". In the remainder of this paper, we decompose strings that capture changes in {P a } a\u2208A in terms of records and record types with labels equal to subsets of A, approaching MSO (under the projections above) bottom-up and perhaps even probabilistically. Given a finite set A, and f \u2208 {bc, d , id}, an (A, f )-string is a string s over the alphabet 2 A such that f (s) = s (meaning s is stutterless for f = bc, or s is -less for",
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"text": "f = d ). An (A, f )-record is a record { i , v i } i\u2208[k]",
"cite_spans": [],
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"text": "such that each label i is a subset of A, and each v i is an ( i , f )-string. We can decompose a string s over 2 A into its {a}-reducts for the (A, id)-record { {a}, \u03c1 {a} (s) } a\u2208A , from which we can reconstruct s by componentwise union & \u2022 of strings of the same length",
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"text": "\u03b1 1 \u2022 \u2022 \u2022 \u03b1 n & \u2022 \u03b1 1 \u2022 \u2022 \u2022 \u03b1 n := (\u03b1 1 \u222a \u03b1 1 ) \u2022 \u2022 \u2022 (\u03b1 n \u222a \u03b1 n )",
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"text": "by repeatedly appealing to",
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"text": "EQUATION",
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"raw_str": "\u03c1 A 1 \u222aA 2 (s) = \u03c1 A 1 (s) & \u2022 \u03c1 A 2 (s) .",
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"text": "For f = bc or d , however, (18) will not do, 4 assuming the (A, f )-record",
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"text": "{ i , v i } i\u2208[k] is understood as describing the set L({ i , v i } i\u2208[k] ) of (A, f )-strings that f -project to each v i L({ i , v i } i\u2208[k] ) := {f (s) | s \u2208 (2 A ) * and (\u2200i \u2208 [k]) f (\u03c1 i (s)) = v i }.",
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"text": "Under this assumption, the (A, bc)-record { {a}, a } a\u2208A describes the set of A-timelines (as defined in section 2). To specify an Allen relation R between a and a , we form the label {a, a } and pair it with the string s aRa from Table 1 . But what if say, we know only that the Allen relation between a and a is either meet, m, or before, b? Then we should pair the label = {a, a } with the set { a a , a a } of (bc, {a, a })-strings picturing a m a and a b a . Mildly generalizing the notions above, let us agree (i) an (A, f )-record type is a record type { i , T i } i\u2208 [k] such that each label i is a subset of A, and each T i is a set of ( i , f )-strings (ii) the language described by an (A, f )-record type",
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"start": 575,
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"start": 231,
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"text": "{ i , T i } i\u2208[k] is the set L({ i , T i } i\u2208[k] ) of (A, f )-strings that for each i \u2208 [k], f -project to some string in T i L({ i , T i } i\u2208[k] ) := {f (s) | s \u2208 (2 A ) * and (\u2200i \u2208 [k]) f (\u03c1 i (s)) \u2208 T i }.",
"cite_spans": [],
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"text": "Different (A, f )-record types can describe the same language, as illustrated by the [k + 1]-timelines in",
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"text": "EQUATION",
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{
"start": 0,
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"ref_id": "EQREF",
"raw_str": "L({ {i}, i } i\u2208[k+1] ) = L({ {i, i + 1}, L i } i\u2208[k] )",
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"text": "where k \u2265 1 and L i is the set of 13 strings, s iRi+1 , one per Allen relation R",
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"text": "L i = { s iRi+1 | R \u2208 AR}.",
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"text": "What is gained by complicating the ([k + 1], bc)-record type on the left side of (19) to that to its right? Labels with two intervals (such as i and i + 1) allow us to represent information updates that eliminate strings from L i . Indeed, this is the basis of interval networks which operate around a transitivity table (Allen, 1983, Figure 4 ) that specifies for every pair (R 1 , R 2 ) of Allen relations, the set t(R 1 , R 2 ) of Allen relations R such that under some {1, 2, 3}-timeline, 1R 1 2, 2R 2 3 and 1R3 t(R 1 , R 2 ) = {R \u2208 AR | there is a {1, 2, 3}-timeline that bc-projects to s 1R 1 2 and s 2R 2 3 and s 1R3 }.",
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"start": 321,
"end": 343,
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"text": "For example, t(m,m) = {b} since 1 2 3 is the one string in the language described by { {1, 2}, 1 2 , {2, 3}, 2 3 } whereas t(m,d) = {o,d,s} means exactly three strings belong to the language described by { {1, 2}, 1 2 , {2, 3}, 3 2, 3 3 } (where s a d a = a a, a a ) . The challenge, in general, is, given a set L of (A, f )-strings, to describe L through an (A, f )-record type { i , T i } i\u2208 [k] such that, if possible, ( \u2020) no two labels in the set { i } i\u2208[k] are \u2286-comparable (minimizing redundancy) ( \u2021) each T i is a singleton {v i } (minimizing branching).",
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"start": 394,
"end": 397,
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"start": 239,
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"text": "(where s a d a = a a, a a )",
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"text": "The antichain condition ( \u2020) on labels mirrors one for Russell instants in section 2, and can be satisfied by keeping only the labels that are \u2286-maximal. ( \u2021) can be a more difficult, if not impossible, demand (Woods and Fernando, 2018) . A measure of non-determinism being unavoidable, L may serve as a sample space on which to define a probability mass function (Fernando and Vogel, 2019) . The strings in L are finite, and hold no mysteries. To make this point forcefully, I close on an aspirational note, brazenly quoting the physicist John Archibald Wheeler on it from bit every it -every particle, every field of force, even the space-time continuum itself -derives its function, its meaning, its very existence entirely -even if in some contexts indirectly -from the apparatus-elicited answers to yes-or-no questions, binary choices, bits. It from bit symbolizes the idea that every item of the physical world has at bottom -a very deep bottom, in most instances -an immaterial source and explanation; that which we call reality arises in the last analysis from the posing of yes-no questions and the registering of equipment-evoked responses; in short, that all things physical are information-theoretic in origin and that this is a participatory universe (Wheeler, 1990, page 5) .",
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{
"start": 210,
"end": 236,
"text": "(Woods and Fernando, 2018)",
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{
"start": 364,
"end": 390,
"text": "(Fernando and Vogel, 2019)",
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"start": 1264,
"end": 1287,
"text": "(Wheeler, 1990, page 5)",
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"text": "Here, it is the value/string v i (or type/language T i ), linked by i in records (or record types), and based (at a shallow bottom) on \"yes-no questions\" P a , the responses to which are registered by the apparatus of MSO in S-steps.",
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"text": "Talk of \"forces\" complements inertia, while \"action\" is in the title ofDavidson (1967) and is likened inAllen and Ferguson (1994) to a program (quite natural to apply). Programs in Dynamic Logic(Harel et al., 2000) underly yet another approach to verb semantics(Naumann, 2001;Pustejovsky and Moszkowicz, 2011), relations with which I hope to take up elsewhere.2 A force that resists change is old hat to readers familiar with, for instance,Talmy (1988).",
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"text": "In terms familiar from, for example,Grenon and Smith (2004), strings that structure occurrents/perdurants along temporal S-steps may arise from strings that structure continuants/endurants along a \u227a-scale. See alsoJackendoff (1996).4 While any finite string is too short to serve as a timeline, it can be extended indefinitely through inverse limits relative to the composition of \u03c1A with bc or d . MSO under these projections has a formulation, spelled out inFernando (2016), as an institution in the sense ofGoguen and Burstall (1992). So too does a finite-state fragment of TTR(Fernando, 2017), although how to relate these institutions category-theoretically remains (as far as I know) to be worked out.",
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"back_matter": [
{
"text": "My thanks to three anonymous referees for their comments. This research is supported by Science Foundation Ireland (SFI) through the CNGL Programme (Grant 12/CE/I2267) in the ADAPT Centre, https://www.adaptcentre.ie. The ADAPT Centre for Digital Content Technology is funded under the SFI Research Centres Programme (Grant 13/RC/2106) and is co-funded under the European Regional Development Fund.",
"cite_spans": [],
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"section": "Acknowledgments",
"sec_num": null
}
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"FIGREF0": {
"num": null,
"type_str": "figure",
"text": "subsets of A. In model-theoretic terms, this suggests construing a string s = \u03b1 1 \u2022 \u2022 \u2022 \u03b1 n of subsets \u03b1 i of A as the model Mod(s) := [n], S n , {[[P a ]] s } a\u2208A with domain/universe [n] := {1, . . . , n} of string positions, interpreting S as the successor relation",
"uris": null
},
"FIGREF1": {
"num": null,
"type_str": "figure",
"text": "records -less strings b( s aRa ), depicting how R orders l(a), l(a ), r(a), r(a ). For example, l(a) r(a) l(a ) r(a ) depicts b's ordering l(a) < r(a) < l(a ) < r(a ) while l(a), l(a ) r(a), r(a ) depicts e's ordering l(a) = l(a ) < r(a) = r(a ).",
"uris": null
},
"FIGREF2": {
"num": null,
"type_str": "figure",
"text": "Box-removal d implements the Aristotelian slogan no time without change under the assumption that (B) all predicates appearing in a box (string symbol) express change.",
"uris": null
},
"FIGREF3": {
"num": null,
"type_str": "figure",
"text": "If d assumes (B), bc assumes strings are built from homogeneous predicates. Stative predicates are commonly assumed to be homogeneous, as in the well-known aspect hypothesis fromDowty (1979) claiming the different aspectual properties of the various kinds of verbs can be explained by postulating a single homogeneous class of predicates -stative predicates -plus three or four sentential operators or connectives. (page 71)",
"uris": null
},
"FIGREF4": {
"num": null,
"type_str": "figure",
"text": "built with predicates P \u2248d approximating D by D \u2022 . Given D \u2022 , (10) says the D \u2022 -degree at y is greater than the D \u2022 -degree d at the S-predecessor x of y. Now, whereas l(a) and r(a) cannot co-occur P l(a) (x) \u2283 \u00acP r(a) (x), we should look out for an opposing force \u2193 D before leaping from ap(\u2191 D) to ef(\u2191 D)",
"uris": null
},
"TABREF1": {
"num": null,
"html": null,
"type_str": "table",
"content": "<table><tr><td>R</td><td>s aRa</td><td>R \u22121</td><td colspan=\"2\">s aR \u22121 a</td><td>R</td><td>s aRa</td><td>R \u22121</td><td>s aR \u22121 a</td></tr><tr><td>b</td><td>a a</td><td>bi</td><td>a</td><td>a</td><td>d</td><td>a a, a a</td><td>di</td><td>a a, a a</td></tr><tr><td>o</td><td>a a, a a</td><td>oi</td><td colspan=\"2\">a a, a a</td><td>s</td><td>a, a a</td><td>si</td><td>a, a a</td></tr><tr><td>m</td><td>a a</td><td>mi</td><td colspan=\"2\">a a</td><td>f</td><td>a a, a</td><td>fi</td><td>a a, a</td></tr><tr><td>e</td><td>a, a</td><td/><td/><td/><td/><td/><td/><td/></tr></table>",
"text": "Allen relations as stutterless strings"
},
"TABREF2": {
"num": null,
"html": null,
"type_str": "table",
"content": "<table/>",
"text": "Allen relations as -less strings, afterDurand and Schwer (2008)"
},
"TABREF3": {
"num": null,
"html": null,
"type_str": "table",
"content": "<table><tr><td/><td>atomic</td><td>extended</td></tr><tr><td colspan=\"2\">+conseq culmination (achievement)</td><td>culminated process (accomplishment)</td></tr><tr><td>a</td><td>pre(a) a</td><td>pre(a), ap(f ) pre(a), ap(f ), ef(f ) ef(f ), a</td></tr><tr><td>\u2212conseq</td><td>point (semelfactive)</td><td>process (activity)</td></tr><tr><td>f</td><td>ap(f ) ef(f )</td><td>ap(f ) ap(f ), ef(f ) ef(f )</td></tr></table>",
"text": "Moens and Steedman (1988)'s reconstruction of ARKV, annotated with strings"
}
}
}
}