text: (cid:80) (cid:80) 5 PaTnrhdoeosLfy.emmOmbmesaetrritvcheaelntlhyaf,otδlδiljiojzwξjijs≤≤imδδmijijξezdijjifafotoerrlayalllflriio,,mjj ffDoorreﬁaallnllizztio≤≤n zz1.≤≤ zz,. tthhaujts∈ηL∂(u(cid:96)Fpj(cid:96))βj≥=(cid:96) ∂∂0τfxj(oa(cid:96)u(cid:96)t.)e.≥,fosra−taisllfyjjin∈∈gLu(cid:96)L(p1αu(cid:96)9pjs))(cid:96)η,.(jt)ha=t is−, ∂1f(cid:96)biny/∂(x7)(cid:96) ≥(re−ca1ll, • Case ∂3x,(cid:96)k(cid:80)∈ Ladj. In this case, inadequate capacity of link This remarkable feature of componentwise monotone sys- (cid:96) k may block ﬂow to link (cid:96), as discussed above. We have t[e3m1]s–i[s33a]n,albougtocuosmtopowneelnl-tkwniosewnmroensoutlotsnficoirtymaolnloowtosnecosynsstiedmers- ∂∂xFk(cid:96) = j∈Lu(cid:96)p βj≤(cid:96) ∂∂fxjokut . Since ∂∂fxjokut ∈ {0, −αjskη(j)/βjk} ation of a much broader class of systems, including trafﬁc a.e., we have ∂F(cid:96) 0 a.e., satisfying (19). networks, which are generally not monotone. • Case 4, k (cid:54)∈ L∂loxck. Then ∂F(cid:96) = 0, trivially satisfying (19). (cid:96) ∂xk Remark 2. Observe that the lower and upper bounds in (24) are achieved for appropriate choice of z and w, thus the The following corollary implies that the one-step reachable approximation of the one-step reachable set is tight. I set of the trafﬁc dynamics from a (closed) box for any given To prove that the trafﬁc network dynamics developed in signaling input s is over-approximated by the union of boxes, Section are componentwise monotone, we ﬁrst require a one box for each i = 1, . . . nD, where each of these boxes technical assumption: is efﬁciently computed by evaluating F at two particular ∈ L ∈ L (cid:96) Assumption 2. For all (cid:96) , points for each (cid:96) . The obtained over-approximation is denoted with the Post operator. This critical result allows efﬁcient computation of a ﬁnite state representation of the c ≤ xcap − βk(cid:96) c ∀k ∈ Lup. (25) trafﬁc dynamics, as detailed in Section . (cid:96) (cid:96) α k (cid:96) k(cid:96) I { | ≤ ≤ } Corollary 1. Consider the set = x x x x¯ ∈ X ∈ L Assumption 2 is a sufﬁcient condition for ensuring that if for x, x , and for each (cid:96) , deﬁne ξ(cid:96)(x, x) = { } { } a link has inadequate capacity and blocks upstream ﬂow, then ξ(cid:96) (x , x ) ∈L , ξ(cid:96)(x,(cid:40)x) = ξ(cid:96) (x , x ) ∈L where this link’s queue will not empty in one time step. This effec- k k k k l(cid:96)oc k k k k l(cid:96)oc tively is an assumption that the time step is sufﬁciently small (cid:40)x if k ∈ Ldown ∪ Lup to appropriately capture the queuing phenomenon. Speciﬁcally, ξ(cid:96) (x , x ) = k ∈ L(cid:96) (cid:96) (26) k k k x if k adj the saturation ﬂow rate c of link (cid:96) is in units of vehicles per k (cid:96) (cid:96) ∈ L ∪ L tsitmepe. sPthepysaicnadl,lyt,huc(cid:96)s,isisreimqupilriecditltyo adefcurnecatsioenwoitfhthdeeccrheoasseend ttiimmee ξ(cid:96)k(xk, xk) = xxk iiff kk ∈ Ld(cid:96)aodwj.n u(cid:96)p (27) step and thus Assumption 2 is satisﬁed when a sufﬁciently k (cid:96) small time step is used for the model. ∈ S I ⊆ I Then for all s , Post( , s) Post( , s) where Theorem 1. The trafﬁc network model is componentwise monotone for any signaling input s ∈ S. In particular, F (cid:91) I (cid:96) Post( , s) := is increasing in x for k downstream or upstream of link (cid:96) or equal to (cid:96), and dekcreasing in xk for k adjacent to link (cid:96). nD {x(cid:48) | F (ξ(cid:96), sloc, di) ≤ x(cid:48) ≤ F (ξ(cid:96), sloc, di) ∀(cid:96) ∈ L}. ∈ S (cid:96) (cid:96) (cid:96) (cid:96) (cid:96) (cid:96) (cid:96) Proof. For ﬁxed s , we show that F(x, s, d) satisﬁes i=1 (28) conditions (19) and (20) of Lemma 1 with x, d replacing z, w. Observe that F is continuous and piecewise differentiable by (8)–(10) and (14), thus it is Lipschitz continuous [34]. Proof. By substituting x, x for z, z and di, di for w, w in (cid:44) (cid:96) (cid:96) { (cid:48) The minimum function in (8) implies that F is differentiable Lemma 2 and deﬁning f(x, d) F(x, s, d), we obtain x = ∈ { } | ∈ I ∈ D } ⊆ { (cid:48) | ≤ (cid:48) ≤ asfoalmtuirsofseytxinhegavue(sr2yt0iwv)e.hNecraoesw.esWc:oensﬁirdsetrh∂aFve(cid:96)/∂∂∂Fdx(cid:96)(cid:96)k. For0(,210),aw.ee. cboyns(1id0e)r, FFfo(cid:96)(l(lxoξ,w(cid:96)s,ss, dl(cid:96)for)co,mdi(cid:96)x)th∀e(cid:96) t∈r,idvLia}l ffoairctaltlhaitx=Po1s,Ft.(cid:96)(.(I.ξ,,(cid:96)ns,D)s.l(cid:96)o=cT,hd∪ei(cid:96)n)cDo{roxxll(cid:48)(cid:96)ar=y • Case 1, k ∈ (Ldown ∪ Lup)\{(cid:96)}. From (8)–(10), link k may F(x, s, d) | x ∈ I, d ∈ D }. i=1 (cid:96) (cid:96) ∈ L i block the outﬂow of link (cid:96) when k down, or link k may contribute to the inﬂow to link (cid:96) if k ∈(cid:96)Lup, thus we have t∂∂hxFek(cid:96) fo∈urt{h0,p−os∂∂sfixb(cid:96)okuitl,itβyk(cid:96)o∂∂cfxckoukutr,s−o∂n∂flx(cid:96)yokut if+kβ∈k(cid:96) ∂L∂(cid:96)fxdkokouwt }n ∩a.eL.upw.hBeuret To applyVthIe. pFoINwIeTrEfuSl TtAooTlEs RoEf PLRTELSEsNynTAthTeIsOisN, we require ∂∂fx(cid:96)okut ≥∈ {0, −α(cid:96)skη((cid:96))/β(cid:96)k} a.e. and ∂∂fxkokut ∈ {(cid:96)0, 1} a.e(cid:96)., thus a ﬁnite state representation of the trafﬁc network model. In ∂F(cid:96) 0 a.e., satisfying (19). general, obtaining ﬁnite state abstractions is a difﬁcult problem • C∂xakse 2, k = (cid:96). We have ∂f(cid:96)out ∈ {0, 1} a.e. and, for j ∈ Lup, and existing techniques do not scale well. In this section, we t∂h∂fxajo(cid:96)tu,t a∈.e{.,0e,iαthjs(cid:96)eη(rj)∂/fβ(cid:96)oujt(cid:96)}=a0.∂ex.o,(cid:96)rho∂wf(cid:96)inev(cid:80)=er,0,Ais.esu.,m∂pftjoiuot n=20enfosur r(cid:96)aelsl eastxbapotelvoeriteaptnhrdeespceronomtpaotpisooennaeonnftewtfhiﬁseceitemrnaoftﬁnmcoetnothenotiwcdiotfyrokrpddroyeptneearrmmtieiicnssi.dnegvaelﬁonpietde j ∈ Lu(cid:96)p. Thus ∂∂∂Fxx(cid:96)(cid:96)(cid:96) ∈ {0, 1, 1∂x+(cid:96) j∈Lu(cid:96)p βj(cid:96)∂∂x∂f(cid:96)xjo(cid:96)ut } a.e. But A. 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text: 8 {I } Q(cid:48) 1: function SUCCESSORS(y, y, ∈ ) returns we obtain a deterministic Rabin automaton that accepts all and { } q{q Q} 2: inputs: y = y ∈L and y = y ∈L, only trajectories that satisfy the LTL speciﬁcation using off- (cid:96) (cid:96) X (cid:96) (cid:96) points in domain the-shelf software. We then construct the synchronous product Q X T 3: , a grid interval partition of of the Rabin automaton and in Deﬁnition 5, resulting aug ∈ L 4: for each (cid:96) do in a nondeterministic product Rabin automaton from which a 5: j := as in (36) control strategy is found by solving a Rabin game [35]. The (cid:96) 6: j := as in (37) result is a control strategy for which trajectories of the trafﬁc 7: end(cid:96)for network are guaranteed to satisfy the LTL speciﬁcation. Q(cid:48) { | ∈ { } ∀ ∈ L} 8: return := (j ) ∈L j j , . . . , ¯j (cid:96) As the discrete state space is ﬁnite, the signaling control (cid:96) (cid:96) (cid:96) (cid:96) (cid:96) 9: end function strategy takes the form of a collection of “lookup” tables Fig. 5. An algorithm for identifying successor states when Q is a gridded over the discrete states of the system, Q, and there is one box partition. such table for each state in the Rabin automaton. Thus, implementing the control strategy requires implementing the {I } underlying deterministic transition system of the speciﬁcation box partition ∈ can be reﬁned to obtain the coarsest q q Q Rabin automaton, which is interpreted as a ﬁnite memory possible gridded box partition with the property that each box I controller that “tracks” progress of the LTL speciﬁcation and is the union of boxes from the reﬁnement. This reﬁnement q updates at each time step. Given the current state of the Rabin is used as an index set; to compute the possible transitions I ∈ Q ∈ S transition system, the controller chooses the signaling input from q for q under signaling s , we compute y and y dictated by the current state of the augmented system Q. as in lines 11 and 12 of the algorithm in Fig. 3, and then use the Q(cid:48) Thus, we obtain a state feedback, ﬁnite memory controller. reﬁnement along with the algorithm in Fig. 5 to determine , Additionally, the controller update only requires knowledge the set of intersected boxes. The reﬁnement does not introduce Q of the currently occupied partition of , and thus does not additional states in the transition system or require addition reach computations; it is only used to efﬁciently determine Q(cid:48). require precise knowledge of the state x. B. Computational Requirements For example, the coarsest reﬁnement of Fig. 2(b) partitions the box labeled q5 into four boxes, which are all labeled q5. This For each q ∈ Q and each s ∈ S, determining the set method will be faster if the total number of intervals in the { (cid:48) | → (cid:48)} I reﬁnement is less than |Q|. q q s q requires·ﬁrst computing Post( q, s), w∈hicLh requires computing F ( ) at 2nD points for each (cid:96) . C. Augmenting the State Space with Signaling · · (cid:96) L Since F ( , s, ) is only a function of the links in loc, each (cid:96) To capture control objectives that include the state of the computation of this function requires time O(1) assuming the signals themselves (which are modeled as inputs in the ﬁnite average number of links at an intersection does not change T I sEtxataempablesstraocftisopneciﬁ)c,awtioensautghmatenretquthiree dtihsicsreatuegmstaetnetiosnpacine-. wOi(t|hL|nneDtw).oTrkhesni,zwe.eTidheunstifPyotshte(seqt, sQ)(cid:48) oisf bcooxmepsuthteadt inintertsimecet I |Q| clude ϕ2 and ϕ3 above or the speciﬁcations “the state of an Post( q, s). As described in Se|Lct|ion , this requires 2 intersection cannot change more than once per nmin time steps” comparisons of vectors of length and thus is done in time |Q||L| or “an input signal cannot remain unchanged for nmax time O( ) via the algorithm in Fig. 4. However, for gridded steps.” In particular, we propose augmenting the ﬁnite state box partitions, Q(cid:48) is computed in time O(|L| max ∈L{N }) (cid:96) (cid:96) abstraction to encompass both the current state of the ﬁnite by the algorithm in Fig. 5. Even for nongridded box partitions, state abstraction and the current state of the trafﬁc signals. Q(cid:48) can be computed in time O(|L| max ∈L{N }) where N is Deﬁnition 5 (Augmented ﬁnite state abstraction). The aug- interpreted as the number of intervals o(cid:96)f link (cid:96)(cid:96) result(cid:81)ing fr(cid:96)om mented ﬁnite state abstraction of the trafﬁc network is the the coarsest reﬁnement of the box partition that results in a T Q S → |Q| transition system aug = ( , , aug) where gridded box partition. For a gridded partition, = ∈L N • Q = Q×S is the set of discrete states consisting of the box and thus the number of boxes grows exponentially(cid:96) with(cid:96) partition index set and the set of allowed input signals, the number of links in the network. For a nongridded box S • is the set of allowed input signals, partition, the number of partitions can be substantially lower. • →aug⊆ Q ×(cid:48) S(cid:48) × ∈Q→is the set of tra(cid:48)nsit(cid:48)ion∈s Qgiven by Since {q(cid:48) | qT →s q(cid:48)} must be c|oQm|p|uSte||dL|for each q and s, ((q, σ), s, (q ,(cid:48)σ ∈))→ aug (cid:48)for (q, σ), (q , σ ) if and constructing requires time O(|Q||2S| 2nD{) wh}e|Ln |using only if (q, s, q ) and σ = s. the algorithm in Fig. 4 or time O( max ∈L N 2nD) (cid:96) (cid:96) for the algorithm in Fig. 5. We brieﬂy compare these computational requirements to VII. SYNTHESIZING CONTROLLERS FROM LTL that of polyhedral methods such as those in [14]. As the SPECIFICATIONS dynamics in (8)–(10) are piecewise afﬁne, such methods can A. Synthesis Summary I in principle be applied here. Computing Post( q, s) requires We omit the details of how a control strategy is synthesized polyhedral afﬁne transformations and polyhedral geometric T |L| from the nondeterministic transition system for a given sums, operations that scale exponentially in [36], [37]. To aug I LTL control objective, as this is well-documented in the litera- determine if Post( q, s) intersects another polytope, geomet- ture, see e.g. [14], [35]. Instead, we summarize the main steps ric differences are required, which again scales exponentially |L| of this synthesis as follows: from the LTL control objective, with . ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

label: paragraph

text: {I

text: function

text: SUCCESSORS(y,

text: y,

text: returns

text: we

text: obtain

text: a

text: deterministic

text: Rabin

text: automaton

text: that

text: accepts

text: all

text: and

text: q{q

text: Q}

text: inputs:

text: y

text: y

text: ∈L

text: and

text: y

text: y

text: ∈L,

text: only

text: trajectories

text: that

text: satisfy

text: the

text: LTL

text: speciﬁcation

text: using

text: off-

text: X

text: points

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text: domain

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text: software.

text: We

text: then

text: construct

text: the

text: synchronous

text: product

text: Q

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text: the

text: Rabin

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text: Deﬁnition

text: resulting

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text: L

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text: a

text: j

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text: control

text: strategy

text: is

text: found

text: by

text: solving

text: a

text: Rabin

text: game

text: The

text: j

text: as

text: in

text: result

text: is

text: a

text: control

text: strategy

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text: which

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text: the

text: trafﬁc

text: network

text: are

text: guaranteed

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text: satisfy

text: the

text: LTL

text: speciﬁcation.

text: L}

text: return

text: (j

text: ∈L

text: j

text: j

text: ¯j

text: As

text: the

text: discrete

text: state

text: space

text: is

text: ﬁnite,

text: the

text: signaling

text: control

text: end

text: function

text: strategy

text: takes

text: the

text: form

text: of

text: a

text: collection

text: of

text: “lookup”

text: tables

text: Fig.

text: An

text: algorithm

text: for

text: identifying

text: successor

text: states

text: when

text: Q

text: is

text: a

text: gridded

text: over

text: the

text: discrete

text: states

text: of

text: the

text: system,

text: Q,

text: and

text: there

text: is

text: one

text: box

text: partition.

text: such

text: table

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text: each

text: state

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text: the

text: Rabin

text: automaton.

text: Thus,

text: implementing

text: the

text: control

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text: {I

text: underlying

text: deterministic

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text: of

text: the

text: speciﬁcation

text: box

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text: be

text: reﬁned

text: to

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text: the

text: coarsest

text: q

text: q

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text: Rabin

text: automaton,

text: which

text: is

text: interpreted

text: as

text: a

text: ﬁnite

text: memory

text: possible

text: gridded

text: box

text: partition

text: with

text: the

text: property

text: that

text: each

text: box

text: I

text: controller

text: that

text: “tracks”

text: progress

text: of

text: the

text: LTL

text: speciﬁcation

text: and

text: is

text: the

text: union

text: of

text: boxes

text: from

text: the

text: reﬁnement.

text: This

text: reﬁnement

text: q

text: updates

text: at

text: each

text: time

text: step.

text: Given

text: the

text: current

text: state

text: of

text: the

text: Rabin

text: is

text: used

text: as

text: an

text: index

text: set;

text: to

text: compute

text: the

text: possible

text: transitions

text: I

text: Q

text: S

text: transition

text: system,

text: the

text: controller

text: chooses

text: the

text: signaling

text: input

text: from

text: q

text: for

text: q

text: under

text: signaling

text: s

text: we

text: compute

text: y

text: and

text: y

text: dictated

text: by

text: the

text: current

text: state

text: of

text: the

text: augmented

text: system

text: Q.

text: as

text: in

text: lines

text: and

text: of

text: the

text: algorithm

text: in

text: Fig.

text: and

text: then

text: use

text: the

text: Thus,

text: we

text: obtain

text: a

text: state

text: feedback,

text: ﬁnite

text: memory

text: controller.

text: reﬁnement

text: along

text: with

text: the

text: algorithm

text: in

text: Fig.

text: to

text: determine

text: Additionally,

text: the

text: controller

text: update

text: only

text: requires

text: knowledge

text: the

text: set

text: of

text: intersected

text: boxes.

text: The

text: reﬁnement

text: does

text: not

text: introduce

text: Q

text: of

text: the

text: currently

text: occupied

text: partition

text: of

text: and

text: thus

text: does

text: not

text: additional

text: states

text: in

text: the

text: transition

text: system

text: or

text: require

text: addition

text: reach

text: computations;

text: it

text: is

text: only

text: used

text: to

text: efﬁciently

text: determine

text: require

text: precise

text: knowledge

text: of

text: the

text: state

text: x.

text: B.

text: Computational

text: Requirements

text: For

text: example,

text: the

text: coarsest

text: reﬁnement

text: of

text: Fig.

text: partitions

text: the

text: box

text: labeled

text: into

text: four

text: boxes,

text: which

text: are

text: all

text: labeled

text: This

text: For

text: each

text: q

text: Q

text: and

text: each

text: s

text: S,

text: determining

text: the

text: set

text: method

text: will

text: be

text: faster

text: if

text: the

text: total

text: number

text: of

text: intervals

text: in

text: the

text: I

text: reﬁnement

text: is

text: less

text: than

text: |Q|.

text: q

text: q

text: s

text: q

text: requires·ﬁrst

text: computing

text: Post(

text: q,

text: s),

text: w∈hicLh

text: requires

text: computing

text: F

text: at

text: points

text: for

text: each

text: C.

text: Augmenting

text: the

text: State

text: Space

text: with

text: Signaling

text: L

text: Since

text: F

text: s,

text: is

text: only

text: a

text: function

text: of

text: the

text: links

text: in

text: loc,

text: each

text: To

text: capture

text: control

text: objectives

text: that

text: include

text: the

text: state

text: of

text: the

text: computation

text: of

text: this

text: function

text: requires

text: time

text: assuming

text: the

text: signals

text: themselves

text: (which

text: are

text: modeled

text: as

text: inputs

text: in

text: the

text: ﬁnite

text: average

text: number

text: of

text: links

text: at

text: an

text: intersection

text: does

text: not

text: change

text: T

text: I

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text: oisf

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text: inintertsimecet

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text: the

text: speciﬁcations

text: “the

text: state

text: of

text: an

text: Post(

text: q,

text: s).

text: As

text: described

text: in

text: Se|Lct|ion

text: this

text: requires

text: intersection

text: cannot

text: change

text: more

text: than

text: once

text: per

text: nmin

text: time

text: steps”

text: comparisons

text: of

text: vectors

text: of

text: length

text: and

text: thus

text: is

text: done

text: in

text: time

text: |Q||L|

text: or

text: “an

text: input

text: signal

text: cannot

text: remain

text: unchanged

text: for

text: nmax

text: time

text: O(

text: via

text: the

text: algorithm

text: in

text: Fig.

text: However,

text: for

text: gridded

text: steps.”

text: In

text: particular,

text: we

text: propose

text: augmenting

text: the

text: ﬁnite

text: state

text: box

text: partitions,

text: is

text: computed

text: in

text: time

text: O(|L|

text: max

text: ∈L{N

text: abstraction

text: to

text: encompass

text: both

text: the

text: current

text: state

text: of

text: the

text: ﬁnite

text: by

text: the

text: algorithm

text: in

text: Fig.

text: Even

text: for

text: nongridded

text: box

text: partitions,

text: state

text: abstraction

text: and

text: the

text: current

text: state

text: of

text: the

text: trafﬁc

text: signals.

text: can

text: be

text: computed

text: in

text: time

text: O(|L|

text: max

text: ∈L{N

text: where

text: N

text: is

text: Deﬁnition

text: (Augmented

text: ﬁnite

text: state

text: abstraction).

text: The

text: aug-

text: interpreted

text: as

text: the

text: number

text: of

text: intervals

text: link

text: mented

text: ﬁnite

text: state

text: abstraction

text: of

text: the

text: trafﬁc

text: network

text: is

text: the

text: the

text: coarsest

text: reﬁnement

text: of

text: the

text: box

text: partition

text: that

text: results

text: in

text: a

text: T

text: Q

text: S

text: |Q|

text: transition

text: system

text: aug

text: aug)

text: where

text: gridded

text: box

text: partition.

text: For

text: a

text: gridded

text: partition,

text: ∈L

text: N

text: Q

text: Q×S

text: is

text: the

text: set

text: of

text: discrete

text: states

text: consisting

text: of

text: the

text: box

text: and

text: thus

text: the

text: number

text: of

text: boxes

text: grows

text: partition

text: index

text: set

text: and

text: the

text: set

text: of

text: allowed

text: input

text: signals,

text: the

text: number

text: of

text: links

text: in

text: the

text: network.

text: For

text: a

text: nongridded

text: box

text: S

text: is

text: the

text: set

text: of

text: allowed

text: input

text: signals,

text: partition,

text: the

text: number

text: of

text: partitions

text: can

text: be

text: substantially

text: lower.

text: →aug⊆

text: Q

text: ∈Q→is

text: the

text: set

text: of

text: Qgiven

text: by

text: Since

text: qT

text: →s

text: must

text: be

text: c|oQm|p|uSte||dL|for

text: each

text: q

text: and

text: s,

text: ((q,

text: σ),

text: s,

text: (q

text: aug

text: (q,

text: σ),

text: (q

text: σ

text: if

text: and

text: constructing

text: requires

text: time

text: wh}e|Ln

text: |using

text: only

text: if

text: (q,

text: s,

text: q

text: and

text: σ

text: s.

text: the

text: algorithm

text: in

text: Fig.

text: or

text: time

text: O(

text: max

text: ∈L

text: N

text: for

text: the

text: algorithm

text: in

text: Fig.

text: We

text: brieﬂy

text: compare

text: these

text: computational

text: requirements

text: to

text: VII.

text: SYNTHESIZING

text: CONTROLLERS

text: FROM

text: LTL

text: that

text: of

text: polyhedral

text: methods

text: such

text: as

text: those

text: in

text: As

text: the

text: SPECIFICATIONS

text: dynamics

text: in

text: are

text: piecewise

text: afﬁne,

text: such

text: methods

text: can

text: A.

text: Synthesis

text: Summary

text: I

text: in

text: principle

text: be

text: applied

text: here.

text: Computing

text: Post(

text: q,

text: s)

text: requires

text: We

text: omit

text: the

text: details

text: of

text: how

text: a

text: control

text: strategy

text: is

text: synthesized

text: polyhedral

text: afﬁne

text: transformations

text: and

text: polyhedral

text: geometric

text: T

text: |L|

text: from

text: the

text: nondeterministic

text: transition

text: system

text: for

text: a

text: given

text: sums,

text: operations

text: that

text: scale

text: exponentially

text: in

text: To

text: aug

text: I

text: LTL

text: control

text: objective,

text: as

text: this

text: is

text: well-documented

text: in

text: the

text: litera-

text: determine

text: if

text: Post(

text: q,

text: s)

text: intersects

text: another

text: polytope,

text: geomet-

text: ture,

text: see

text: e.g.

text: Instead,

text: we

text: summarize

text: the

text: main

text: steps

text: ric

text: differences

text: are

text: required,

text: which

text: again

text: scales

text: exponentially

text: |L|

text: of

text: this

text: synthesis

text: as

text: follows:

text: from

text: the

text: LTL

text: control

text: objective,

text: with

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

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text: ##LTLine##

text: ##LTLine##


text: We may assume without loss of generality that each V is a ﬁnite union of intervals. n To see this, enumerate the intervals that constitute V and notice that the total length of n the ﬁrst k intervals is a monotone sequence bounded above by one. We may thus truncate the sequence of intervals such that the length of the resulting ﬁnite union of intervals is arbitrarily close to the limit length. Truncate each V to within (cid:15)/2n+1 of this limit. By n discarding the descendants of the truncated tails we ensure that the resulting sequence of open sets is still nested. The lengths of the ﬁnite unions of intervals that remain at each depth are all at least (cid:15)/2. We may further assume that there are no terminating nodes (nodes whose descendants end at a ﬁnite depth). They may be safely discarded as the nonterminating nodes of depth (cid:48) n wholly contain V (cid:48) for suﬃciently large n . So, all intervals can be assumed to have n descendants at every deeper level. The case where there is an inﬁnite path down the tree such that the lengths of the intervals traversed by the path stays above a ﬁxed number is easy. The midpoints of the intervals traversed have a cluster point; it is in the intersection of all the V s. We may thus n assume that as we travel down any path, the lengths of the intervals traversed vanish. One last reduction: all the nodes split (have at least two descendants at suﬃciently large depths). Each V is the disjoint union of S , the splitting nodes, and O , the nodes with n n n (cid:48) only one descendant at each further depth. The descendants of O at depth n vanish in (cid:48) n length as n increases. This means the descendants of S , and thus S itself, are longer than n n some ﬁxed uniform bound. We can discard O at every level. n Now, every node is nonterminating and splits. Note that any node N splits into two descendants and that one of these subintervals splits, implying that N has three descendants at some depth. The midpoints of the outermost of these nodes deﬁne a closed subinterval of (cid:84) N that wholly contains the middle descendant. Starting with any initial node and inductively applying this construction to each such yielded middle node gives a nested sequence of closed ∞ sets whose nonempty intersection is contained in V . n=1 n Figure 3: In a tree where every node is nonterminating and splits, every interval contains a closed set which in turn contains a descendant node. Acknowledgements. I would like to thank Andres del Junco for interesting me in this problem during a real analysis class. In addition, I am grateful to George Elliott and Dror Bar-Natan for their guidance and encouragement and to Alex Bloemendal, Larry Guth, and the referees for their helpful comments. I would also like to acknowledge the support of the the Department of Mathematics at the University of Toronto. References [1] R. Gordon, A convergence theorem for the Riemann integral, Math. Mag. 73 (2000) 141–147. [2] W. A. J. Luxemburg, Arzel`a’s dominated convergence theorem for the Riemann integral, Amer. Math. Monthly 78 (1971) 970–979. 3 ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: We

text: may

text: assume

text: without

text: loss

text: of

text: generality

text: that

text: each

text: V

text: is

text: a

text: ﬁnite

text: union

text: of

text: intervals.

text: n

text: To

text: see

text: this,

text: enumerate

text: the

text: intervals

text: that

text: constitute

text: V

text: and

text: notice

text: that

text: the

text: total

text: length

text: of

text: n

text: the

text: ﬁrst

text: k

text: intervals

text: is

text: a

text: monotone

text: sequence

text: bounded

text: above

text: by

text: one.

text: We

text: may

text: thus

text: truncate

text: the

text: sequence

text: of

text: intervals

text: such

text: that

text: the

text: length

text: of

text: the

text: resulting

text: ﬁnite

text: union

text: of

text: intervals

text: is

text: arbitrarily

text: close

text: to

text: the

text: limit

text: length.

text: Truncate

text: each

text: V

text: to

text: within

text: of

text: this

text: limit.

text: By

text: n

text: discarding

text: the

text: descendants

text: of

text: the

text: truncated

text: tails

text: we

text: ensure

text: that

text: the

text: resulting

text: sequence

text: of

text: open

text: sets

text: is

text: still

text: nested.

text: The

text: lengths

text: of

text: the

text: ﬁnite

text: unions

text: of

text: intervals

text: that

text: remain

text: at

text: each

text: depth

text: are

text: all

text: at

text: least

text: We

text: may

text: further

text: assume

text: that

text: there

text: are

text: no

text: terminating

text: nodes

text: (nodes

text: whose

text: descendants

text: end

text: at

text: a

text: ﬁnite

text: depth).

text: They

text: may

text: be

text: safely

text: discarded

text: as

text: the

text: nonterminating

text: nodes

text: of

text: depth

text: n

text: wholly

text: contain

text: V

text: for

text: suﬃciently

text: large

text: n

text: So,

text: all

text: intervals

text: can

text: be

text: assumed

text: to

text: have

text: n

text: descendants

text: at

text: every

text: deeper

text: level.

text: The

text: case

text: where

text: there

text: is

text: an

text: inﬁnite

text: path

text: down

text: the

text: tree

text: such

text: that

text: the

text: lengths

text: of

text: the

text: intervals

text: traversed

text: by

text: the

text: path

text: stays

text: above

text: a

text: ﬁxed

text: number

text: is

text: easy.

text: The

text: midpoints

text: of

text: the

text: intervals

text: traversed

text: have

text: a

text: cluster

text: point;

text: it

text: is

text: in

text: the

text: intersection

text: of

text: all

text: the

text: V

text: s.

text: We

text: may

text: thus

text: n

text: assume

text: that

text: as

text: we

text: travel

text: down

text: any

text: path,

text: the

text: lengths

text: of

text: the

text: intervals

text: traversed

text: vanish.

text: One

text: last

text: reduction:

text: all

text: the

text: nodes

text: split

text: (have

text: at

text: least

text: two

text: descendants

text: at

text: suﬃciently

text: large

text: depths).

text: Each

text: V

text: is

text: the

text: disjoint

text: union

text: of

text: S

text: the

text: splitting

text: nodes,

text: and

text: O

text: the

text: nodes

text: with

text: n

text: n

text: n

text: only

text: one

text: descendant

text: at

text: each

text: further

text: depth.

text: The

text: descendants

text: of

text: O

text: at

text: depth

text: n

text: vanish

text: in

text: n

text: length

text: as

text: n

text: increases.

text: This

text: means

text: the

text: descendants

text: of

text: S

text: and

text: thus

text: S

text: itself,

text: are

text: longer

text: than

text: n

text: n

text: some

text: ﬁxed

text: uniform

text: bound.

text: We

text: can

text: discard

text: O

text: at

text: every

text: level.

text: n

text: Now,

text: every

text: node

text: is

text: nonterminating

text: and

text: splits.

text: Note

text: that

text: any

text: node

text: N

text: splits

text: into

text: two

text: descendants

text: and

text: that

text: one

text: of

text: these

text: subintervals

text: splits,

text: implying

text: that

text: N

text: has

text: three

text: descendants

text: at

text: some

text: depth.

text: The

text: midpoints

text: of

text: the

text: outermost

text: of

text: these

text: nodes

text: deﬁne

text: a

text: closed

text: subinterval

text: of

text: N

text: that

text: wholly

text: contains

text: the

text: middle

text: descendant.

text: Starting

text: with

text: any

text: initial

text: node

text: and

text: inductively

text: applying

text: this

text: construction

text: to

text: each

text: such

text: yielded

text: middle

text: node

text: gives

text: a

text: nested

text: sequence

text: of

text: closed

text: sets

text: whose

text: nonempty

text: intersection

text: is

text: contained

text: in

text: V

text: n

text: Figure

text: In

text: a

text: tree

text: where

text: every

text: node

text: is

text: nonterminating

text: and

text: splits,

text: every

text: interval

text: contains

text: a

text: closed

text: set

text: which

text: in

text: turn

text: contains

text: a

text: descendant

text: node.

text: Acknowledgements.

text: I

text: would

text: like

text: to

text: thank

text: Andres

text: del

text: Junco

text: for

text: interesting

text: me

text: in

text: this

text: problem

text: during

text: a

text: real

text: analysis

text: class.

text: In

text: addition,

text: I

text: am

text: grateful

text: to

text: George

text: Elliott

text: and

text: Dror

text: Bar-Natan

text: for

text: their

text: guidance

text: and

text: encouragement

text: and

text: to

text: Alex

text: Bloemendal,

text: Larry

text: Guth,

text: and

text: the

text: referees

text: for

text: their

text: helpful

text: comments.

text: I

text: would

text: also

text: like

text: to

text: acknowledge

text: the

text: support

text: of

text: the

text: the

text: Department

text: of

text: Mathematics

text: at

text: the

text: University

text: of

text: Toronto.

text: References

text: R.

text: Gordon,

text: A

text: convergence

text: theorem

text: for

text: the

text: Riemann

text: integral,

text: Math.

text: Mag.

text: W.

text: A.

text: J.

text: Luxemburg,

text: Arzel`a’s

text: dominated

text: convergence

text: theorem

text: for

text: the

text: Riemann

text: integral,

text: Amer.

text: Math.

text: Monthly

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text: 2 an event horizon nor a singularity. Instead the result our starting point is a generalization of the spherically will be a completely non-singular spacetime that con- symmetric Einstein action that allows us to use quantum tains a closed dynamical horizon bounding a compact movitated corrections that resolve the singularity in the trapped region. At late times there will therefore be no vacuum solution. Since the class of lagrangians we con- impediment to information about the collapsed matter sider obey a Birkhoﬀ theorem, this guarantees that grav- escaping to null inﬁnity. Such a scenario was discussed itational collapse will yield an exterior spacetime that in [5, 6] and ﬁrst made explicit by Sean Hayward [9]. asymptotes to the corresponding non-singular solution. More recently, a compact, dynamical trapping horizon We therefore expect a complete absence of singularities was realized via the numerical simulation of the spheri- in the dynamical evolution of the collapsing matter as cal collapse of a massless scalar ﬁeld in [10]. In this work found in [11]. This will in turn permit us to examine via the singularity resolution and energy loss were modelled rigorous calculations, albeit in a simplistic model, the by introducing explicit modiﬁcations to the gravitational question posed above: How does singularity resolution potential in the equations of motion. Since the equations aﬀect the end point of gravitational collapse and Hawk- were not derived from a diﬀeomorphism invariant action, ing radiation? they did not preserve the constraint algebra and were Recently several other models have been proposed to non-conservative. A more systematic energy conserving investigate non–singular, radiating black holes [22–25] us- model was later constructed [11] and used to show that ing diﬀerent methods to model the radiation and singu- non-singular black holes could indeed be formed via grav- larity resolution than those used in this work. Using the itational collapse4 provided that the eﬀective quantum Polyakov term to model the radiation back reaction has corrections to the gravitational potential were introduced the advantage that it is rigorous, at least in 2D. It is within the framework of a (spherical symmetry preserv- also useful to have a variety of plausible models to hunt ing) diﬀeomorphism invariant action. The equations in for generic features of the formation and evapouration this model were derived from a variational principle and process. hence are conservative, but they did not contain a mech- anism for describing Hawking radiation. In the present paper, we describe the model and de- In the following, we continue the above program by rive the equations of motion both in conformal gauge and constructing a set of dynamical equations for four D (and in P-G type coordinates. The organization is as follows: higher) gravity that lead to non-singular collapse, are Section discusses the classical action (without radiat- energy conserving and include radiation back-reaction. ing terms). Section discusses the addition of the radi- Modelling the eﬀect of Hawking radiation is diﬃcult and ation corrections via the generalized Polyakov action in so far it is not well understood in more than two di- local form. Section derives the equations of motion in mensions. In two dimensions, however, the conformal conformal gauge and discusses singularity resolution in anomaly can be calculated at one loop order and inte- the energy momentum tensor while Section presents the grated to derive the non(cid:90)–local Polyakov action [15], Hamiltonian analysis and the equations of motion in P- G type coordinates. Section closes with conclusions and √ prospects. The numerical simulations of these equations ∼ − − R 1 R I d2x g , (1) are relegated to a subsequent paper. Poly D2 R where g is the determinant of the 2D metric, g , µν is the Ricci scalar calculated with g and D is the co- µν variant derivative compatible with g . This one loop µν eﬀective action is exact in the large N limit, where N is II. EFFECTIVE GRAVITATIONAL ACTION the number of conformally coupled scalar ﬁelds. It has been studied extensively in the context of 2D models for black hole evaporation [16–19]. It has also been used in a We start from Einstein gravity in n space-time dimen- toy model to simulate Hawking radiation in four dimen- sions minimally co(cid:90)upled (cid:112)to a m(cid:16)assless scalar ﬁeld, wh(cid:17)ose sional, spherically symmetric Einstein gravity [20, 21]. action is: This is the approach we will adopt. The new, and cru- 1 − R − cial features of our analysis are two-fold: ﬁrst we will I = dnx g (g) 8πG(n)(Dψ)2 (n) 16πG(n) formulate the theory in terms of a local, diﬀeomorphism (2) invariant eﬀective action. Our equations are therefore where G(n) is the higher dimensional gravitational con- energy conserving and allow for the use of slicings that R stant, (g¯) is the Ricci scalar calculated using the n- extend into the horizon. Secondly, the action which is dimensional metric, g¯, D¯ is the derivative compatible with g¯ and ψ is a scalar ﬁeld. After imposing spherical √ symmetry, integrating out the angular variables and ab- 4 See also [12] and [13] for closely related work and [14] for early sorbing a factor of 8πG(n) into the scalar ﬁeld to make work on singularity resolved black holes. it dimensionless, this action takes the form [26, 27]: ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: an

text: event

text: horizon

text: nor

text: a

text: singularity.

text: Instead

text: the

text: result

text: our

text: starting

text: point

text: is

text: a

text: generalization

text: of

text: the

text: spherically

text: will

text: be

text: a

text: completely

text: non-singular

text: spacetime

text: that

text: con-

text: symmetric

text: Einstein

text: action

text: that

text: allows

text: us

text: to

text: use

text: quantum

text: tains

text: a

text: closed

text: dynamical

text: horizon

text: bounding

text: a

text: compact

text: movitated

text: corrections

text: that

text: resolve

text: the

text: singularity

text: in

text: the

text: trapped

text: region.

text: At

text: late

text: times

text: there

text: will

text: therefore

text: be

text: no

text: vacuum

text: solution.

text: Since

text: the

text: class

text: of

text: lagrangians

text: we

text: con-

text: impediment

text: to

text: information

text: about

text: the

text: collapsed

text: matter

text: sider

text: obey

text: a

text: Birkhoﬀ

text: theorem,

text: this

text: guarantees

text: that

text: grav-

text: escaping

text: to

text: inﬁnity.

text: Such

text: a

text: scenario

text: was

text: discussed

text: itational

text: collapse

text: will

text: yield

text: an

text: exterior

text: spacetime

text: that

text: in

text: and

text: ﬁrst

text: made

text: explicit

text: by

text: Sean

text: Hayward

text: asymptotes

text: to

text: the

text: corresponding

text: non-singular

text: solution.

text: More

text: recently,

text: a

text: compact,

text: dynamical

text: trapping

text: horizon

text: We

text: therefore

text: expect

text: a

text: complete

text: absence

text: of

text: singularities

text: was

text: realized

text: via

text: the

text: numerical

text: simulation

text: of

text: the

text: spheri-

text: in

text: the

text: dynamical

text: evolution

text: of

text: the

text: collapsing

text: matter

text: as

text: cal

text: collapse

text: of

text: a

text: massless

text: scalar

text: ﬁeld

text: in

text: In

text: this

text: work

text: found

text: in

text: This

text: will

text: in

text: turn

text: permit

text: us

text: to

text: examine

text: via

text: the

text: singularity

text: resolution

text: and

text: energy

text: loss

text: were

text: modelled

text: rigorous

text: calculations,

text: albeit

text: in

text: a

text: simplistic

text: model,

text: the

text: by

text: introducing

text: explicit

text: modiﬁcations

text: to

text: the

text: gravitational

text: question

text: posed

text: above:

text: How

text: does

text: singularity

text: resolution

text: potential

text: in

text: the

text: equations

text: of

text: motion.

text: Since

text: the

text: equations

text: aﬀect

text: the

text: end

text: point

text: of

text: gravitational

text: collapse

text: and

text: Hawk-

text: were

text: not

text: derived

text: from

text: a

text: diﬀeomorphism

text: invariant

text: action,

text: ing

text: radiation?

text: they

text: did

text: not

text: preserve

text: the

text: constraint

text: algebra

text: and

text: were

text: Recently

text: several

text: other

text: models

text: have

text: been

text: proposed

text: to

text: non-conservative.

text: A

text: more

text: systematic

text: energy

text: conserving

text: investigate

text: non–singular,

text: radiating

text: black

text: holes

text: us-

text: model

text: was

text: later

text: constructed

text: and

text: used

text: to

text: show

text: that

text: ing

text: diﬀerent

text: methods

text: to

text: model

text: the

text: radiation

text: and

text: singu-

text: non-singular

text: black

text: holes

text: could

text: indeed

text: be

text: formed

text: via

text: grav-

text: larity

text: resolution

text: than

text: those

text: used

text: in

text: this

text: work.

text: Using

text: the

text: itational

text: provided

text: that

text: the

text: eﬀective

text: quantum

text: Polyakov

text: term

text: to

text: model

text: the

text: radiation

text: back

text: reaction

text: has

text: corrections

text: to

text: the

text: gravitational

text: potential

text: were

text: introduced

text: the

text: advantage

text: that

text: it

text: is

text: rigorous,

text: at

text: least

text: in

text: It

text: is

text: within

text: the

text: framework

text: of

text: a

text: (spherical

text: symmetry

text: preserv-

text: also

text: useful

text: to

text: have

text: a

text: variety

text: of

text: plausible

text: models

text: to

text: hunt

text: ing)

text: diﬀeomorphism

text: invariant

text: action.

text: The

text: equations

text: in

text: for

text: generic

text: features

text: of

text: the

text: formation

text: and

text: evapouration

text: this

text: model

text: were

text: derived

text: from

text: a

text: variational

text: principle

text: and

text: process.

text: hence

text: are

text: conservative,

text: but

text: they

text: did

text: not

text: contain

text: a

text: mech-

text: anism

text: for

text: describing

text: Hawking

text: radiation.

text: In

text: the

text: present

text: paper,

text: we

text: describe

text: the

text: model

text: and

text: de-

text: In

text: the

text: following,

text: we

text: continue

text: the

text: above

text: program

text: by

text: rive

text: the

text: equations

text: of

text: motion

text: both

text: in

text: conformal

text: gauge

text: and

text: constructing

text: a

text: set

text: of

text: dynamical

text: equations

text: for

text: four

text: D

text: (and

text: in

text: P-G

text: type

text: coordinates.

text: The

text: organization

text: is

text: as

text: follows:

text: higher)

text: gravity

text: that

text: lead

text: to

text: non-singular

text: collapse,

text: are

text: Section

text: discusses

text: the

text: classical

text: action

text: (without

text: radiat-

text: energy

text: conserving

text: and

text: include

text: radiation

text: back-reaction.

text: ing

text: terms).

text: Section

text: discusses

text: the

text: addition

text: of

text: the

text: radi-

text: Modelling

text: the

text: eﬀect

text: of

text: Hawking

text: radiation

text: is

text: diﬃcult

text: and

text: ation

text: corrections

text: via

text: the

text: generalized

text: Polyakov

text: action

text: in

text: so

text: far

text: it

text: is

text: not

text: well

text: understood

text: in

text: more

text: than

text: two

text: di-

text: local

text: form.

text: Section

text: derives

text: the

text: equations

text: of

text: motion

text: in

text: mensions.

text: In

text: two

text: dimensions,

text: however,

text: the

text: conformal

text: conformal

text: gauge

text: and

text: discusses

text: singularity

text: resolution

text: in

text: anomaly

text: can

text: be

text: calculated

text: at

text: one

text: loop

text: order

text: and

text: inte-

text: the

text: energy

text: momentum

text: tensor

text: while

text: Section

text: presents

text: the

text: grated

text: to

text: derive

text: the

text: Polyakov

text: action

text: Hamiltonian

text: analysis

text: and

text: the

text: equations

text: of

text: motion

text: in

text: P-

text: G

text: type

text: coordinates.

text: Section

text: closes

text: with

text: conclusions

text: and

text: prospects.

text: The

text: numerical

text: simulations

text: of

text: these

text: equations

text: R

text: R

text: I

text: g

text: are

text: relegated

text: to

text: a

text: subsequent

text: paper.

text: Poly

text: R

text: where

text: g

text: is

text: the

text: determinant

text: of

text: the

text: metric,

text: g

text: µν

text: is

text: the

text: Ricci

text: scalar

text: calculated

text: with

text: g

text: and

text: D

text: is

text: the

text: co-

text: µν

text: variant

text: derivative

text: compatible

text: with

text: g

text: This

text: one

text: loop

text: µν

text: eﬀective

text: action

text: is

text: exact

text: in

text: the

text: large

text: N

text: limit,

text: where

text: N

text: is

text: II.

text: EFFECTIVE

text: GRAVITATIONAL

text: ACTION

text: the

text: number

text: of

text: conformally

text: coupled

text: scalar

text: ﬁelds.

text: It

text: has

text: been

text: studied

text: extensively

text: in

text: the

text: context

text: of

text: models

text: for

text: black

text: hole

text: evaporation

text: It

text: has

text: also

text: been

text: used

text: in

text: a

text: We

text: start

text: from

text: Einstein

text: gravity

text: in

text: n

text: space-time

text: dimen-

text: toy

text: model

text: to

text: simulate

text: Hawking

text: radiation

text: in

text: four

text: dimen-

text: sions

text: minimally

text: a

text: scalar

text: ﬁeld,

text: sional,

text: spherically

text: symmetric

text: Einstein

text: gravity

text: action

text: is:

text: This

text: is

text: the

text: approach

text: we

text: will

text: adopt.

text: The

text: new,

text: and

text: cru-

text: R

text: cial

text: features

text: of

text: our

text: analysis

text: are

text: two-fold:

text: ﬁrst

text: we

text: will

text: I

text: dnx

text: g

text: (g)

text: (n)

text: formulate

text: the

text: theory

text: in

text: terms

text: of

text: a

text: local,

text: diﬀeomorphism

text: invariant

text: eﬀective

text: action.

text: Our

text: equations

text: are

text: therefore

text: where

text: G(n)

text: is

text: the

text: higher

text: dimensional

text: gravitational

text: con-

text: energy

text: conserving

text: and

text: allow

text: for

text: the

text: use

text: of

text: slicings

text: that

text: R

text: stant,

text: (g¯)

text: is

text: the

text: Ricci

text: scalar

text: calculated

text: using

text: the

text: n-

text: extend

text: into

text: the

text: horizon.

text: Secondly,

text: the

text: action

text: which

text: is

text: dimensional

text: metric,

text: g¯,

text: D¯

text: is

text: the

text: derivative

text: compatible

text: with

text: g¯

text: and

text: ψ

text: is

text: a

text: scalar

text: ﬁeld.

text: After

text: imposing

text: spherical

text: symmetry,

text: integrating

text: out

text: the

text: angular

text: variables

text: and

text: ab-

text: See

text: also

text: and

text: for

text: closely

text: related

text: work

text: and

text: for

text: early

text: sorbing

text: a

text: factor

text: of

text: into

text: the

text: scalar

text: ﬁeld

text: to

text: make

text: work

text: on

text: singularity

text: resolved

text: black

text: holes.

text: it

text: dimensionless,

text: this

text: action

text: takes

text: the

text: form

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 11 R = 0 then neither is the metric. So, we can resolve and found (in agreement with [20]) that the energy mo- both the singularity in the energy momentum tensor and mentum tensor is singular at the origin. We found, how- the one in the metric (in the case where (9) is satisﬁed) ever, that we could remove this singularity (as well as by choosing the functions V (R), φ(R) and h(R) in the the singularity in the vacuum solution) by appropriate (cid:48) action so that the factors containing j (= φ ) are not sin- choice of coeﬃcients in the action without violating the gular at the origin. Note that this choice will not violate Bianchi identities since our equations of motion were de- the Bianchi identities since we derived our equations of rived from a variational principal. We then performed motion from a variational principal. a detailed Hamiltonian analysis, including a prescription With the gauge choices, (87) and (90), the equations of of the boundary conditions and corresponding boundary motion for the remaining phase space variables can now terms needed to make the variational principle well de- be written as ﬁned. From this Hamiltonian we imposed suitable gauge ﬁxing conditions and derived the equations of motion in a family of non-null coordinates. The fact that these are − P (cid:48) ψ˙ = N ψ + N ψ , (102) ﬁrst order in time derivatives makes them well suited to 2B r numerical simulations of black hole formation. We then chose a gauge such that the metric is well deﬁned at hori- zon formation, which is well suited to the investigation of the dynamics past horizon formation. − (cid:48) (cid:48) P˙ψ =( 2NBψ + NrPψ) , (103) At ﬁrst glance the Lagrangian (27) that is our start- ing point contains many arbitrary functions. However, as we saw, the requirement that singularities be removed is quite restrictive. As shown in Section in order for the eﬀective stress energy tensor to be regular it is necessary z˙1 = −NS3 − NWPz2 + Nrz1(cid:48) , (104) that j = ω2 = (φ(cid:48))2 at R = 0. In addition, φ(cid:48) must go to zero at the origin at least as fast as R2, as can be seen from (51). Finally, the remaining freedom in j, φ and V can be ﬁxed so that the vacuum solution approaches the − (cid:48) (cid:48) (cid:48) form derived via general arguments in [34]. This leaves P˙ =( W(Nz + 2N ) + N P ) z1 2 r z1 the freedom in the matter coupling B(R). The simplest − − − NWb(R 2 1), (105) choice is B(R) = Rn 2, i.e. the classical form obtained ,y by dimensional reduction which, as shown in [11] yields non-singular collapse. b(R) and c(R) are normally taken to be zero in two dimensional models, but, when the cou- pling is not conformal these functions are determined by − − NP (cid:48) z˙ = NS z1 + N z (106) our choice of B(R). It would be interesting to see what 2 3 W r 2 eﬀect these terms have on the structure of the evaporat- ing, non-singular black hole spacetime. In an upcoming and paper we will present the results of numerical simulations using these equations. − (cid:48) (cid:48) (cid:48) VII. ACKNOWLEDGEMENTS P˙ = ( W(Nz + 2N ) + N P ) . (107) z2 1 r z2 We thank Nils Deppe for useful discussions. This work has been funded by the Fondecyt grant 3140123. These equations of motion, (102), (103), (104), (105), The Centro de Estudios Cient´ıﬁcos (CECs) is funded by (106), (107) along with the consistency conditions, (95), the Chilean Government through the Centers of Excel- (93), the constraints, (100), (101) and the deﬁnitions, lence Base Financing Program of Conicyt. This work (96), (97), (98), (99) can be used to evolve appropriate was also funded in part by the Natural Sciences and initial conditions forward in time to show the formation Engineering Research Council of Canada. Support was of a black hole with a radiating term. also provided by WestGrid (www.westgrid.ca), Compute VI. CONCLUSIONS Canada/Calcul Canada (www.computecanada.ca), and the Perimeter Institute for Theoretical Physics (funded In this work we deﬁned an action which mimics spher- by Industry Canada and the Province of Ontario Min- ically symmetric, dimensionally reduced gravity with a istry of Research and Innovation). radiating term. Although the radiating term was bor- Appendix A: Loop Quantum Gravity Black Hole rowed from the conformal anomaly and is only valid in two dimensional physics we incorporated terms which ac- count for the non-minimal coupling (in 2-D) of the mat- An example of a non-singular, single horizon black hole ter ﬁeld. We derived Lagrange’s equations in null gauge in four dimensions that can be derived as a solution to ##LTLine## ##LTLine## ##LTLine##

text: R

text: then

text: neither

text: is

text: the

text: metric.

text: So,

text: we

text: can

text: resolve

text: and

text: found

text: (in

text: agreement

text: with

text: that

text: the

text: energy

text: mo-

text: both

text: the

text: singularity

text: in

text: the

text: energy

text: momentum

text: tensor

text: and

text: mentum

text: tensor

text: is

text: singular

text: at

text: the

text: origin.

text: We

text: found,

text: how-

text: the

text: one

text: in

text: the

text: metric

text: (in

text: the

text: case

text: where

text: is

text: satisﬁed)

text: ever,

text: that

text: we

text: could

text: remove

text: this

text: singularity

text: (as

text: well

text: as

text: by

text: choosing

text: the

text: functions

text: V

text: (R),

text: φ(R)

text: and

text: h(R)

text: in

text: the

text: the

text: singularity

text: in

text: the

text: vacuum

text: solution)

text: by

text: appropriate

text: action

text: so

text: that

text: the

text: factors

text: containing

text: j

text: φ

text: are

text: not

text: sin-

text: choice

text: of

text: coeﬃcients

text: in

text: the

text: action

text: without

text: violating

text: the

text: gular

text: at

text: the

text: origin.

text: Note

text: that

text: this

text: choice

text: will

text: not

text: violate

text: Bianchi

text: identities

text: since

text: our

text: equations

text: of

text: motion

text: were

text: de-

text: the

text: Bianchi

text: identities

text: since

text: we

text: derived

text: our

text: equations

text: of

text: rived

text: from

text: a

text: variational

text: principal.

text: We

text: then

text: performed

text: motion

text: from

text: a

text: variational

text: principal.

text: a

text: detailed

text: Hamiltonian

text: analysis,

text: including

text: a

text: prescription

text: With

text: the

text: gauge

text: choices,

text: and

text: the

text: equations

text: of

text: of

text: the

text: boundary

text: conditions

text: and

text: corresponding

text: boundary

text: motion

text: for

text: the

text: remaining

text: phase

text: space

text: variables

text: can

text: now

text: terms

text: needed

text: to

text: make

text: the

text: variational

text: principle

text: well

text: de-

text: be

text: written

text: as

text: ﬁned.

text: From

text: this

text: Hamiltonian

text: we

text: imposed

text: suitable

text: gauge

text: ﬁxing

text: conditions

text: and

text: derived

text: the

text: equations

text: of

text: motion

text: in

text: a

text: family

text: of

text: non-null

text: coordinates.

text: The

text: fact

text: that

text: these

text: are

text: P

text: ψ˙

text: N

text: ψ

text: N

text: ψ

text: ﬁrst

text: order

text: in

text: time

text: derivatives

text: makes

text: them

text: well

text: suited

text: to

text: r

text: numerical

text: simulations

text: of

text: black

text: hole

text: formation.

text: We

text: then

text: chose

text: a

text: gauge

text: such

text: that

text: the

text: metric

text: is

text: well

text: deﬁned

text: at

text: hori-

text: zon

text: formation,

text: which

text: is

text: well

text: suited

text: to

text: the

text: investigation

text: of

text: the

text: dynamics

text: past

text: horizon

text: formation.

text: P˙ψ

text: NrPψ)

text: At

text: ﬁrst

text: glance

text: the

text: Lagrangian

text: that

text: is

text: our

text: start-

text: ing

text: point

text: contains

text: many

text: arbitrary

text: functions.

text: However,

text: as

text: we

text: saw,

text: the

text: requirement

text: that

text: singularities

text: be

text: removed

text: is

text: quite

text: restrictive.

text: As

text: shown

text: in

text: Section

text: in

text: order

text: for

text: the

text: eﬀective

text: stress

text: energy

text: tensor

text: to

text: be

text: regular

text: it

text: is

text: necessary

text: that

text: j

text: ω2

text: at

text: R

text: In

text: addition,

text: must

text: go

text: to

text: zero

text: at

text: the

text: origin

text: at

text: least

text: as

text: fast

text: as

text: as

text: can

text: be

text: seen

text: from

text: Finally,

text: the

text: remaining

text: freedom

text: in

text: j,

text: φ

text: and

text: V

text: can

text: be

text: ﬁxed

text: so

text: that

text: the

text: vacuum

text: solution

text: approaches

text: the

text: form

text: derived

text: via

text: general

text: arguments

text: in

text: This

text: leaves

text: P˙

text: W(Nz

text: N

text: P

text: r

text: the

text: freedom

text: in

text: the

text: matter

text: coupling

text: B(R).

text: The

text: simplest

text: NWb(R

text: choice

text: is

text: B(R)

text: Rn

text: i.e.

text: the

text: classical

text: form

text: obtained

text: ,y

text: by

text: dimensional

text: reduction

text: which,

text: as

text: shown

text: in

text: yields

text: non-singular

text: collapse.

text: b(R)

text: and

text: c(R)

text: are

text: normally

text: taken

text: to

text: be

text: zero

text: in

text: two

text: dimensional

text: models,

text: but,

text: when

text: the

text: cou-

text: pling

text: is

text: not

text: conformal

text: these

text: functions

text: are

text: determined

text: by

text: NP

text: z˙

text: NS

text: N

text: z

text: our

text: choice

text: of

text: B(R).

text: It

text: would

text: be

text: interesting

text: to

text: see

text: what

text: W

text: r

text: eﬀect

text: these

text: terms

text: have

text: on

text: the

text: structure

text: of

text: the

text: evaporat-

text: ing,

text: non-singular

text: black

text: hole

text: spacetime.

text: In

text: an

text: upcoming

text: and

text: paper

text: we

text: will

text: present

text: the

text: results

text: of

text: numerical

text: simulations

text: using

text: these

text: equations.

text: VII.

text: ACKNOWLEDGEMENTS

text: P˙

text: W(Nz

text: N

text: P

text: r

text: We

text: thank

text: Nils

text: Deppe

text: for

text: useful

text: discussions.

text: This

text: work

text: has

text: been

text: funded

text: by

text: the

text: Fondecyt

text: grant

text: These

text: equations

text: of

text: motion,

text: The

text: Centro

text: de

text: Estudios

text: Cient´ıﬁcos

text: (CECs)

text: is

text: funded

text: by

text: along

text: with

text: the

text: consistency

text: conditions,

text: the

text: Chilean

text: Government

text: through

text: the

text: Centers

text: of

text: Excel-

text: the

text: constraints,

text: and

text: the

text: deﬁnitions,

text: lence

text: Base

text: Financing

text: Program

text: of

text: Conicyt.

text: This

text: work

text: can

text: be

text: used

text: to

text: evolve

text: appropriate

text: was

text: also

text: funded

text: in

text: part

text: by

text: the

text: Natural

text: Sciences

text: and

text: initial

text: conditions

text: forward

text: in

text: time

text: to

text: show

text: the

text: formation

text: Engineering

text: Research

text: Council

text: of

text: Canada.

text: Support

text: was

text: of

text: a

text: black

text: hole

text: with

text: a

text: radiating

text: term.

text: also

text: provided

text: by

text: WestGrid

text: (www.westgrid.ca),

text: Compute

text: VI.

text: CONCLUSIONS

text: Canada/Calcul

text: Canada

text: (www.computecanada.ca),

text: and

text: the

text: Perimeter

text: Institute

text: for

text: Theoretical

text: Physics

text: (funded

text: In

text: this

text: work

text: we

text: deﬁned

text: an

text: action

text: which

text: mimics

text: spher-

text: by

text: Industry

text: Canada

text: and

text: the

text: Province

text: of

text: Ontario

text: Min-

text: ically

text: symmetric,

text: dimensionally

text: reduced

text: gravity

text: with

text: a

text: istry

text: of

text: Research

text: and

text: Innovation).

text: radiating

text: term.

text: Although

text: the

text: radiating

text: term

text: was

text: bor-

text: Appendix

text: A:

text: Loop

text: Quantum

text: Gravity

text: Black

text: Hole

text: rowed

text: from

text: the

text: conformal

text: anomaly

text: and

text: is

text: only

text: valid

text: in

text: two

text: dimensional

text: physics

text: we

text: incorporated

text: terms

text: which

text: ac-

text: count

text: for

text: the

text: non-minimal

text: coupling

text: (in

text: of

text: the

text: mat-

text: An

text: example

text: of

text: a

text: non-singular,

text: single

text: horizon

text: black

text: hole

text: ter

text: ﬁeld.

text: We

text: derived

text: Lagrange’s

text: equations

text: in

text: gauge

text: in

text: four

text: dimensions

text: that

text: can

text: be

text: derived

text: as

text: a

text: solution

text: to

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 7 (cid:112) − − − G(n) = (n 2)(−R + 2f R ) (47) (DR)2 = Ry2 + Λ 2R,x2, (53) vv R ,vv ,v ,v − − − g D2R = ∂ (ΛR ) + ∂ (ΛN R + Λ 1NR ), (2) t ,y x r ,y ,x (cid:20) (cid:18) (cid:19)(cid:21) (54) − − (n 2) − (n 3) e2f − where we deﬁne the operator, , y acting on some ﬁeld, G(n) = R R R uv R ,uv R 4 ,u ,v β by (48) − − − β := N 1(β N β ). (55) ,y ,t r ,x Concentrating on the u v component, using the equa- tion of motion (36) to solve for R and the expression ,uv for the mass (B8) (cid:18) (cid:19) Using the metric (52) the Ricci scalar can be written as jω2 − M (DR)2 = (cid:48) 1 (49) √ φ 2 j − R { − − − − g = 2 Nr,xN,yΛN 1 2Nr,xΛ,y Λ(Nr,x),y = 4e 2fR R (50) − − − } ,u ,v + N 1Λ(N )2 + NΛ (N Λ 1) (56) r,x ,yy ,x ,x − (where a factor of ln 2 has been absorbed into M) gives To write the action in a form that lends itself to Hamil- the following expression for the eﬀective stress tensor (cid:40) (cid:32) (cid:33) tonian analysis we deﬁne − − (cid:48) G(unv) = (nφ(cid:34)˜(cid:48)R2) e42f Mj(cid:18) φ˜(cid:48)(cid:48) −(cid:18)(n R3)φ˜(cid:19)(cid:19) (cid:35)(cid:41) Φ(R(t, x), t, x) := φ(R) + W(z1 + z2) + Wc(R) = φ˜(R) + W(z + z ) (57) 1 2 − e2f (cid:48)(cid:48) − M + W 4f c + b 1 . (51) ,uv 4 j and In the above we have assumed only that (9) is satisﬁed, (cid:48) which also implies that j = φ = ω2. This is necessary − h (R(t, x), z (t, x)) := h(R) + Wb(R)(z ln µ2) (58) in order to eliminate terms in the stress energy tensor z 1 1 that are singular at R = 0. These conditions are sat- isﬁed for the two horizon quantum corrected black hole described by (15)– (19). Assuming reasonable regularity With thes(cid:90)e deﬁnitions the action looks like conditions on the mass function and c and b at the origin, the singularity observed by Piran and Ayal in the eﬀec- √ tive stress tensor is resolved as long as the singularity in I(cid:110)= 1− d2x −g× (cid:111) 1/j is resolved. Similar statements apply to the diagonal ln 2 components of the energy momentum tensor as we can R Φ + h (DR)2 + V + WD z DAz + B(Dψ)2 see by comparing (47) to (42). z A 2 1 (59) V. HAMILTON’S EQUATIONS IN NON–NULL CO–ORDINATES We ﬁrst work with the term containing the Ricci scalar. From (56) we can calculate this term up to total deriva- To ﬁnd the equations of motion in non-null co–ordinates. tives (t.d.) to be We start with the general, ADM metric [44], √ − R − − − ds2 = −N2dt2 + Λ2(N dt + dx)2. (52) g Φ = 2Φ,y(Nr,xΛ NΛ,y) 2N(Φ,xΛ 1),x + t.d. r (60) For later use, we compute the following quantities: Using this the action (59) becomes ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

label: equation

text: G(n)

text: (n

text: R

text: Λ

text: vv

text: R

text: ,vv

text: ,v

text: ,v

text: g

text: (ΛR

text: (ΛN

text: R

text: Λ

text: t

text: ,y

text: x

text: r

text: ,y

text: ,x

text: (n

text: (n

text: where

text: we

text: deﬁne

text: the

text: operator,

text: y

text: acting

text: on

text: some

text: ﬁeld,

text: G(n)

text: R

text: R

text: R

text: uv

text: R

text: ,uv

text: R

text: ,u

text: ,v

text: β

text: by

text: β

text: N

text: N

text: β

text: ,y

text: ,t

text: r

text: ,x

text: Concentrating

text: on

text: the

text: u

text: v

text: component,

text: using

text: the

text: equa-

text: tion

text: of

text: motion

text: to

text: solve

text: for

text: R

text: and

text: the

text: expression

text: ,uv

text: for

text: the

text: mass

text: Using

text: the

text: metric

text: the

text: Ricci

text: scalar

text: can

text: be

text: written

text: as

text: M

text: φ

text: j

text: R

text: g

text: Nr,xN,yΛN

text: Λ(Nr,x),y

text: R

text: ,u

text: ,v

text: N

text: NΛ

text: (N

text: Λ

text: r,x

text: ,yy

text: ,x

text: ,x

text: (where

text: a

text: factor

text: of

text: ln

text: has

text: been

text: absorbed

text: into

text: M)

text: gives

text: To

text: write

text: the

text: action

text: in

text: a

text: form

text: that

text: lends

text: itself

text: to

text: Hamil-

text: the

text: following

text: expression

text: for

text: the

text: eﬀective

text: stress

text: tensor

text: tonian

text: analysis

text: we

text: deﬁne

text: G(unv)

text: Φ(R(t,

text: x),

text: t,

text: x)

text: φ(R)

text: Wc(R)

text: φ˜(R)

text: W(z

text: z

text: M

text: W

text: c

text: b

text: ,uv

text: j

text: and

text: In

text: the

text: above

text: we

text: have

text: assumed

text: only

text: that

text: is

text: satisﬁed,

text: which

text: also

text: implies

text: that

text: j

text: φ

text: This

text: is

text: necessary

text: h

text: (R(t,

text: x),

text: z

text: (t,

text: x))

text: h(R)

text: Wb(R)(z

text: ln

text: in

text: order

text: to

text: eliminate

text: terms

text: in

text: the

text: stress

text: energy

text: tensor

text: z

text: that

text: are

text: singular

text: at

text: R

text: These

text: conditions

text: are

text: sat-

text: isﬁed

text: for

text: the

text: two

text: horizon

text: quantum

text: corrected

text: black

text: hole

text: described

text: by

text: Assuming

text: reasonable

text: regularity

text: With

text: deﬁnitions

text: the

text: action

text: looks

text: like

text: conditions

text: on

text: the

text: mass

text: function

text: and

text: c

text: and

text: b

text: at

text: the

text: origin,

text: the

text: singularity

text: observed

text: by

text: Piran

text: and

text: Ayal

text: in

text: the

text: eﬀec-

text: tive

text: stress

text: tensor

text: is

text: resolved

text: as

text: long

text: as

text: the

text: singularity

text: in

text: 1−

text: −g×

text: is

text: resolved.

text: Similar

text: statements

text: apply

text: to

text: the

text: diagonal

text: ln

text: components

text: of

text: the

text: energy

text: momentum

text: tensor

text: as

text: we

text: can

text: R

text: Φ

text: h

text: V

text: WD

text: z

text: DAz

text: see

text: by

text: comparing

text: to

text: z

text: A

text: V.

text: HAMILTON’S

text: EQUATIONS

text: IN

text: NON–NULL

text: CO–ORDINATES

text: We

text: ﬁrst

text: work

text: with

text: the

text: term

text: containing

text: the

text: Ricci

text: scalar.

text: From

text: we

text: can

text: calculate

text: this

text: term

text: up

text: to

text: total

text: deriva-

text: To

text: ﬁnd

text: the

text: equations

text: of

text: motion

text: in

text: non-null

text: co–ordinates.

text: tives

text: (t.d.)

text: to

text: be

text: We

text: start

text: with

text: the

text: general,

text: ADM

text: metric

text: R

text: dt

text: g

text: Φ

text: NΛ,y)

text: t.d.

text: r

text: For

text: later

text: use,

text: we

text: compute

text: the

text: following

text: quantities:

text: Using

text: this

text: the

text: action

text: becomes

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

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text: 10 ALESSIO FIGALLI A´LVARO PELAYO deﬁned by the quotient P P vol ( ) ∈ euc Ω( ) := [0, 1], (cid:110) voleuc(∆) (cid:111) and the density of ∆ is deﬁned by P | P Ω(∆) := sup Ω( ) is an admissible packing of ∆ . An optimal packing (also called a maximal packing) is an admissible ball packing at which this density is achieved. The optimal density function assigns to a Delzant polytope the density of one of its optimal packings (such packing always exists, see [16]). Figure 4 shows an example of an admissible packing and a non admissible one; at the manifold level, the admissible one corresponds to a symplectic CP ×CP toric ball packing of 1 1 by two disjoint balls, while the non admissible one cannot be interpreted in this way because the shaded triangle does not correspond to an equivariantly and symplectially embedded ball. Both packings in Figure 5 are admissible. (0,2) (1,2) (0,2) (1,2) (2,1) (2,1) (0,0) (2,0) (0,0) (2,0) Figure 5. Admissible packings of the polygon of a b√low up of the product of two symplectic spheres of radius 1/ 2. ⊂ R Let I n be an interval with rational slope. The rational-length Z lengthQ(I) of I is the (unique) number (cid:96) such that I is AGL(n, )-congruent to a length (cid:96) interval on a coordinate axis. As it is shown by the following result, an admissible simplex is parametrized by its center and its radius. ∈ Lemma 6.1 ([13]). Let ∆ be a Delzant polytope and let v ∆ be a vertex of ∆. We denote the n edg(cid:110)es leaving v by e1, . . . , en. Th(cid:111)en: v v (i) r is given by v r = min lengthQ(e1), . . . , lengthQ(en) . v v v (ii) There exists an admissible simplex Σ(v, r) centered at v of radius ≤ ≤ r if and only if we have that 0 r r . Moreover Σ(v, r) is the v unique such admissible simplex. ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: ALESSIO

text: FIGALLI

text: A´LVARO

text: PELAYO

text: deﬁned

text: by

text: the

text: quotient

text: P

text: P

text: vol

text: euc

text: Ω(

text: voleuc(∆)

text: and

text: the

text: density

text: of

text: is

text: deﬁned

text: by

text: P

text: P

text: Ω(∆)

text: sup

text: Ω(

text: is

text: an

text: admissible

text: packing

text: of

text: An

text: optimal

text: packing

text: (also

text: called

text: a

text: maximal

text: packing)

text: is

text: an

text: admissible

text: ball

text: packing

text: at

text: which

text: this

text: density

text: is

text: achieved.

text: The

text: optimal

text: density

text: function

text: assigns

text: to

text: a

text: Delzant

text: polytope

text: the

text: density

text: of

text: one

text: of

text: its

text: optimal

text: packings

text: (such

text: packing

text: always

text: exists,

text: see

text: Figure

text: shows

text: an

text: example

text: of

text: an

text: admissible

text: packing

text: and

text: a

text: non

text: admissible

text: one;

text: at

text: the

text: manifold

text: level,

text: the

text: admissible

text: one

text: corresponds

text: to

text: a

text: symplectic

text: CP

text: ×CP

text: toric

text: ball

text: packing

text: of

text: by

text: two

text: disjoint

text: balls,

text: while

text: the

text: non

text: admissible

text: one

text: cannot

text: be

text: interpreted

text: in

text: this

text: way

text: because

text: the

text: shaded

text: triangle

text: does

text: not

text: correspond

text: to

text: an

text: equivariantly

text: and

text: symplectially

text: embedded

text: ball.

text: Both

text: packings

text: in

text: Figure

text: are

text: admissible.

text: Figure

text: Admissible

text: packings

text: of

text: the

text: polygon

text: of

text: a

text: b√low

text: up

text: of

text: the

text: product

text: of

text: two

text: symplectic

text: spheres

text: of

text: radius

text: R

text: Let

text: I

text: n

text: be

text: an

text: interval

text: with

text: rational

text: slope.

text: The

text: rational-length

text: Z

text: lengthQ(I)

text: of

text: I

text: is

text: the

text: (unique)

text: number

text: such

text: that

text: I

text: is

text: AGL(n,

text: )-congruent

text: to

text: a

text: length

text: interval

text: on

text: a

text: coordinate

text: axis.

text: As

text: it

text: is

text: shown

text: by

text: the

text: following

text: result,

text: an

text: admissible

text: simplex

text: is

text: parametrized

text: by

text: its

text: center

text: and

text: its

text: radius.

text: Lemma

text: Let

text: be

text: a

text: Delzant

text: polytope

text: and

text: let

text: v

text: be

text: a

text: vertex

text: of

text: We

text: denote

text: the

text: n

text: leaving

text: v

text: by

text: en.

text: v

text: v

text: (i)

text: r

text: is

text: given

text: by

text: v

text: r

text: min

text: lengthQ(en)

text: v

text: v

text: v

text: (ii)

text: There

text: exists

text: an

text: admissible

text: simplex

text: Σ(v,

text: r)

text: centered

text: at

text: v

text: of

text: radius

text: r

text: if

text: and

text: only

text: if

text: we

text: have

text: that

text: r

text: r

text: Moreover

text: Σ(v,

text: r)

text: is

text: the

text: v

text: unique

text: such

text: admissible

text: simplex.

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: Algorithm 1 captioncolorSimple Coding Scheme for Θ(log n)-bit alphabet Σ 321... ← 1: Π n-round protocol to be simulated + ﬁnal conﬁrmation steps ← (cid:6)(cid:112) (cid:7) 2: hash hash family from Lemma 4.1 with p = 1/n5 and o = s = Θ(log n) ← ← ← (cid:100) (cid:101) ← ∅ 3: Initialize Parameters: rc Θ(1); r r(cid:15)c ; Rtotal n/r + 32n(cid:15) ; T 4: for R = 1 to R do total ← 5: S s uniformly random bits (cid:46) Veriﬁcation Phase | | (cid:48) (cid:48) (cid:48) 6: Send (S, hash (T), T ); Receive (S , H , l ) ← S ← | | T 7: H hash (cid:48)(T); l T T S (cid:48) 8: if H = H then (cid:46) Computation Phase T T 9: continue computation of Π for r communications and record those in T 10: else 11: do r dummy communications keeping T unchanged (cid:54) (cid:48) ≥ (cid:48) 12: if H = H and l l then (cid:46) Transition Phase T T 13: rollback computation of Π and transcript T by r steps 14: Output the outcome of Π corresponding to transcript T 4.3 Proof of Correctness Theorem 4.2. Suppose any n-round protocol Π using an alphabet Σ of bit-size Θ(log n). Algorithm − 1 is a computationally eﬃcient randomized coding scheme which given Π, with probability 1 − 2 Θ(n(cid:15)), robustly simulates it over an√y fully adversarial error channel with alphabet Σ and error rate (cid:15).√The simulation uses n(1 + Θ( (cid:15))) rounds and therefore achieves a communication rate of − 1 Θ( (cid:15)). Proof Outline We show that the algorithm terminates correctly by deﬁning an appropriate potential Φ. We prove that any iteration without an error or hash collision increases the potential by at least one while any error or hash collision reduces the potential by at most some ﬁxed constant. Lastly, we show that with very high probability the number of hash collisions is at most O((cid:15)n) and therefore negligible. This guarantees an overall potential increase that suﬃces to show that the algorithm terminates correctly after the ﬁxed R number of iterations. total Potential The potential Φ is based on the transcript T of both parties. We use TA and TB to denote the transcript of Alice and Bob respectively. We ﬁrst deﬁne the following intermediate quantities: The agreem(cid:22)ent l+ between the two transcripts at Alice and Bob is th(cid:23)e number of blocks (cid:8) (cid:9) of length r in which they agree, that is, 1 (cid:48) ∈ {| | | |} (cid:48) (cid:48) l+ = max l [1, min TA , TB ] s.th. TA[1, l ] = TB[1, l ] . r 13 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Algorithm

text: captioncolorSimple

text: Coding

text: Scheme

text: for

text: Θ(log

text: n)-bit

text: alphabet

text: Σ

text: Π

text: n-round

text: protocol

text: to

text: be

text: simulated

text: ﬁnal

text: conﬁrmation

text: steps

text: hash

text: hash

text: family

text: from

text: Lemma

text: with

text: p

text: and

text: o

text: s

text: Θ(log

text: n)

text: Initialize

text: Parameters:

text: rc

text: r

text: Rtotal

text: n/r

text: T

text: for

text: R

text: to

text: R

text: do

text: total

text: S

text: s

text: uniformly

text: random

text: bits

text: Veriﬁcation

text: Phase

text: Send

text: (S,

text: hash

text: (T),

text: T

text: Receive

text: (S

text: H

text: l

text: S

text: T

text: H

text: hash

text: l

text: T

text: T

text: S

text: if

text: H

text: H

text: then

text: Computation

text: Phase

text: T

text: T

text: continue

text: computation

text: of

text: Π

text: for

text: r

text: communications

text: and

text: record

text: those

text: in

text: T

text: else

text: do

text: r

text: dummy

text: communications

text: keeping

text: T

text: unchanged

text: if

text: H

text: H

text: and

text: l

text: l

text: then

text: Transition

text: Phase

text: T

text: T

text: rollback

text: computation

text: of

text: Π

text: and

text: transcript

text: T

text: by

text: r

text: steps

text: Output

text: the

text: outcome

text: of

text: Π

text: corresponding

text: to

text: transcript

text: T

text: Proof

text: of

text: Correctness

text: Theorem

text: Suppose

text: any

text: n-round

text: protocol

text: Π

text: using

text: an

text: alphabet

text: Σ

text: of

text: bit-size

text: Θ(log

text: n).

text: Algorithm

text: is

text: a

text: computationally

text: eﬃcient

text: randomized

text: coding

text: scheme

text: which

text: given

text: Π,

text: with

text: probability

text: robustly

text: simulates

text: it

text: over

text: an√y

text: fully

text: adversarial

text: error

text: channel

text: with

text: alphabet

text: Σ

text: and

text: error

text: rate

text: simulation

text: uses

text: Θ(

text: rounds

text: and

text: therefore

text: achieves

text: a

text: communication

text: rate

text: of

text: Θ(

text: Proof

text: Outline

text: We

text: show

text: that

text: the

text: algorithm

text: terminates

text: correctly

text: by

text: deﬁning

text: an

text: appropriate

text: potential

text: Φ.

text: We

text: prove

text: that

text: any

text: iteration

text: without

text: an

text: error

text: or

text: hash

text: collision

text: increases

text: the

text: potential

text: by

text: at

text: least

text: one

text: while

text: any

text: error

text: or

text: hash

text: collision

text: reduces

text: the

text: potential

text: by

text: at

text: most

text: some

text: ﬁxed

text: constant.

text: Lastly,

text: we

text: show

text: that

text: with

text: very

text: high

text: probability

text: the

text: number

text: of

text: hash

text: collisions

text: is

text: at

text: most

text: and

text: therefore

text: negligible.

text: This

text: guarantees

text: an

text: overall

text: potential

text: increase

text: that

text: suﬃces

text: to

text: show

text: that

text: the

text: algorithm

text: terminates

text: correctly

text: after

text: the

text: ﬁxed

text: R

text: number

text: of

text: iterations.

text: total

text: Potential

text: The

text: potential

text: Φ

text: is

text: based

text: on

text: the

text: transcript

text: T

text: of

text: both

text: parties.

text: We

text: use

text: TA

text: and

text: TB

text: to

text: denote

text: the

text: transcript

text: of

text: Alice

text: and

text: Bob

text: respectively.

text: We

text: ﬁrst

text: deﬁne

text: the

text: following

text: intermediate

text: quantities:

text: The

text: l+

text: between

text: the

text: two

text: transcripts

text: at

text: Alice

text: and

text: Bob

text: is

text: number

text: of

text: blocks

text: of

text: length

text: r

text: in

text: which

text: they

text: agree,

text: that

text: is,

text: l+

text: max

text: l

text: min

text: TA

text: TB

text: s.th.

text: l

text: l

text: r

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##