Incremental semantic scales by strings
Tim Fernando
Computer Science Department Trinity College Dublin
Dublin, Ireland
Abstract
Scales for natural language semantics are analyzed as moving targets, perpetually under construction and subject to ad-justment. Projections, factorizations and constraints are described on strings of bounded but refinable granularities, shap-ing types by the processes that put seman-tics in flux.
1 Introduction
An important impetus for recent investigations into type theory for natural language semantics is the view of “semantics in flux,” correcting “the im-pression” from, for example, Montague 1973 “of natural languages as being regimented with mean-ings determined once and for all” (Cooper 2012, page 271). The present work concerns scales for temporal expressions and gradable predicates. Two questions that loom large from the perspec-tive of semantics in flux are: how to construct scales and align them against one another (e.g. Klein and Rovatsos 2011). The formal study car-ried out below keeps scales as simple as possi-ble, whilst allowing for necessary refinements and adjustments. The basic picture is that a scale is a moving target finitely approximable as a string over an alphabet which we can expand to refine granularity. Reducing a scale to a string comes, however, at a price; indivisible points must give way to refinable intervals (embodying underspec-ification).
Arguments for a semantic reorientation around intervals (away from points) are hardly new. Best known within linguistic semantics perhaps are those in tense and aspect from Bennett and Partee 1972, which seem to have met less resistance than arguments in the degree literature from Kennedy 2001 and Schwarzschild and Wilkinson 2002 (see Solt 2013). At the center of the present argument
for intervals is a notion of finite approximabil-ity, plausibly related to cognition. What objection might there be to it? The fact that no finite lin-ear order is dense raises the issue of compatibility between finite approximability and density — no small worry, given the popularity of dense linear orders for time (e.g. Kamp and Reyle 1993, Pratt-Hartmann 2005, Klein 2009) as well as measure-ment (e.g. Fox and Hackl 2006).
Fortunately, finite linear orders can be orga-nized into a system of approximations converging at the limit to a dense linear order. The present work details ways to form such systems and lim-its, with density reanalyzed as refinability of ar-bitrary finite approximations. A familiar example provides some orientation.
Example A (calendar) We can represent a cal-endar year as the string
smo := Jan Feb Mar · · · Dec
of length 12, or, were we interested also in days d1,d2. . .,d31, the string
smo,dy := Jan,d1 Jan,d2 · · · Jan,d31 Feb,d1 · · · Dec,d31
of length 365 for a non-leap year (Fernando 2011).1 In contrast to the points in the real line
R, a box can split, as Jan insmodoes (30 times) to
Jan,d1 Jan,d2 · · · Jan,d31
in smo,dy, on introducing days d1, d2,. . ., d31 into the picture. Reversing direction and gener-alizing from
mo := {Jan,Feb,. . .Dec}
1We draw boxes (instead of the usual curly braces{and})
around sets-as-symbols, stringing together “snapshots” much like a cartoon/film strip.
to any setA, we define the functionρAon strings (of sets) to componentwise intersect withA
ρA(α1· · ·αn) := (α1∩A)· · ·(αn∩A)
(throwing out non-A’s from each box) so that
ρmo(smo,dy) = Jan31 Feb28· · · Dec31.
Next, the block compression bc(s) of a string s compresses all repeating blocks αn (for n ≥ 1) of a boxαin a stringstoαfor
bc(s) :=
bc(αs0) ifs=ααs0
α bc(βs0) ifs=αβs0 with
α6=β
s otherwise
so that ifbc(s) = α1· · ·αnthenαi 6= αi+1 fori from 1 ton−1. In particular,
bc(Jan31 Feb28· · · Dec31) = smo.
Writing bcA for the function mapping s to bc(ρA(s)), we have
bcmo(smo,dy) = smo.
In general, we can refine a string sA of granu-larity A to one sA0 of granularity A0 ⊇ A with bcA(sA0) =sA. Iterating over a chain
A⊆A0 ⊆A00⊆ · · · ,
we can glue together stringssA, sA0, sA00, . . .such that
bcX(sX0) = sX forX∈ {A, A0, A00, . . .}.
Details in section 2.
We shall refer to the expressions we can put in a box asfluents(short for temporal propositions), and assume they are the elements of a setΦ. While the setΦof fluents might be infinite, we restrict the boxes that we string together to finite sets of flu-ents. WritingFin(Φ)for the set of finite subsets ofΦ and2X for the powerset ofX (i.e. the set ofX’s subsets), we will organize the strings over the infinite alphabet Fin(Φ) around finite alpha-bets2A, forA∈Fin(Φ)
Fin(Φ)∗ = [
A∈F in(Φ)
(2A)∗.
In addition to projectingFin(Φ)down to2Afor some A ∈ Fin(Φ), we can build up, forming
the componentwise unions of stringsα1· · ·αnand β1· · ·βnof the same numbernof sets for their su-perposition
α1· · ·αn&β1· · ·βn := (α1∪β1)· · ·(αn∪βn)
and superposing languagesLandL0 overFin(Φ)
by superposing strings in L and L0 of the same
length
L&L0 := {s&s0|s∈L, s0 ∈L0and
length(s)= length(s0)}
(Fernando 2004). For example,
smo,dy = ρmo(smo,dy) &ρdy(smo,dy)
where dy := {d1,d2. . . ,d31}. More generally, writingLAfor the image ofLunderρA
LA := {ρA(s)|s∈L},
observe that for L ⊆ (2B)∗ and A ⊆ B, L is
included in the superposition ofLAandLB−A
L ⊆ LA&LB−A.
The next step is to identify a languageL0such that
L = (LA&LB−A) ∩L0 (1)
other than L0 = L. For a decomposition (1) of
Linto (generic) contextual constraintsL0separate
from the (specific) components LA and LB−A, it will be useful to sharpen LA, LB−A and &, factoring in bc and variants of bc (not to mention
∩). Measurements ranging from crude compar-isons (of order) to quantitative judgments (mul-tiplying unit magnitudes with real numbers) can be expressed through fluents in Φ. We interpret the fluents relative to suitable strings inFin(Φ)∗,
presented below in category-theoretic terms con-nected to type theory (e.g. Mac Lane and Moerdijk 1992). Central to this presentation is the notion of a presheaf on Fin(Φ)— a functor from the op-posite category Fin(Φ)op (a morphism in which is a pair (B, A) of finite subsets of Φ such that A⊆B) to the categorySetof sets and functions. The Fin(Φ)-indexed family of functions bcA (for A∈ Fin(Φ)) provides an important example that we generalize in section 2.
Example B (continuous change) The pair (a), (b) below superposes two events, soup cooling and an hour passing, in different ways (Dowty 1979). (a) The soup cooled in an hour.
(b) The soup cooled for an hour.
A common intuition is that in an hour requires an event that culminates, while for an hour re-quires a homogeneous event. In the case of (a), the culmination may be that some threshold tem-perature (supplied by context) was reached, while in (b), the homogeneity may be the steady drop in temperature over that hour. We might track soup cooling by a descending sequence of degrees, d1 > d2 > · · · > dn, with d1 at the beginning of the hour, and dn at the end. Butnosample of finite size n can be complete. To overcome this limitation, it is helpful to construe the ith box in a string as a description of an intervalIi over the real lineR. We call a sequenceI1· · ·In of inter-vals asegmentationifSni=1Iiis an interval and for
1≤i < n,Ii< Ii+1, where<isfull precedence
I < I0 iff (∀r∈I)(∀r0 ∈I0)r < r0.
Now, assuming an assignment of degreessDg(r)
to real numbersr representing temporal instants, the idea is to define satisfaction|=between inter-valsI and fluentssDg < daccording to
I |=sDg < d iff (∀r∈I)sDg(r)< d
and similarly for d ≤ sDg. We then lift |= to segmentations I1· · ·In and strings α1· · ·αn ∈
Fin(Φ)nof the same lengthnsuch that
I1· · ·In|=α1· · ·αn iff whenever1≤i≤n andϕ∈Ii, Ii|=ϕi
and analyze (a) above as (c) below, where d is the contextually given threshold required byin an hour, and x is the start of that hour, the end of which is marked by hour(x).
(c) x, d≤sDg d≤sDg hour(x), sDg < d
All fluentsϕin (c) have the stative property
(†) for all intervalsI andI0 whose unionI ∪I0
is an interval,
I∪I0|=ϕ iff I |=ϕandI0 |=ϕ
(Dowty 1979). (†) holds also for the fluents in the string (d) below for (b), where the subinterval relationvis inclusion restricted to intervals,
I |= [w]ϕ iff (∀I0vI)I0 |=ϕ
andsDg↓is the fluent
∃x(sDg < x∧Prev(x≤sDg))
saying the degree drops (with I |= Prev(ϕ) iff I0I |= ϕ for some I0 < I such that I ∪I0 is
an interval).
(d) x [w]sDg↓ hour(x),[w]sDg↓
(†) is intimately related to block compression bc (Fernando 2013b), supporting derivations of (c) and (d) by a modification&bcof&defined in§2.3 below.
Our third example directly concerns computa-tional processes, which we take up in section 3.
Example C(finite automata) Given a finite al-phabet A, a (non-deterministic) finite automaton
A over A is a quadruple (Q, δ, F, q0) consisting of a finite set Q of states, a transition relation δ ⊆ Q×A×Q, a subsetF ofQ consisting of final (accepting) states, and an initial stateq0 ∈Q.
Aacceptsa stringa1· · ·an∈A∗precisely if there is a stringq1· · ·qn∈Qnsuch that
qn∈F and δ(qi−1, ai, qi)for1≤i≤n (2)
(whereq0 isA’s designated initial state). The ac-cepting runs of Aare strings of the form
a1, q1 · · · an, qn ∈ (2A∪Q)∗
satisfying (2). While we can formulate such runs as strings over the alphabetA×Q, we opt for the alphabet 2A∪Q (formed from A∪Q ∈ Fin(Φ)) to link up smoothly with examples where more than one automata may be running, not all neces-sarily known nor in perfect harmony with others. Such examples are arguably of linguistic interest, the so-calledImperfective Paradox(Dowty 1979) being a case in point (Fernando 2008). That said, the attention below is largely on certain category-theoretic preliminaries for type theory.2
We adopt the following notational conventions. Given a functionf and a setX, we write
2Only the most rudimentary category-theoretic notions
- f Xforfrestricted toX∩domain(f)
- image(f)for{f(x)|x∈domain(f)}
- fXforimage(f X)
- f−1Xfor{x∈domain(f)|f(x)∈X} and if g is a function for which image(f) ⊆
domain(g),
- f;gforf composed (left to right) withg
(f;g)(x) := g(f(x))
for allx∈domain(f). We sayf is a functiononXif
domain(f) =X⊇image(f)
— i.e., f : X → X. The kernel of f, ker(f), is the equivalence relation ondomain(f) that holds betweens, s0 such thatf(s) =f(s0). Clearly,
ker(f) ⊆ ker(f;g)
whenf;gis defined.
2 Some presheaves onFin(Φ)
Given a functionf onFin(Φ)∗andA ∈Fin(Φ),
let us writefAfor the functionρA;f onFin(Φ)∗
fA(s) := f(ρA(s))
(recallingρA(α1· · ·αn) := (α1∩A)· · ·(αn∩A) and generalizingbcAfrom Example A). To extract a presheaf on Fin(Φ) from the Fin(Φ)-indexed family of functionsfA, certain requirements onf are helpful. Toward that end, let us agree that
- f preserves a function g with domain
Fin(Φ)∗ ifg=f;g
- fisidempotentiffpreserves itself (i.e.,f =
f;f)
- thevocabulary voc(s)of s∈Fin(Φ)∗is the
set of fluents that occur ins
voc(α1· · ·αn) := n [
i=1 αi
whences∈voc(s)∗.
Note that for idempotentf,image(f) consists of canonical representativesf(s)ofker(f)’s equiva-lence classes{s0 ∈Fin(Φ)∗|f(s0) =f(s)}.
2.1 Φ-preserving functions
A functionfonFin(Φ)∗isΦ-preservingiff
pre-servesvocandfA, for allA∈Fin(Φ). Note that bcisΦ-preserving, as is the identity functionidon
Fin(Φ)∗.
Proposition 1. If f is Φ-preserving then f is idempotent and
fB;fA = fA∩B
for allA, B∈Fin(Φ).
LetPf be the function with domain
Fin(Φ)∪ {(B, A)∈Fin(Φ)×Fin(Φ)|A⊆B}
mappingA∈Fin(Φ)tof(2A)∗
Pf(A) := {f(s)|s∈(2A)∗}
and a Fin(Φ)op-morphism (B, A) to the restric-tion offAtoPf(B)
Pf(B, A) := fAPf(B).
Corollary 2. If f is Φ-preserving then Pf is a presheaf onFin(Φ).
Apart frombc, we get a Φ-preserving function by stripping off any initial or final empty boxes
unpad(s) :=
unpad(s0) ifs= s0or
elses=s0
s otherwise
so thatunpad(s) neither begins nor ends with . Notice thatbc;unpad=unpad;bc.
Proposition 3. Iff andgareΦ-preserving and f;g=g;f, thenf;gisΦ-preserving.
2.2 The Grothendieck construction
Given a presheafF onFin(Φ), thecategoryR F of elements of F (also known as theGrothendieck constructionforF) has
- objectsP (A, s) ∈ Fin(Φ)× F(A) (making X∈F in(Φ)F(X)the set of objects in
R F)
- morphisms(B, s0, A, s) from objects(B, s0)
to(A, s)whenA⊆BandF(B, A)(s0) =s
(e.g. Mac Lane and Moerdijk 1992). Letπfbe the left projection
fromR Pf back to Fin(Φ). Theinverse limit of
Pf,lim←−Pf, is the set of (R Pf)-valued presheaves ponFin(Φ)(i.e. functorsp:Fin(Φ)op→R P
f) that are inverted byπf
πf(p(A)) = A for allA∈Fin(Φ).
That is, p(A) = (A, sA) for somesA ∈ f(2A)∗
such that
(‡) sA=fA(sB)wheneverA⊆B ∈Fin(Φ).
(‡) is the essential restriction that lim←−Pf adds to objects {sX}X∈F in(Φ) of the dependent type Q
X∈F in(Φ)Pf(X).
2.3 Superposition and non-determinism
Taking the presheaf Pid induced by the identity function id on Fin(Φ)∗, observe that in R Pid,
there is a product of
(∅, )and({ϕ}, ϕ )
but not of
({ϕ}, )and({ϕ}, ϕ ).
The tag A in (A, s) differentiating (∅, ) from
({ϕ}, )cannot be ignored when forming prod-ucts inRPid. A necessary and sufficient condition for(A, s)and(B, s0)to have a product is
ρB(s) = ρA(s0)
presupposed by the pullback of
(A, s) → (A∩B, ρB(s)) ← (B, s0). By comparison, the superpositions&s0 exists (as
a string) if and only if
ρ∅(s) = ρ∅(s0)
for
(voc(s), s) → (∅, ρ∅(s)) ← (voc(s0), s0)
(or length(s)= length(s0)asρ
∅(s) = length(s)).
Products inRPidare superpositions, but superpo-sitions need not be products.
Next, we step from id to other Φ-preserving functions f such as bc and bc;unpad. A pair
(A, s) and (B, s0) of R P
f-objects may fail to have a productnotbecause there isnoR Pf-object
(A∪B, s00)such that
(A, s) ← (A∪B, s00) → (B, s0)
but too many non-isomorphic choices for suchs00.
Consider the case ofbc;unpad, with(∅, )terminal inR Pbc;unpad(whereis the null string of length 0). For distinct fluentsaandb ∈ Φ, there are 13 stringss∈Pbc;unpad({a, b})such that
({a}, a)←({a, b}, s)→({b}, b))
corresponding to the 13 interval relations in Allen 1983 (Fernando 2007).
The explosion of solutionss00 ∈P
f(A∪B)to the equations
fA(s00) =s and fB(s00) =s0
given
(A, s) → (A∩B, fB(s)) ← (B, s0)
(i.e.,fB(s) = fA(s0)) is paralleled by the trans-formation, underf, of a languageLto
Lf := f−1fL
used to turn the superpositionL&L0 of languages
LandL0into
L&f L0 := f(Lf &L0f).
Forf := bc;unpad, the set a &f b consists of the 13 strings mentioned above. (We follow the usual practice of conflating a stringswith the sin-gleton language{s}whenever convenient.)
Stepping from strings to languages, we lift the presheaf Pf to the presheaf Qf mapping A ∈ Fin(Φ)to
Qf(A) := {fL|L⊆(2A)∗}
and aFin(Φ)op-morphism(B, A)to the function
Qf(B, A) := (λL∈Qf(B))fAL
sending L ∈ Qf(B) tofAL ∈ Qf(A). Then, for non-identity morphisms betweenRQf-objects
(A, L) and (A, L0) where L ⊆ L0, we add
in-clusions from (A, L) to (A, L0) to the R Qf
-morphisms for the categoryC(Φ, f)with
- objects the same as those inR Qf, and
- morphisms (B, L0, A, L) from objects
(B, L0) to (A, L) whenever A ⊆ B and
As is the case withR Qf-morphisms, the sources (domains) of C(Φ, f)-morphisms entail their tar-gets (codomains). To make these entailments pre-cise, we can identify the space of possible worlds with the inverse limit ofPf, and reduce(A, L)to
[[A, L]]f :={p∈lim←−Pf |
(∃s∈L)p(A) = (A, s)}.
The inclusion
[[B, L0]]f ⊆ [[A, L]]f
can then be pronounced:(B, L0)f-entails(A, L).
Proposition 4. Letfbe aΦ-preserving function and(A, L)and(B, L0)beR Qf-objects such that
A ⊆ B. (B, L0) f-entails (A, L) iff there is a
C(Φ, f)-morphism from(B, L0)to(A, L).
Relaxing the assumption A ⊆ B, one can also check that forf ∈ {bc,unpad,(bc;unpad)}, pull-backs of
(A, L)→(A∩B,(f∅L)∩f∅L0)←(B, L0)
inC(Φ, f)are given by
(A, L)←(A∪B, L&fL0)→(B, L0) (3)
although (3) need not hold forL&fL0 to be well-defined.
3 Constraints and finite automata
We now bring finite automata into the picture, re-calling from section 1 Example C’s superpositions
a1 · · · an & q1 · · · qn (4)
wherea1· · ·an is accepted by a finite automaton
Agoing through the sequence q1· · ·qn of (inter-nal) states. We can assume the tape alphabetA ⊇ {a1, . . . , an} and the state set Q ⊇ {q1, . . . , qn} are two disjoint subsets of the setΦof fluents; flu-ents inA are “observable” (on a tape), while flu-ents inQare “hidden” (inside a black box). Dis-joint though they may be,AandQare tightly cou-pled byA’s transition tableδ ⊆Q×A×Q(not to mention the other components ofA, its initial and final states). That coupling can hardly be recreated by superposition&(or some simple modification
&f) without the help of some machinery encoding δ. But first, there is the small matter of formulat-ing the map a1· · ·an 7→ a1 · · · an implicit in (4) above as a natural transformation.
3.1 Bottom⊥naturally
If the functionηAsuch that fora1· · ·an∈A∗,
ηA(a1· · ·an) = a1 · · · an
is to be the A-th component of a natural trans-formation η : S ⇒ Pid, we need to specify the presheaf S on Fin(Φ). To form a function
S(B, A) : S(B) → S(A) forA ⊆ B ∈ Fin(Φ)
withB∗ ⊆ S(B)andA∗ ⊆ S(A), it is handy to
introduce a bottom⊥forB−A, adjoining⊥to a finite subsetX ofΦforX⊥ := X+{⊥}before
forming the strings inS(X) :=X⊥∗. We then set
S(B, A) :B⊥∗→A⊥∗
S(B, A)() :=
S(B, A)(βs) :=
βS(B, A)(s) ifβ∈A⊥
⊥S(B, A)(s) otherwise
(e.g. S({a, b},{a})(ba⊥) = ⊥a⊥) and let ηA : A⊥∗→(2A)∗mapto itself, and
ηA(αs) :=
ηA(s) ifα=⊥
α ηA(s) otherwise
(e.g.η{a}(⊥a⊥) = a ).
Proposition 5. η is a natural transformation fromStoPid.
3.2 Another presheaf and category
Turning now to finite automata, we recall a funda-mental result about languages that are regular (i.e., accepted by finite automata),3 the B¨uchi-Elgot-Trakhtenbrot theorem (e.g. Thomas 1997)
for every finite alphabetA 6= ∅, a language L⊆A+is regular iff there is a sentenceϕof MSOAsuch that
L = {s∈A+|s|= Aϕ}.
MSOA is Monadic Second Order logic with a unary relation symbolUafor each a ∈ A, plus a binary relation symbolSfor successors. The pred-icate|=Atreats a string a1a2· · ·an overA as an MSOA-model with universe{1,2, . . . , n}, Ua as its subset{i|ai=a}, andSas
{(1,2),(2,3), . . . ,(n−1, n)}
3Whether or not this sense ofregular has an interesting
so that, for instance,
a1· · ·an|=A∃x∃yS(x, y) iff n≥2 (5)
for all finite A 6= ∅. Notice that no a ∈ A is required to interpret∃x∃yS(x, y), which after all is an MSO∅-sentence suited to strings⊥n∈S(∅). Furthermore, fora6=band{a, b} ⊆A,
nostring inA+satisfies∃xU
a(x)∧Ub(x) (6)
which makes it awkward to extend|=Ato formulas with free variables (requiring variable assignments on top of strings inA+).
A simple way to accommodate variables is to include them inA and interpret MSOA-formulas not over A+ but over (2A)+, lifting |=
A from stringssoverAto a predicate|=Aon strings over
2Asuch that
s|=Aϕ iff ηA(s)|=Aϕ (7)
for every MSOA-sentence ϕ (Fernando 2013a). For alls∈(2A)+, we set
s|=AS(x, y) iff ρ{x,y}(s)∈ ∗ x y ∗ (8)
forA⊇ {x, y}, and
s|=AUa(x) iff ρ{a,x}(s)∈Eaa, x Ea (9)
forA⊇ {a, x}, whereEa:= ( + a)∗. We must be careful to incorporate into the clauses defining s |=A ϕ the presupposition that each first-order variable x free in ϕ occurs uniquely ins — i.e. s|=Ax=xwhere
s|=Ax=y iff ρ
{x,y}(s)∈ ∗ x, y ∗ (10)
for x, y ∈ A. In particular, we restrict negation
¬ϕto strings|=A-satisfyingx=x, for each first-order variablexfree inϕ. We can then put
s|=A∃x ϕ iff (∃s0)ρA(s0) =ρA(s)
ands0 |=A∪{x}ϕ
and similarly for second-order existential quantifi-cation. The equivalence (5) above then becomes
s|=A∃x∃yS(x, y) iff ρ
∅(s)∈ + (11)
and in place of (6), we have
s|=A∃xUa(x)∧Ub(x) iff ρ{a,b}(s)∈
(2{a,b})∗ a, b(2{a,b})∗ (12)
fora, b∈A.
Working back from (7)
s|=Aϕ iff ηA(s)|=Aϕ
to the B¨uchi-Elgot-Trakhtenbrot theorem, one can check that for every finite Aand MSOA-formula ϕ, the set
LA(ϕ) := {s∈(2A)+|s|=Aϕ}
of strings over2Athat|=A-satisfyϕis regular, us-ing the fact that for all A0 ⊆ A, the restriction
of ρA0 to (2A)∗ is computable by a finite state transducer. But forA ⊆ Φ,4 ρ
A0 (2A)∗ is just
Pid(A, A0). In recognition of the role of these functions in|=A, we effectivize the presheafQ
id from§2.3as follows. LetRΦ be the presheaf on Fin(Φ)mapping
- A∈Fin(Φ)to the set of languages over the alphabet2Athat are regular
RΦ(A) := {L∈Qid(A)|Lis regular}
and
- aFin(Φ)op-morphism (B, A)to the restric-tion ofQid(B, A)toRΦ(B)
RΦ(B, A) := (λL∈RΦ(B))ρAL. R
RΦ-objects are then pairs (A, L) where A ∈ Fin(Φ)and L is a regular language over the al-phabet 2A, while RRΦ-morphisms are quadru-ples(B, L, A, ρAL)from(B, L)to(A, ρAL)for A⊆B ∈Fin(Φ). To account for the Boolean op-erations in MSO (as opposed to the predications (8)– (10) involving ρA), we add inclusions for a categoryR(Φ)with
- the same objects asRRΦ
- morphisms all of those in C(Φ, id) be-tween objects in RRΦ — i.e., quadruples
(B, L0, A, L) such that A ⊆ B ∈ Fin(Φ),
L0 ⊆ (2B)∗ is regular,L⊆(2A)∗is regular,
andρAL0 ⊆L.
Let us agree to write
(B, L0);(A, L)
4Note an MSO
to mean (B, L0, A, L) is a R(Φ)-morphism.
Clearly, fors∈(2A)+,A0 ⊆AandL⊆(2A0
)+, ρA0(s)∈L iff (A,{s});(A0, L). In particular, forx∈Aands∈(2A)+,
s|=Ax=x iff (A,{s});({x}, ∗ x ∗)
and similarly for x = x replaced by the differ-ent MSOA-formulas specified in clauses (8)–(12) above. The MSOA-sentence
spec(A) := ∀x _ a∈A
(Ua(x)∧ ^
b∈A−{a}
¬Ub(x))
associating a uniquea ∈ A with each string po-sition (presupposed in|=Abut not in|=A) fits the same pattern
s|=Aspec(A) iff ρA(s)∈ {a |a∈A}+
iff (A∪voc(s),{s});
(A,{ a |a∈A}+) iff ρA(s)∈ηAA+. Let us define a strings∈Fin(Φ)+to be
- A-specifiedifs|=Aspec(A)
- A-underspecifiedifρA(s)∈ηA(A⊥+−A+)
- A-overspecifiedifρA(s)6∈image(ηA) so that fora 6= a0 andA = {a, a0}, a a is A
-specified, a isA-underspecified, and a, a0 a
is A-overspecified. Given a finite automaton A
overA with setQ of states, its setAcRun(A) of accepting runs (Example C) is both A-specified andQ-specified, providedA∩Q=∅(and other-wise risks being A-overspecified). The language accepted by A is theηA−1-image of the language ρAAcRun(A) that is Q-underspecified, in accor-dance with the intuition that the states are hidden. From the regularity of AcRun(A), however, it is clear that we can make these states visible, with
AcRun(A)as the language accepted by a finite au-tomaton A0 (over 2A∪Q) that may (or may not) have the same setQof states.
The mapsρA and inclusions⊆underlying the morphisms of R(Φ) represent the two ways in-formation may grow from R RΦ-objects (A, L) to(B, L0) — expansivelywith A ⊆ B and L =
ρAL0, andeliminativelywithL0 ⊆LandA=B. The same notion of f-entailment defined in§2.3
through the sets[[A, L]]f applies, but we have been careful here to fixf toid, in view of
Proposition 6. For A ⊆ B ∈ Fin(Φ), ϕ an MSOA-formula ands∈(2B)+,
s|=Bϕ iff ρ
A(s)|=Aϕ.
Proposition 6 says thats |=B ϕdepends only on the part ρA(s) of s mentioned in ϕ. It is a par-ticular instance of thesatisfaction conditionin in-stitutions, expressing the invariance of truth under change of notation (Goguen and Burstall 1992). Proposition 6 breaks down if we replace ρA by bcA orunpadA, as can be seen with A = ∅, and ϕ=∃x∃yS(x, y), for which recall (11).
3.3 Varying grain and span
Troublesome as they are, the maps bcA and
unpadAhave some use. Just as we can vary tem-poral grain throughbc(Examples A and B in sec-tion 1), we can vary temporal span throughunpad. For instance, we can combine runs of automataA1 overA1andA2overA2in
L(A1,A2) := AcRun(A1) &unpadAcRun(A2) with the subscript unpad on & relaxing the re-quirement thatA1andA2start and finish together (running in lockstep throughout). Fori∈ {1,2}, andQithe state set forAi,
AcRun(Ai) = unpadAi∪QiL(A1,A2) assuming the sets A1, A2, Q1 and Q2 are pair-wise disjoint. The disjointness assumption rules out any communication (or interference) between
A1 and A2. As subsets of one large set Φ of fluents, however, it is perfectly natural for these sets to intersect (and communicate through a com-mon vocabulary), and we might express very par-tial constraints involving them through, for ex-ample, MSO-formulas. Recalling the definition
LA(ϕ) :={s∈(2A)+|s|=Aϕ}, we can rewrite the satisfaction condition
s|=B ϕ iff f
A(s)|=Aϕ
on MSOA-formulasϕ,A⊆B ∈Fin(Φ)ands∈
(2B)+as
LB(ϕ) = {s∈(2B)+|fA(s)∈ LA(ϕ)}. This equation lifts any regular languageLA(ϕ)to a regular languageLB(ϕ), providedfis computed by a finite-state transducer (as in the case ofbcor
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