Predication via Finite-State Methods

Transcription

1 Predication via Finite-State Methods 1. Introduction: Less is More Tim Fernando ESSLLI 2017, Toulouse Course themes: finite-state approximability bounded granularity Today: extension/instance/node }{{} instantiation vs mechanism/string/label }{{} characterization 1 / 25 Predicates and instances analysis G. Carlson (1) Socrates is human S h individual Socrates is mortal Socrates is widespread Socrates is an omnivore (2) Humans are mortal h m Humans are widespread h w kind Humans are omnivores generic (3) Socrates was thirsty temporal stage Socrates was walking his dog Socrates was running a mile Semantic networks (Woods), frames (Fillmore, Barsalou/Düsseldorf), records (Cooper), attribute value structures... as finite automata (with strings for instances)

2 Attribute values and strings name nationality dob pat irish day 30 month june year 1990 name pat, nationality irish, dob day 30, dob month june, dob year 1990 Form strings over the alphabet Σ = {name, pat, nationality,..., 1990} = Attributes Values A domain D of objects d D described by L(d) Σ a D-indexed family L : D 2 Σ of Σ-languages. Brzozowski automaton over Σ A state is a subset of Σ, and for L, L Σ, L a L L = {s Σ as L} L is accepting ɛ L Brzozowski s-derivative of L so that s 1 L := {s Σ ss L} ɛ 1 L = L (as) 1 L = s 1 (a 1 L) L s L L = s 1 L s L ɛ s 1 L Myhill-Nerode: L is regular iff {s 1 L s Σ } is finite

3 Relations, type symbols, extensions and records n-ary relation R with attributes att 1,..., att n R att 1 att n d d 1 d n For d 1,..., d n, R Σ, define extension.. ext L (R(d 1 d n )) := {d D {R, att 1 d 1,..., att n d n } L(d)} ext L (R) := ext L (R(d 1 d n )) d 1 d n D n. Or add A 1,..., A n Σ with A i L(d i ) whenever ext L (R(d 1 d n )) ext L (R A1 A n ) := {d D {R, att 1 A 1,..., att n A n } L(d)} Record type [att 1 :A 1,..., att n :A n ] with extension ext L ([att 1 :A 1,..., att n :A n ]) := {d D {att 1 A 1,..., att n A n } L(d)}. Subsets of D from languages over Σ A language L Σ describes the D-subset L L := {d D L L(d)} E.g. {att 1 A 1,..., att n A n } L = ext L ([att 1 :A 1,..., att n :A n ]) {R, att 1 A 1,..., att n A n } L = ext L (R A1 A n ) {R, att 1 d 1,..., att n d n } L = ext L (R(d 1 d n )) Clearly, L L = L L L L but = may fail because D is not large enough L L(d) d L L L(d) L d L L

4 Languages from subsets of D A subset E of D is described by the Σ-language E L := L(d) d E with (antitone) Galois connection L E L E L L Formal Concept Analysis: a concept is a pair (E, L) s.t. L = E L (intent) and E = L L (extent). concepts (E, L) and (E, L ) can be partially ordered E E L L E consists of objects in D L consists of strings in Σ, not just symbols in Σ. NEXT: focus on concepts given by objects form strings sharpening partial order on concepts Concepts from objects in D An object d D names the concept (L(d) L, L(d)). An L-extensional account of is-a d is-a L d d L(d ) L L(d) L L(d ) L L(d ) L(d) supporting inheritance along is-a s L(d ) d is-a L d s L(d) e.g. bird flies Tweety is-a L bird Tweety flies Complications 1 defeasibility: penguins are birds that don t fly 2 other sorts of predication: kind-level, stage-level 3 dependence on choice of L : D 2 Σ

5 Reconstruing is-a: defeasibility & strings Replace L(d) by a set Φ[d] of predicates that hold of d ψ Φ[d ] d is-a d ψ Φ[d] ψ Φ[d] ψ Ψ 1 defeasibility: introduce negation not-opposed(ψ, d) 2 other sorts: Ψ Φ 3 dependence on L: form Φ[d] institutionally (Goguen) Ensure {ψ, ψ} Φ[d] by taking is-a to be a successor relation on finitely many d s for a string Φ[d 1 ] Φ[d 2 ] Φ[d n ] with d i is-a d i+1. Similar strings for stage-level predication, with prev (and inertia) in place of is-a (and inheritance). Woods intensional subsumption (contra is-a L ) - more on Thursday (Lecture 4) Subatomic semantics (Parsons 1990) the study of those formulas of English that are treated as atomic formulas in most logical investigations of English including predication/modification (Davidson 1967) Jones did it slowly, deliberately, in the bathroom, with a knife, at midnight and tense & aspect - Reichenbach relates R to { S for tense E for aspect - Inside E: e.g. (non-)entailments (Aristotle... Dowty 1979) Al was running (towards home) Al ran (towards home) Al was running home Al ran home

6 Strings for natural language semantics W. Klein The expression of time in natural languages relates a clause-internal temporal structure to a clause-external temporal structure. The latter may shrink to a single interval, for example, the time at which the sentence is uttered; but this is just a special case. The clause-internal temporal structure may also be very simple it may be reduced to a single interval without any further differentiation, the time of the situation ; but if this ever happens, it is only a borderline case. As a rule, the clause-internal structure is much more complex. Ed exhaled E S H. Reichenbach tense aspect it rained E,R S R S E,R it has rained E R,S R,S E R Open-endedness & institutions more than one symbol in a box alphabet as powerset Σ 2 Σ := {A A Σ} add symbols, possibly lengthening string e.g. real line R from finite sets of rational numbers Formulate Σ as a signature in an institution (Goguen & Burstall) model M given by a functor Mod M = Σ ϕ signature Σ in a category Sign sentence ϕ given by a functor sen Mark Steedman: temporality is not just about timelines Greg Carlson: rules & regulations Tess is eating dal Tess eats dal From episodes/particulars to generics/types (frames) models-as-strings models-as-finite automata

7 Strings built from fluents representing states tense aspect it rained E,R S R S E,R it has rained E R,S R,S E R E,R,S as temporal propositions or fluents (for short) Interpret a fluent ϕ over an interval I I = ϕ leading to filmstrips such as months in a year Jan Feb Dec + d1,d2,... d31 Jan,d1 Jan,d2 Jan,d31 Feb,d1 Dec,d31 where, for example, Jan represents a state I = Jan ( t I ) {t} = Jan Against such strings Bach 1986 (NLM, page 587) There is apparently a strong tendency to think that states are somehow basic, a sort of filmstrip view of reality which I do not share. If anything quite the opposite seems to be true. It took about two millenia to come up with satisfactory ways of coping with Zeno s questions about what it could possibly mean to be in a state of motion at an instant or how you could possibly add together dimensionless instants to get changes (you can t). Krifka 1998 (page 98) the filmstrip model of change... arguably is not the way movement and change is conceptualized (cf. Jackendoff 1996).

8 Jackendoff 1996 on filmstrips I wish to reject this snapshot conceptualization, on the grounds that it misrepresents the essential continuity of events of motion. For one thing, aside from the beginning and end points, the choice of a finite set of subevents is altogether arbitrary. How many subevents are there, and how is one to choose them? Notice that to stipulate the subevents as equally spaced, for instance one second or 3.5 milliseconds apart, is as arbitrary and unmotivated as any other choice. Another difficulty with a snapshot conceptualization concerns the representation of nonbounded events (activities) such as John ran along the river (for hours). A finite sequence of subevents necessarily has a specified beginning and ending, so it cannot encode the absence of endpoints. And excluding the specified endpoints simply exposes other specified subevents, which thereby become new endpoints. Thus encoding nonbounded events requires major surgery in the semantic representation. page 316 Calendar & block compression days in a year months in a year Jan,d1 Jan,d2 Dec,d31 ρ months Jan 31 Feb 28 Dec 31 bc Jan Feb Dec ρ A see only A ρ A (α 1 α 2 α n ) := (α 1 A)(α 2 A) (α n A) bc compress α + to α [ no identical adjacent boxes ] bc(s) := s if length(s) 1 { bc(αα bc(αs) if α = α s) := α bc(α s) otherwise α 1 α n is stutterless if α i α i+1 for 1 i < n (fixed pt of bc) bc A is ρ A ; bc [ vocabulary ; ontology ]

9 The real line R & inverse limits R and numbers a < a < a < a a a a a a For any set X, let Fin(X ) be the set of finite subsets of X. Let lim (bc A ) A Fin(Q) be the set of f : Fin(Q) Fin(Q) s.t. ( A Fin(Q))( B A) f (B) = bc B (f (A)). E.g. str : Fin(Q) Fin(Q) + s.t. str({a 1,..., a n }) := a 1 a n where a 1 < < a n and also for C Q, str C : Fin(Q) Fin(C) + s.t. str C (A) := str(a C). Dedekind cuts & prefixes Map a real number to the set of rational numbers less than it left Q : R 2 Q, r {q Q q < r} Let Cut be the set of C Q s.t. C Q ( a C)( a Q) a < a implies a C (lower half) ( a C)( a C) a < a (else 2 copies of Q) Fact. left Q : R, < = Cut, (extends to ordered field) = {str C C Cut}, where for f, f lim (bc A ) A Fin(Φ), f f f f and ( A Fin(Φ)) f (A) prefix f (A) and for strings s, s, s prefix s ( x) sx = s.

10 Where are we? We formed the continuum R from stutterless strings in Fin(Q) + }{{} {bc(s) s Fin(Q) + } Next: consider intervals - Reichenbachian progressive R inside E - subinterval property (Bennett and Partee 1972/8) Ed slept from 3 to 6 }{{}}{{} Ed slept from 3 to 5 }{{}}{{} ϕ I ϕ I I = ϕ and I I implies I = ϕ Ed slept from 3 to 5pm, Ed slept from 4 to 6pm Ed slept from 3 to 6pm. From points to intervals Fix a (strict) linear order < on some non-empty set T of points. We lift < to non-empty subsets I, I of T via I < I ( t I )( t I ) t < t for the left and right sides of I left(i ) := {t T {t} < I } right(i ) := {t T I < {t}}. The set Ivl of intervals (wrt <) is Ivl := {I T T left(i ) I right(i )} { } and a satisfaction relation = between T and a set Φ of fluents ϕ can be lifted to intervals I via I = ϕ ( t I ) t = ϕ.

11 Fluents over intervals & Dowty s aspect hypothesis Interpret fluents from a set Φ over a Φ-timeline A = T,, = - a linear order on a non-empty set T of points - a binary relation = between intervals I and fluents in Φ Given = between Ivl and Φ, call ϕ pointwise if I = ϕ ( t I ) {t} = ϕ. Widespread assumption (Taylor 1977, Dowty 1979,...). Fluents expressing statives (predicates over states) are pointwise. Dowty s Aspect Hypothesis (1979) 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 and connectives Negation & homogeneity The negation ϕ of a pointwise ϕ need not be pointwise I = ϕ not I = ϕ An interval I is ϕ-homogeneous if ϕ is satisfied by either all or none of the subintervals of I ( I I ) I = ϕ ( I I ) I = ϕ (where the subinterval relation is restricted to Ivl). N.B. I is ϕ-homogeneous I is ϕ-homogeneous Wikipedia on Ramsey s theorem: one will find monochromatic cliques in any edge colouring of a sufficiently large complete graph. Plan. Segment an interval into homogeneous subintervals string

12 Segmentations A segmentation of I is a <-increasing sequence I 1 I n covering I I 1 I n I n I i = I and for 1 i < n, I i < I i+1 i=1 i.e. a finite partition of I ordered by <. A seg is a segmentation I 1 I n of n i=1 I i. A ϕ-segmentation is a seg I 1 I n s.t. each I i is ϕ-homogeneous. A seg I 1 I n tracks ϕ if for all I n i=1 I i, I = ϕ I {I i 1 i n and I i = ϕ}. Fact. For pointwise ϕ, a seg is a ϕ-segmentation iff it tracks ϕ. Alternations When does an interval I have a ϕ-segmentation? A (ϕ, n)-alternation of I is a string t 1 t n I n s.t. for 1 i < n, t i < t i+1 and {t i } = ϕ {t i+1 } = ϕ (e.g. {t 1 } = ϕ, {t 2 } = ϕ, {t 3 } = ϕ,...). I is ϕ-alternation bounded (a.b.) if there is an integer n > 0 s.t. no (ϕ, n)-alternation of I exists. Fact. For pointwise ϕ, I has a ϕ-segmentation iff I is ϕ-a.b. Stepping up from a fluent ϕ to a set A of fluents, we call a seg an A-segmentation if it is a ϕ-segmentation for each ϕ A. I is A-segmentable if there is an A-segmentation of I. Fact. Given a finite set A of pointwise fluents and an interval I, I is A-segmentable I is ϕ-a.b. for all ϕ A.

13 From A-segmentations to strings in (2 A ) + The A-diagram of a seg I 1 I n is the string A (I 1 I n ) := {ϕ A I 1 = ϕ} {ϕ A I n = ϕ}. Fact. If an interval I is A-segmentable, there is a unique string s I A s.t. for all A-segmentations I 1 I n of I, bc( A (I 1 I n )) = s I A. Moreover, s I A is the A-diagram of the shortest A-segmentation of I. Recall: an A-segmentation I 1 I n tracks each ϕ A for all I n i=1 I i, I = ϕ I {I i 1 i n and I i = ϕ}. Block compression bc implements the Aristotelian insight More tomorrow. No time without change.