text: ∈ { } · · · · · · A for all 1 , where = and = and the are in . We now i , . . . , n u x1 x v y1 y x , y i n i ‘ j j prove that the PCP instance has a solution if and only if the preﬁx-PCP instance has a solution. · · · · · · · · · Suppose that there exists a solution to the PCP instance, that is = . i1 i u u v v Then clearly the preﬁx-PCP also has a solmution, given by · · · i=1 im · · · i1 . im A U U Y B V V Z Conversely, suppose that the preﬁx-PCP instance has a soi1lutii2on. Biymconstruic1tiio2n, becimause of · · · · · · # and *, there must exist a preﬁx-PCP solution of the form = , where i1 · · · i is a preﬁx of i1 · · · i 0 or vice-versa. But the pAaii1rUs i(2U , VU)imdYo noBt ic1oVnit2ain Vainmy0Z#, so the pair (m ) is used exactly omnce, both after th pair and thei 0ith pair, so = 0 and Y, Z m m m m the PCP instance has a solution. 2.3 Undecidable properties of multi-tape automata M A M Let be a -tape automaton on alphabet , and let be a state of . State is universal d q q if every sequence in AN is q-accepted by M. A ﬁnite sequence x ∈ A? is a universal preﬁx for state if for every inﬁnite sequence ∈ AN, the inﬁnite sequence is -accepted by M. q y xy q M { } Example 2.4. Let be a 1-tape, 1-state automaton on alphabet 0 1 with three transitions , M labelled by 1, 10 and 00. The single state of is not universal because every sequence starting with 01 is rejected, but the word 1 is a universal preﬁx: any sequence starting with 1 is accepted, × × because any ﬁnite segment 10n1 is accepted by transitions 1, 00 k, 1 if n = 2k or 10, 00 k, 1 if = 2 + 1, and any inﬁnite tail of 0’s or 1’s is obviously accepted. Hence there exist some n k multi-tape automata without universal states but that admit universal preﬁxes. The self-aﬃne set associated with this automaton is discussed in Example 3.5. Theorem 2.5. It is undecidable whether a given state of a given -tape automaton is universal. d This problem remains undecidable if we restrict to 2-tape automata with 3 states. Proof. We reduce preﬁx-PCP, which is undecidable thanks to Lemma 2.3. Let ( ), . . . , u1, v1 B ( ) be an instance of preﬁx-PCP where the , are words over . We deﬁne a 2- u , v u v n n M i i M × tape automaton on 3 states (denoted by ). The alphabet of is , with X, U, V A1 A2 { } B ∪ { } B = 1 and = # , where is the size of the preﬁx-PCP instance, is the A1 , . . . , n A2 n B M alphabet of words and # is a new symbol not in . The transitions of are u , v i i −|→ −|→ ∈ (1) X i ui U and U i ui U for every i A1; −|→ −|→ ∈ (2) X i vi V and V i vi V for every i A1; (3) −i|→u for every ∈ and ∈ + such that U X i A1 u A2 (i) | | (cid:54) | |, u u i (ii) is not a preﬁx of , u u i (iii) does not begin with #; u (4) −i|→u for every ∈ and ∈ + such that (i) and (ii) above hold; X X i A1 u A2 (5) −i→|v for every ∈ and ∈ + such that V X i A1 v A2 (i) | | (cid:54) | |, v v i (ii) is not a preﬁx of , v v i (iii) does not begin with #; v (6) −i→|v for every ∈ and ∈ + such that (i) and (ii) above hold. X X i A1 v A2 We now prove that the preﬁx-PCP instance ( ) ( ) has a solution if and only if u1, v1 , . . . , u , v M n n state is not universal in . X (⇒) Suppose that the preﬁx-PCP instance admits a solution: there exist 0 (cid:62) 1 and two m, m · · · · · · · · · · · · · · · words and such that = and one of the two words and i·1· · imis a pir1eﬁx iomf0the other. uWi1ithouuitm lossvio1f gevniemr0ality we can assume that i1 (cid:62) im0 i1 i 0 m m m 4 ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: A

text: for

text: all

text: where

text: and

text: and

text: the

text: are

text: in

text: We

text: now

text: i

text: n

text: u

text: x

text: v

text: y

text: x

text: y

text: i

text: n

text: i

text: j

text: j

text: prove

text: that

text: the

text: PCP

text: instance

text: has

text: a

text: solution

text: if

text: and

text: only

text: if

text: the

text: preﬁx-PCP

text: instance

text: has

text: a

text: solution.

text: Suppose

text: that

text: there

text: exists

text: a

text: solution

text: to

text: the

text: PCP

text: instance,

text: that

text: is

text: i

text: u

text: u

text: v

text: v

text: Then

text: clearly

text: the

text: preﬁx-PCP

text: also

text: has

text: a

text: solmution,

text: given

text: by

text: im

text: im

text: A

text: U

text: U

text: Y

text: B

text: V

text: V

text: Z

text: Conversely,

text: suppose

text: that

text: the

text: preﬁx-PCP

text: instance

text: has

text: a

text: becimause

text: of

text: and

text: there

text: must

text: exist

text: a

text: preﬁx-PCP

text: solution

text: of

text: the

text: form

text: where

text: i

text: is

text: a

text: preﬁx

text: of

text: i

text: or

text: vice-versa.

text: But

text: the

text: VU)imdYo

text: noBt

text: so

text: the

text: pair

text: (m

text: is

text: used

text: exactly

text: omnce,

text: both

text: after

text: th

text: pair

text: and

text: thei

text: pair,

text: so

text: and

text: Y,

text: Z

text: m

text: m

text: m

text: m

text: the

text: PCP

text: instance

text: has

text: a

text: solution.

text: Undecidable

text: properties

text: of

text: multi-tape

text: automata

text: M

text: A

text: M

text: Let

text: be

text: a

text: -tape

text: automaton

text: on

text: alphabet

text: and

text: let

text: be

text: a

text: state

text: of

text: State

text: is

text: universal

text: d

text: q

text: q

text: if

text: every

text: sequence

text: in

text: AN

text: is

text: q-accepted

text: by

text: M.

text: A

text: ﬁnite

text: sequence

text: x

text: A?

text: is

text: a

text: universal

text: preﬁx

text: for

text: state

text: if

text: for

text: every

text: inﬁnite

text: sequence

text: AN,

text: the

text: inﬁnite

text: sequence

text: is

text: -accepted

text: by

text: M.

text: q

text: y

text: xy

text: q

text: M

text: Example

text: Let

text: be

text: a

text: automaton

text: on

text: alphabet

text: with

text: three

text: transitions

text: M

text: labelled

text: by

text: and

text: The

text: single

text: state

text: of

text: is

text: not

text: universal

text: because

text: every

text: sequence

text: starting

text: with

text: is

text: rejected,

text: but

text: the

text: word

text: is

text: a

text: universal

text: preﬁx:

text: any

text: sequence

text: starting

text: with

text: is

text: accepted,

text: because

text: any

text: ﬁnite

text: segment

text: is

text: accepted

text: by

text: transitions

text: k,

text: if

text: n

text: or

text: k,

text: if

text: and

text: any

text: inﬁnite

text: tail

text: of

text: or

text: is

text: obviously

text: accepted.

text: Hence

text: there

text: exist

text: some

text: n

text: k

text: multi-tape

text: automata

text: without

text: universal

text: states

text: but

text: that

text: admit

text: universal

text: preﬁxes.

text: The

text: self-aﬃne

text: set

text: associated

text: with

text: this

text: automaton

text: is

text: discussed

text: in

text: Example

text: Theorem

text: It

text: is

text: undecidable

text: whether

text: a

text: given

text: state

text: of

text: a

text: given

text: -tape

text: automaton

text: is

text: universal.

text: d

text: This

text: problem

text: remains

text: undecidable

text: if

text: we

text: restrict

text: to

text: automata

text: with

text: states.

text: Proof.

text: We

text: reduce

text: preﬁx-PCP,

text: which

text: is

text: undecidable

text: thanks

text: to

text: Lemma

text: Let

text: B

text: be

text: an

text: instance

text: of

text: preﬁx-PCP

text: where

text: the

text: are

text: words

text: over

text: We

text: deﬁne

text: a

text: u

text: v

text: u

text: v

text: n

text: n

text: M

text: i

text: i

text: M

text: tape

text: automaton

text: on

text: states

text: (denoted

text: by

text: The

text: alphabet

text: of

text: is

text: with

text: X,

text: U,

text: V

text: B

text: B

text: and

text: where

text: is

text: the

text: size

text: of

text: the

text: preﬁx-PCP

text: instance,

text: is

text: the

text: n

text: n

text: B

text: M

text: alphabet

text: of

text: words

text: and

text: is

text: a

text: new

text: symbol

text: not

text: in

text: The

text: transitions

text: of

text: are

text: u

text: v

text: i

text: i

text: X

text: i

text: ui

text: U

text: and

text: U

text: i

text: ui

text: U

text: for

text: every

text: i

text: X

text: i

text: vi

text: V

text: and

text: V

text: i

text: vi

text: V

text: for

text: every

text: i

text: −i|→u

text: for

text: every

text: and

text: such

text: that

text: U

text: X

text: i

text: u

text: (i)

text: u

text: u

text: i

text: (ii)

text: is

text: not

text: a

text: preﬁx

text: of

text: u

text: u

text: i

text: (iii)

text: does

text: not

text: begin

text: with

text: u

text: −i|→u

text: for

text: every

text: and

text: such

text: that

text: (i)

text: and

text: (ii)

text: above

text: hold;

text: X

text: X

text: i

text: u

text: −i→|v

text: for

text: every

text: and

text: such

text: that

text: V

text: X

text: i

text: v

text: (i)

text: v

text: v

text: i

text: (ii)

text: is

text: not

text: a

text: preﬁx

text: of

text: v

text: v

text: i

text: (iii)

text: does

text: not

text: begin

text: with

text: v

text: −i→|v

text: for

text: every

text: and

text: such

text: that

text: (i)

text: and

text: (ii)

text: above

text: hold.

text: X

text: X

text: i

text: v

text: We

text: now

text: prove

text: that

text: the

text: preﬁx-PCP

text: instance

text: has

text: a

text: solution

text: if

text: and

text: only

text: if

text: u

text: v

text: M

text: n

text: n

text: state

text: is

text: not

text: universal

text: in

text: X

text: Suppose

text: that

text: the

text: preﬁx-PCP

text: instance

text: admits

text: a

text: solution:

text: there

text: exist

text: and

text: two

text: m,

text: m

text: words

text: and

text: such

text: that

text: and

text: one

text: of

text: the

text: two

text: words

text: and

text: i·1·

text: imis

text: a

text: other.

text: we

text: can

text: assume

text: that

text: i

text: m

text: m

text: m

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: • (1) and (2) like in the proof of Theorem 2.5, without allowing any symbol & or *; • (3), (4), (5), (6) like in the proof of Theorem 2.5, where symbols & or * are allowed, except in the ﬁrst letter of or ; u v plus the following transitions: (7) −a→|& , −a→|& and −a→|& for every ∈ ; X X U X V X a A1 (8) −&→|a , −&→|a and −&→|a for every ∈ \ {*}; X X U X V X a A2 (9) −a|→*b , −a|→*b and −a|→*b for every ∈ and ∈ . X X U X V X a A1 b A2 We now prove that the preﬁx-PCP instance ( ) ( ) has a solution if and only if u1, v1 , . . . , u , v n n state does not have any universal preﬁx. X (⇒) Suppose that the preﬁx-PCP instance has a solution: there exist 0 (cid:62) 1 and two m, m · · · · · · · · · · · · · · · words and such that = and one of the two words and i·1· · imis a prie1ﬁx oimf t0 he other. Cuoi1nsideurimthe fvoi1llowivnigm0claim. i1 im i1 i 0 m ∈ ∈ · · · | · · · Claim. Let x A?1 and y A?2 be such that x&& y&& is X-accepted by at most k (cid:62) 1 diﬀerent runs of M. Then there exist x0 ∈ A?1 and y0 ∈ A?2 such that 0&& · · · | 0&& · · · is -accepted by at most − 1 diﬀerent runs. xx yy X k This claim implies that does not have any universal preﬁx, i.e., that for every ﬁnite words X ∈ ∈ | x A?1 and y A?2, there exists a conﬁguration starting with x y that is not X-accepted. Indeed, for every such , there can be only ﬁnitely many diﬀerent accepting runs (say ) for x, y k · · · | · · · M | && && , because eventually loops on state with transition & &. So it suﬃces to x y X | apply the claim times to obtain a conﬁguration starting with which is not -accepted. k x y X ∈ ∈ We now prove the claim, using the preﬁx-PCP solution. Let x A?1 and y A?2 be such · · · | · · · that && && is -accepted by diﬀerent runs. Denote by the ﬁnite preﬁxes x y X k R1, . . . , R M · · · | · · · k· · · ∈ of the ·k· ·runs, each cut when reaches the && && ·p·a·rt. Let s∈= i1 im A?1 and let t =\u{1# &u*im},, twhhainckhsctaontbhee wmroitdtieﬁncaintiotnhemfoardme tto=thae1*inas2t*ance*.a|t|−1* A?2, where each ai is in A2 , , M Let be the distance between the two tapes heads when has completed the ﬁnite run ‘ . (Note that the ﬁrst head is always behind the second one because it can only move by one R1 cell at at time.) Without loss of generality we can assume that is the run for which such R1 an is minimal. We now construct a conﬁguration which will “block” any run starting with ‘ c , without giving the other runs any possibilities for new nondeterministic branching. Let R1 0 (cid:62) 0 such that (on the ﬁrst tape) begins positions behind (on the second tape) in the L, L s ‘ t conﬁguration · · · | 0 · · · c = x&Ls&& y&L t#&& , M so that during any run starting with , starts reading and # exactly at the same time. R1 s t It follows that cannot be extended to an accepting run for , because corresponds to a R1 c s, t preﬁx-PCP solution, similarly as in the proof of Theorem 2.5. The same is true for any other run for which such an is the same as . R ‘ R1 i Let us now consider another accepting run . By minimality of , the distance between the R ‘ M · · · | i· · · two tapes heads when ﬁrst reaches && && during run is strictly larger than . We R ‘ i now prove that can be extended in a unique way to an accepting run for . Indeed, any run R c M i of starting with must evolve in the following way: R i • when starts being read the second tape, is not yet being read on the ﬁrst tape, so at t s M this time is reading & on the ﬁrst tape and on the second tape; a1 • M the only possible transition is (8), so moves one step on both tapes, and is now reading * on the second tape; • M the only possible transition is (9), so moves one step on the ﬁrst tape and two steps on the second, and is again reading * on the second tape; 6

label: list

label: paragraph

text: and

text: like

text: in

text: the

text: proof

text: of

text: Theorem

text: without

text: allowing

text: any

text: symbol

text: or

text: like

text: in

text: the

text: proof

text: of

text: Theorem

text: where

text: symbols

text: or

text: are

text: allowed,

text: except

text: in

text: the

text: ﬁrst

text: letter

text: of

text: or

text: u

text: v

text: plus

text: the

text: following

text: transitions:

text: −a→|&

text: −a→|&

text: and

text: −a→|&

text: for

text: every

text: X

text: X

text: U

text: X

text: V

text: X

text: a

text: −&→|a

text: −&→|a

text: and

text: −&→|a

text: for

text: every

text: X

text: X

text: U

text: X

text: V

text: X

text: a

text: −a|→*b

text: −a|→*b

text: and

text: −a|→*b

text: for

text: every

text: and

text: X

text: X

text: U

text: X

text: V

text: X

text: a

text: b

text: We

text: now

text: prove

text: that

text: the

text: preﬁx-PCP

text: instance

text: has

text: a

text: solution

text: if

text: and

text: only

text: if

text: u

text: v

text: n

text: n

text: state

text: does

text: not

text: have

text: any

text: universal

text: preﬁx.

text: X

text: Suppose

text: that

text: the

text: preﬁx-PCP

text: instance

text: has

text: a

text: solution:

text: there

text: exist

text: and

text: two

text: m,

text: m

text: words

text: and

text: such

text: that

text: and

text: one

text: of

text: the

text: two

text: words

text: and

text: i·1·

text: imis

text: a

text: oimf

text: he

text: other.

text: im

text: i

text: m

text: Claim.

text: Let

text: x

text: and

text: y

text: be

text: such

text: that

text: x&&

text: y&&

text: is

text: X-accepted

text: by

text: at

text: most

text: k

text: diﬀerent

text: runs

text: of

text: M.

text: Then

text: there

text: exist

text: and

text: such

text: that

text: is

text: -accepted

text: by

text: at

text: most

text: diﬀerent

text: runs.

text: xx

text: yy

text: X

text: k

text: This

text: claim

text: implies

text: that

text: does

text: not

text: have

text: any

text: universal

text: preﬁx,

text: i.e.,

text: that

text: for

text: every

text: ﬁnite

text: words

text: X

text: x

text: and

text: y

text: there

text: exists

text: a

text: conﬁguration

text: starting

text: with

text: x

text: y

text: that

text: is

text: not

text: X-accepted.

text: Indeed,

text: for

text: every

text: such

text: there

text: can

text: be

text: only

text: ﬁnitely

text: many

text: diﬀerent

text: accepting

text: runs

text: (say

text: for

text: x,

text: y

text: k

text: M

text: because

text: eventually

text: loops

text: on

text: state

text: with

text: transition

text: So

text: it

text: suﬃces

text: to

text: x

text: y

text: X

text: apply

text: the

text: claim

text: times

text: to

text: obtain

text: a

text: conﬁguration

text: starting

text: with

text: which

text: is

text: not

text: -accepted.

text: k

text: x

text: y

text: X

text: We

text: now

text: prove

text: the

text: claim,

text: using

text: the

text: preﬁx-PCP

text: solution.

text: Let

text: x

text: and

text: y

text: be

text: such

text: that

text: is

text: -accepted

text: by

text: diﬀerent

text: runs.

text: Denote

text: by

text: the

text: ﬁnite

text: preﬁxes

text: x

text: y

text: X

text: k

text: R

text: M

text: k·

text: of

text: the

text: ·k·

text: ·runs,

text: each

text: cut

text: when

text: reaches

text: the

text: ·p·a·rt.

text: Let

text: s∈=

text: im

text: and

text: let

text: t

text: &u*im},,

text: twhhainckhsctaontbhee

text: wmroitdtieﬁncaintiotnhemfoardme

text: where

text: each

text: ai

text: is

text: in

text: M

text: Let

text: be

text: the

text: distance

text: between

text: the

text: two

text: tapes

text: heads

text: when

text: has

text: completed

text: the

text: ﬁnite

text: run

text: (Note

text: that

text: the

text: ﬁrst

text: head

text: is

text: always

text: behind

text: the

text: second

text: one

text: because

text: it

text: can

text: only

text: move

text: by

text: one

text: cell

text: at

text: at

text: time.)

text: Without

text: loss

text: of

text: generality

text: we

text: can

text: assume

text: that

text: is

text: the

text: run

text: for

text: which

text: such

text: an

text: is

text: minimal.

text: We

text: now

text: construct

text: a

text: conﬁguration

text: which

text: will

text: “block”

text: any

text: run

text: starting

text: with

text: c

text: without

text: giving

text: the

text: other

text: runs

text: any

text: possibilities

text: for

text: new

text: nondeterministic

text: branching.

text: Let

text: such

text: that

text: (on

text: the

text: ﬁrst

text: tape)

text: begins

text: positions

text: behind

text: (on

text: the

text: second

text: tape)

text: in

text: the

text: L,

text: L

text: s

text: t

text: conﬁguration

text: c

text: x&Ls&&

text: y&L

text: t#&&

text: M

text: so

text: that

text: during

text: any

text: run

text: starting

text: with

text: starts

text: reading

text: and

text: exactly

text: at

text: the

text: same

text: time.

text: s

text: t

text: It

text: follows

text: that

text: cannot

text: be

text: extended

text: to

text: an

text: accepting

text: run

text: for

text: because

text: corresponds

text: to

text: a

text: c

text: s,

text: t

text: preﬁx-PCP

text: solution,

text: similarly

text: as

text: in

text: the

text: proof

text: of

text: Theorem

text: The

text: same

text: is

text: true

text: for

text: any

text: other

text: run

text: for

text: which

text: such

text: an

text: is

text: the

text: same

text: as

text: R

text: i

text: Let

text: us

text: now

text: consider

text: another

text: accepting

text: run

text: By

text: minimality

text: of

text: the

text: distance

text: between

text: the

text: R

text: M

text: i·

text: two

text: tapes

text: heads

text: when

text: ﬁrst

text: reaches

text: during

text: run

text: is

text: strictly

text: larger

text: than

text: We

text: R

text: i

text: now

text: prove

text: that

text: can

text: be

text: extended

text: in

text: a

text: unique

text: way

text: to

text: an

text: accepting

text: run

text: for

text: Indeed,

text: any

text: run

text: R

text: c

text: M

text: i

text: of

text: starting

text: with

text: must

text: evolve

text: in

text: the

text: following

text: way:

text: R

text: i

text: when

text: starts

text: being

text: read

text: the

text: second

text: tape,

text: is

text: not

text: yet

text: being

text: read

text: on

text: the

text: ﬁrst

text: tape,

text: so

text: at

text: t

text: s

text: M

text: this

text: time

text: is

text: reading

text: on

text: the

text: ﬁrst

text: tape

text: and

text: on

text: the

text: second

text: tape;

text: M

text: the

text: only

text: possible

text: transition

text: is

text: so

text: moves

text: one

text: step

text: on

text: both

text: tapes,

text: and

text: is

text: now

text: reading

text: on

text: the

text: second

text: tape;

text: M

text: the

text: only

text: possible

text: transition

text: is

text: so

text: moves

text: one

text: step

text: on

text: the

text: ﬁrst

text: tape

text: and

text: two

text: steps

text: on

text: the

text: second,

text: and

text: is

text: again

text: reading

text: on

text: the

text: second

text: tape;


text: • · · · this continues until the whole t = a1*a2* *a| |−1* has been read on the second tape, and M | t is deterministically looping on & &. From this analysis, it follows that can be extended in a unique way to an accepting run for . R c i | − Hence is a conﬁguration starting with with at most 1 accepting runs, because every c x y k accepting run for must start with an , each of which can be extended in at most one way c R if i ∈ {2, . . . , k}, or in no way at all if i =i 1. Thus the claim is proved by taking x0 = &Ls and y0 = &L0t#. ⇐ ( ) Suppose that no solution exists for the preﬁx-PCP instance. The strategy described in ⇐ the “ ” direction of the proof of Theorem 2.5 can be applied to prove that state is universal, X with the additional case that if the tape begins by & or *, then the transition (7), (8) or (9) can always be used. Remark 2.7. In the reduction made in the above proof of Theorem 2.6, if state has a universal X preﬁx, then in fact is universal. Also, in this case, it is easy to see that any ﬁnite word X satisfying (i), (ii) and (iii) of transition (3) is a universal preﬁx for (and ), so , (and ) U V X U V M have a common universal preﬁx Hence we have the following: given a 2-tape automaton on 3 M states and two states of , it is undecidable if and have a common universal preﬁx. q, r q r 3 Aﬃne GIFS associated with multi-tape automata M A × × Let be a -tape automaton on alphabet = . We want to give a “numerical d A1 . . . A interpretation” to a ﬁnite word u ∈ A? or to an inﬁnite conﬁgduration c ∈ AN. We must ﬁrst ∈ { } specify, for each 1 , a numerical interpretation of the letters of by choosing a k , . . . , n A bijection δ : A → {0, . . . , |A | − 1}. We theXn deﬁne ∆ : A? → R by k k k k k k | |− ∆ (u) = δ (u ) A i. k k i k 1(cid:54) (cid:54)| | i u · · · ∈ · · · Equivalently, for u = u1 u An, the number ∆ (u) is represented by 0.δ (u1) δ (u ) in Tbahsee d|Aomk|.aiFnisnoaflly∆, letan∆d:∆A+1ca×nn. .n.a×tAuk r+dal→ly bRed ebxetdeneﬁdnekeddtboy ∆N(wan1,d. .A.N, w, rde)s=pe(c∆ti1vk(ewly1.), . . . ,k∆dn(wd)). A k k { | | − } In the examples that will follow, if the alphabets are all of the form 0 1 and A , . . . , A → { | | − } k k the maps : 0 1 are not speciﬁed, we will assume for convenience that they δ A , . . . , A k k k are identity mappings. M Q A × · · · × Deﬁnition 3.1. Let be a -tape automaton on state set and alphabet = . d A1 A M Q n The GIFS associated with is the GIFS deﬁned by the graph with vertex set and, for G M → every transition R = (q, r, (w1, . . . , w )) of , an edge (q, r) labelled by the map f : [0, 1]d  d  R [0, 1]d deﬁned by   |A1|−|w1| 0  f (x) = ... x + ∆(w1, . . . , w ). R d 0 |Ad|−|wd| Example 3.2. Let M be a 2-tape automaton on alphabet A = {0 1} × {0 1}, and let ∈ AN , , c M be conﬁguration. If contains a transition = ( (1011 11)), then applying the contracting R q, r, , ! ∈ ! map on ∆( ) = (0 0 ) [0 1]2 has the following eﬀect: f c .x1x2 . . . , .y1y2 . . . , R 1 16 0 0 (∆( )) = / !.x1x 2 . . . +!∆(1 011 11) ! fR c 0 1/4 0.y1u2 . . . , 0 0000 0 1011 0 1011 = . x1x2 . . . + . = . x1x2 . . . 0 00 0 11 0 11 . . y1u2 . . . . . y1u2 . . . 7 ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: this

text: continues

text: until

text: the

text: whole

text: t

text: *a|

text: has

text: been

text: read

text: on

text: the

text: second

text: tape,

text: and

text: M

text: t

text: is

text: deterministically

text: looping

text: on

text: From

text: this

text: analysis,

text: it

text: follows

text: that

text: can

text: be

text: extended

text: in

text: a

text: unique

text: way

text: to

text: an

text: accepting

text: run

text: for

text: R

text: c

text: i

text: Hence

text: is

text: a

text: conﬁguration

text: starting

text: with

text: with

text: at

text: most

text: accepting

text: runs,

text: because

text: every

text: c

text: x

text: y

text: k

text: accepting

text: run

text: for

text: must

text: start

text: with

text: an

text: each

text: of

text: which

text: can

text: be

text: extended

text: in

text: at

text: most

text: one

text: way

text: c

text: R

text: if

text: i

text: k},

text: or

text: in

text: no

text: way

text: at

text: all

text: if

text: i

text: =i

text: Thus

text: the

text: claim

text: is

text: proved

text: by

text: taking

text: &Ls

text: and

text: Suppose

text: that

text: no

text: solution

text: exists

text: for

text: the

text: preﬁx-PCP

text: instance.

text: The

text: strategy

text: described

text: in

text: the

text: direction

text: of

text: the

text: proof

text: of

text: Theorem

text: can

text: be

text: applied

text: to

text: prove

text: that

text: state

text: is

text: universal,

text: X

text: with

text: the

text: additional

text: case

text: that

text: if

text: the

text: tape

text: begins

text: by

text: or

text: then

text: the

text: transition

text: or

text: can

text: always

text: be

text: used.

text: Remark

text: In

text: the

text: reduction

text: made

text: in

text: the

text: above

text: proof

text: of

text: Theorem

text: if

text: state

text: has

text: a

text: universal

text: X

text: preﬁx,

text: then

text: in

text: fact

text: is

text: universal.

text: Also,

text: in

text: this

text: case,

text: it

text: is

text: easy

text: to

text: see

text: that

text: any

text: ﬁnite

text: word

text: X

text: satisfying

text: (i),

text: (ii)

text: and

text: (iii)

text: of

text: transition

text: is

text: a

text: universal

text: preﬁx

text: for

text: (and

text: so

text: (and

text: U

text: V

text: X

text: U

text: V

text: M

text: have

text: a

text: common

text: universal

text: preﬁx

text: Hence

text: we

text: have

text: the

text: following:

text: given

text: a

text: automaton

text: on

text: M

text: states

text: and

text: two

text: states

text: of

text: it

text: is

text: undecidable

text: if

text: and

text: have

text: a

text: common

text: universal

text: preﬁx.

text: q,

text: r

text: q

text: r

text: Aﬃne

text: GIFS

text: associated

text: with

text: multi-tape

text: automata

text: M

text: A

text: Let

text: be

text: a

text: -tape

text: automaton

text: on

text: alphabet

text: We

text: want

text: to

text: give

text: a

text: “numerical

text: d

text: A

text: interpretation”

text: to

text: a

text: ﬁnite

text: word

text: u

text: A?

text: or

text: to

text: an

text: inﬁnite

text: conﬁgduration

text: c

text: AN.

text: We

text: must

text: ﬁrst

text: specify,

text: for

text: each

text: a

text: numerical

text: interpretation

text: of

text: the

text: letters

text: of

text: by

text: choosing

text: a

text: k

text: n

text: A

text: bijection

text: δ

text: A

text: |A

text: We

text: theXn

text: deﬁne

text: A?

text: R

text: by

text: k

text: k

text: k

text: k

text: k

text: k

text: (u)

text: δ

text: (u

text: A

text: i.

text: k

text: k

text: i

text: k

text: i

text: u

text: Equivalently,

text: for

text: u

text: u

text: An,

text: the

text: number

text: (u)

text: is

text: represented

text: by

text: δ

text: (u

text: in

text: Tbahsee

text: d|Aomk|.aiFnisnoaflly∆,

text: .n.a×tAuk

text: r+dal→ly

text: bRed

text: ebxetdeneﬁdnekeddtboy

text: .A.N,

text: w,

text: ,k∆dn(wd)).

text: A

text: k

text: k

text: In

text: the

text: examples

text: that

text: will

text: follow,

text: if

text: the

text: alphabets

text: are

text: all

text: of

text: the

text: form

text: and

text: A

text: A

text: k

text: k

text: the

text: maps

text: are

text: not

text: speciﬁed,

text: we

text: will

text: assume

text: for

text: convenience

text: that

text: they

text: δ

text: A

text: A

text: k

text: k

text: k

text: are

text: identity

text: mappings.

text: M

text: Q

text: A

text: Deﬁnition

text: Let

text: be

text: a

text: -tape

text: automaton

text: on

text: state

text: set

text: and

text: alphabet

text: d

text: A

text: M

text: Q

text: n

text: The

text: GIFS

text: associated

text: with

text: is

text: the

text: GIFS

text: deﬁned

text: by

text: the

text: graph

text: with

text: vertex

text: set

text: and,

text: for

text: G

text: M

text: every

text: transition

text: R

text: (q,

text: r,

text: w

text: of

text: an

text: edge

text: (q,

text: r)

text: labelled

text: by

text: the

text: map

text: f

text: d

text: R

text: deﬁned

text: by

text: f

text: (x)

text: x

text: w

text: R

text: d

text: |Ad|−|wd|

text: Example

text: Let

text: M

text: be

text: a

text: automaton

text: on

text: alphabet

text: A

text: and

text: let

text: AN

text: c

text: M

text: be

text: conﬁguration.

text: If

text: contains

text: a

text: transition

text: then

text: applying

text: the

text: contracting

text: R

text: q,

text: r,

text: map

text: on

text: has

text: the

text: following

text: eﬀect:

text: f

text: c

text: R

text: fR

text: c

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: · · · This suggests that applying a sequence of mappings (∆( )) corresponds to concate- f f c nating the words associated with the transitions inR1the nuRmn erical interpretation ∆( ) of a R c n conﬁguration . This is the key idea to establish a correspondence between the GIFS of an c automaton and its accepted conﬁgurations. This is formalized in the next proposition. M M Proposition 3.3. Let be a 2-tape automaton and let be a state of . The GIFS attractor q of M associated with q is equal to the set {∆(c) ∈ Rd : c ∈ AN is q-accepted by M}. ∈ Proof. Let x T[0, 1]d. It follows from a standard fact in the theory of iterated function ∈ systems [Fal03, Chapter 9] that if and only if there is an inﬁnite run ( ) starting at x X R (cid:62)1 · · · q n nM wq istuhchrutnhat x.=Monr(cid:62)e1ofvRer1, byfRdne(ﬁ[0n,it1i]odn), wofhethreefGRnIFiSs tohfe Mma,pfpoirngevoefrtyhesuGcIhFSruonf ( a)ssoc,iatthede R R (cid:62)1 n · · · | · · · | · · · n n conﬁguration = is such that = ∆( ), where the are given c w1 1w2 1 w1 w2 x c w by the transitions ( , , ( ,d)) fo,dr all (cid:62) 1, so the proposition is provednb,kecause is q , r , w 1, . . . , w n c n n n, n,d a -accepted conﬁguration. q M { } Example 3.4. Let be the 1-state, 2-tape automaton on alphabet 0 1 with transitions , | | | M 7→ 0x 07→, 0x2 1+, (112,00.),Txh7→e itx2e+ra(t0e,d12)fuanncdtiiotncasnysetaesmilyabsesosceeiantethdawt tithhe assoccoiantseidstasttorfacthtoer mthaepSsiexrpińskx2i, triangle (see Figure 1). M { } Example 3.5. The 1-tape, 1-state automaton on alphabet 0 1 with three transitions 1, , 10 and 00 (given in Example 2.4) is an example of a non-universal automaton which admits M universal preﬁxes. This reﬂects in the attractor associated with in the following way: it is not equal to [0 1] but it has nonempty interior. This can be proved either by SProposition 3.7, or , by proving directly that a conﬁguration ∈ {0 1}N is accepted by M if and only if it does not x , start with 02k+11 for some k (cid:62) 0, which implies that the attractor is equal to (cid:62)0[2−2k−1, 2−2k]. k ∈ Remark 3.6. Given a d-tape automaton and a point x [0, 1]d, if there exists two conﬁgurations 0 such that = ∆( ) = ∆( 0) and such that the tapes components ∈ N and 0 ∈ N diﬀer c, c x c c c A c A for some k ∈ {1, . . . , d}, then c and c0 are both stationary, ending wkith 0kω or (|Ak | − 1k)ω. In particular, ∆ : AN → Rd is ﬁnitke-to-onke. k The next proposition establishes the desired correspondence between word-theoretical proper- ties of multi-tape automata and topological properties of the associated self-aﬃne attractors. M A M Proposition 3.7. Let be a -tape automaton on alphabet , let be a state of , and let d q be the associated GIFS attractor. We have: X q (1) q is universal if and only if X = [0, 1]d, q (2) has a universal preﬁx if and only if has nonemtpy interior. q X q Proof. (1) If state q is universal the expansion of every element of [0, 1]d is q-accepted so X = [0, 1]d thanks to Proposition 3.3. Conversely, suppose that there exists an inﬁnite sequence q that is not -accepted. By a compactness argument, there must exist a preﬁx of such that c q w c 0 is not -accepted for any inﬁnite sequence 0. Thanks to Remark 3.6, by choosing 0 with wc q c c no tape components ending by 0ω or (|A | − 1)ω, the sequence wc0 is the only sequence such that = ∆( 0), so ∆( 0) ∈ becausek otherwise 0 would be -accepted. It follows that x wc wc / X wc q 6 q X = [0, 1]d. q ∈ A (2) For a ﬁnite word w ?, deﬁne the cylinder [w] to be equal to the set of conﬁgurations ⊆ that start with . If admits a universal preﬁx , then ∆([ ]) by Proposition 3.3, so w q w w X X q ⊆ q has nonempty interior. Conversely, suppose that there exists a nonempty open set , and U X ∈ A ⊆ q let w ? be a ﬁnite word such that ∆([w]) U. By a reasoning similar as in the proof of (1), we can prove that is a universal preﬁx for . w q 8 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: This

text: suggests

text: that

text: applying

text: a

text: sequence

text: of

text: mappings

text: corresponds

text: to

text: concate-

text: f

text: f

text: c

text: nating

text: the

text: words

text: associated

text: with

text: the

text: transitions

text: nuRmn

text: erical

text: interpretation

text: of

text: a

text: R

text: c

text: n

text: conﬁguration

text: This

text: is

text: the

text: key

text: idea

text: to

text: establish

text: a

text: correspondence

text: between

text: the

text: GIFS

text: of

text: an

text: c

text: automaton

text: and

text: its

text: accepted

text: conﬁgurations.

text: This

text: is

text: formalized

text: in

text: the

text: next

text: proposition.

text: M

text: M

text: Proposition

text: Let

text: be

text: a

text: automaton

text: and

text: let

text: be

text: a

text: state

text: of

text: The

text: GIFS

text: attractor

text: q

text: of

text: M

text: associated

text: with

text: q

text: is

text: equal

text: to

text: the

text: set

text: {∆(c)

text: Rd

text: c

text: AN

text: is

text: q-accepted

text: by

text: M}.

text: Proof.

text: Let

text: x

text: It

text: follows

text: from

text: a

text: standard

text: fact

text: in

text: the

text: theory

text: of

text: iterated

text: function

text: systems

text: Chapter

text: that

text: if

text: and

text: only

text: if

text: there

text: is

text: an

text: inﬁnite

text: run

text: starting

text: at

text: x

text: X

text: R

text: q

text: n

text: nM

text: wq

text: istuhchrutnhat

text: wofhethreefGRnIFiSs

text: tohfe

text: Mma,pfpoirngevoefrtyhesuGcIhFSruonf

text: a)ssoc,iatthede

text: R

text: R

text: n

text: n

text: n

text: conﬁguration

text: is

text: such

text: that

text: where

text: the

text: are

text: given

text: c

text: x

text: c

text: w

text: by

text: the

text: transitions

text: ,d))

text: fo,dr

text: all

text: so

text: the

text: proposition

text: is

text: provednb,kecause

text: is

text: q

text: r

text: w

text: w

text: n

text: c

text: n

text: n

text: n,

text: n,d

text: a

text: -accepted

text: conﬁguration.

text: q

text: M

text: Example

text: Let

text: be

text: the

text: automaton

text: on

text: alphabet

text: with

text: transitions

text: M

text: 7→

text: tithhe

text: assoccoiantseidstasttorfacthtoer

text: triangle

text: (see

text: Figure

text: M

text: Example

text: The

text: automaton

text: on

text: alphabet

text: with

text: three

text: transitions

text: and

text: (given

text: in

text: Example

text: is

text: an

text: example

text: of

text: a

text: non-universal

text: automaton

text: which

text: admits

text: M

text: universal

text: preﬁxes.

text: This

text: reﬂects

text: in

text: the

text: attractor

text: associated

text: with

text: in

text: the

text: following

text: way:

text: it

text: is

text: not

text: equal

text: to

text: but

text: it

text: has

text: nonempty

text: interior.

text: This

text: can

text: be

text: proved

text: either

text: by

text: SProposition

text: or

text: by

text: proving

text: directly

text: that

text: a

text: conﬁguration

text: is

text: accepted

text: by

text: M

text: if

text: and

text: only

text: if

text: it

text: does

text: not

text: x

text: start

text: with

text: for

text: some

text: k

text: which

text: implies

text: that

text: the

text: attractor

text: is

text: equal

text: to

text: k

text: Remark

text: Given

text: a

text: d-tape

text: automaton

text: and

text: a

text: point

text: x

text: if

text: there

text: exists

text: two

text: conﬁgurations

text: such

text: that

text: and

text: such

text: that

text: the

text: tapes

text: components

text: N

text: and

text: N

text: diﬀer

text: c,

text: c

text: x

text: c

text: c

text: c

text: A

text: c

text: A

text: for

text: some

text: k

text: d},

text: then

text: c

text: and

text: are

text: both

text: stationary,

text: ending

text: wkith

text: or

text: (|Ak

text: In

text: particular,

text: AN

text: Rd

text: is

text: ﬁnitke-to-onke.

text: k

text: The

text: next

text: proposition

text: establishes

text: the

text: desired

text: correspondence

text: between

text: word-theoretical

text: proper-

text: ties

text: of

text: multi-tape

text: automata

text: and

text: topological

text: properties

text: of

text: the

text: associated

text: self-aﬃne

text: attractors.

text: M

text: A

text: M

text: Proposition

text: Let

text: be

text: a

text: -tape

text: automaton

text: on

text: alphabet

text: let

text: be

text: a

text: state

text: of

text: and

text: let

text: d

text: q

text: be

text: the

text: associated

text: GIFS

text: attractor.

text: We

text: have:

text: X

text: q

text: q

text: is

text: universal

text: if

text: and

text: only

text: if

text: X

text: q

text: has

text: a

text: universal

text: preﬁx

text: if

text: and

text: only

text: if

text: has

text: nonemtpy

text: interior.

text: q

text: X

text: q

text: Proof.

text: If

text: state

text: q

text: is

text: universal

text: the

text: expansion

text: of

text: every

text: element

text: of

text: is

text: q-accepted

text: so

text: X

text: thanks

text: to

text: Proposition

text: Conversely,

text: suppose

text: that

text: there

text: exists

text: an

text: inﬁnite

text: sequence

text: q

text: that

text: is

text: not

text: -accepted.

text: By

text: a

text: compactness

text: argument,

text: there

text: must

text: exist

text: a

text: preﬁx

text: of

text: such

text: that

text: c

text: q

text: w

text: c

text: is

text: not

text: -accepted

text: for

text: any

text: inﬁnite

text: sequence

text: Thanks

text: to

text: Remark

text: by

text: choosing

text: with

text: wc

text: q

text: c

text: c

text: no

text: tape

text: components

text: ending

text: by

text: 0ω

text: or

text: (|A

text: the

text: sequence

text: is

text: the

text: only

text: sequence

text: such

text: that

text: so

text: becausek

text: otherwise

text: would

text: be

text: -accepted.

text: It

text: follows

text: that

text: x

text: wc

text: wc

text: X

text: wc

text: q

text: q

text: X

text: q

text: A

text: For

text: a

text: ﬁnite

text: word

text: w

text: deﬁne

text: the

text: cylinder

text: [w]

text: to

text: be

text: equal

text: to

text: the

text: set

text: of

text: conﬁgurations

text: that

text: start

text: with

text: If

text: admits

text: a

text: universal

text: preﬁx

text: then

text: by

text: Proposition

text: so

text: w

text: q

text: w

text: w

text: X

text: X

text: q

text: q

text: has

text: nonempty

text: interior.

text: Conversely,

text: suppose

text: that

text: there

text: exists

text: a

text: nonempty

text: open

text: set

text: and

text: U

text: X

text: A

text: q

text: let

text: w

text: be

text: a

text: ﬁnite

text: word

text: such

text: that

text: ∆([w])

text: U.

text: By

text: a

text: reasoning

text: similar

text: as

text: in

text: the

text: proof

text: of

text: we

text: can

text: prove

text: that

text: is

text: a

text: universal

text: preﬁx

text: for

text: w

text: q

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 2 P. A. SISSOKHO ∈ F A zero-sum sequence over G is a sequence S (G ) such that σ(S) = 0. Such a 0 0 sequence is called minimal if it does not contain a proper non-trivial zero-sum subsequence. Then, the submonoid B B { ∈ F } = (G ) = S (G ) : σ(S) = 0 0 0 0 F A B B of (G ) is a Krull monoid (e.g., see [15]). The set ( ) of the atoms of is the set of 0 B 0 A B 0 all minimal zero-sum sequences in . A characterization of ( ) would shed some light B 0 0 on the factorization properties of (e.g., see [12, 13]). 0 · · ∈ A B Given a minimal zero-sum sequence S = s . . . s ( ), we are interested in bounding ∈ 1 t 0 its length in function of its terms s for i [1, t]. We are also interested in ﬁnding a natural A B i B structure for ( ) when G (and thus, ) is ﬁnite. 0 0 B 0 The study of zero-sum sequences in (G), when G a ﬁnite cyclic group, is a very active area of research (e.g., see [2, 5, 6, 9, 18, 19, 22]) with applications to Factorization Theory (e.g., see [3, 10, 11, 12]). Similar, but less extensive, investigations have been carried out when G is an inﬁnite cyclic group (e.g., see [4, 7, 13, 14]). ∈ B Z | | | | For all S ( ) with S ﬁnite and S > 1, there exist positive integers a , . . . , a and ≤ ≤ ≤ ≤ 1 n b , . . . , b with a . . . a and b . . . b , such that 1 m 1 (cid:89)n 1 (cid:89) m n − m − · − (2) S+ = a[xi], S = ( b )[yj], and S = S+ S , i j i=1 j=1 ∈ ∈ where x and y are positive integers for all i [1, n] and j [1, m]. i j In his work on Diophantine linear equations, Lambert [17] proved the following theorem. Z | | Theorem 1.1 (Lambert [17]). Let S be a minimal zero-sum sequence over with S ﬁnite | | and S > 1. If S is as in (2), then | | ≤ (cid:107) −(cid:107) | −| ≤ (cid:107) (cid:107) S+ S ∞ = b and S S+ ∞ = a . m n This was reformulated and reproved in the language of sequences by Baginski et al. [4]. Perhaps due to inconsistent notation across various areas, Theorem 1.1 has been indepen- dently rediscovered by Diaconis et al. [8], and Sahs et al. [21]. Currently, the best bounds | | | −| for S+ and S are due to Henk-Weismantel [16]. They proved the following theorem for which Theorem 1.1 is a special case upon setting (cid:96) = m and k = n. (cid:106) (cid:107) (cid:108) (cid:109) (cid:80) (cid:80) Z Theorem 1.2 (Henk-Weismantel [16]). Let S be a minimal zero-sum sequence over with | | | | S ﬁnite and S > 1. If S(cid:106) is as(cid:107)in (2), then (cid:108) (cid:109) (cid:80) (cid:80) | | ≤ − − − − ∈ (J ) : S+ b (cid:96) 1 b(cid:96) bj y + m bj b(cid:96) y for all (cid:96) [1, m], (cid:96) (cid:96) j=1 an j j=(cid:96)+1 a1 j | −| ≤ − − − − ∈ (I ) : S a k 1 ak ai x + n ai ak x for all k [1, n]. k k i=1 bm i i=k+1 b1 i In this paper, we improve on Theorem 1.2 by proving the following theorem. ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: P.

text: A.

text: SISSOKHO

text: F

text: A

text: zero-sum

text: sequence

text: over

text: G

text: is

text: a

text: sequence

text: S

text: (G

text: such

text: that

text: σ(S)

text: Such

text: a

text: sequence

text: is

text: called

text: minimal

text: if

text: it

text: does

text: not

text: contain

text: a

text: proper

text: non-trivial

text: zero-sum

text: subsequence.

text: Then,

text: the

text: submonoid

text: B

text: B

text: F

text: (G

text: S

text: (G

text: σ(S)

text: F

text: A

text: B

text: B

text: of

text: (G

text: is

text: a

text: Krull

text: monoid

text: (e.g.,

text: see

text: The

text: set

text: of

text: the

text: atoms

text: of

text: is

text: the

text: set

text: of

text: B

text: A

text: B

text: all

text: minimal

text: zero-sum

text: sequences

text: in

text: A

text: characterization

text: of

text: would

text: shed

text: some

text: light

text: B

text: on

text: the

text: factorization

text: properties

text: of

text: (e.g.,

text: see

text: A

text: B

text: Given

text: a

text: minimal

text: zero-sum

text: sequence

text: S

text: s

text: s

text: we

text: are

text: interested

text: in

text: bounding

text: t

text: its

text: length

text: in

text: function

text: of

text: its

text: terms

text: s

text: for

text: i

text: t].

text: We

text: are

text: also

text: interested

text: in

text: ﬁnding

text: a

text: natural

text: A

text: B

text: i

text: B

text: structure

text: for

text: when

text: G

text: (and

text: thus,

text: is

text: ﬁnite.

text: B

text: The

text: study

text: of

text: zero-sum

text: sequences

text: in

text: (G),

text: when

text: G

text: a

text: ﬁnite

text: cyclic

text: group,

text: is

text: a

text: very

text: active

text: area

text: of

text: research

text: (e.g.,

text: see

text: with

text: applications

text: to

text: Factorization

text: Theory

text: (e.g.,

text: see

text: Similar,

text: but

text: less

text: extensive,

text: investigations

text: have

text: been

text: carried

text: out

text: when

text: G

text: is

text: an

text: inﬁnite

text: cyclic

text: group

text: (e.g.,

text: see

text: B

text: Z

text: For

text: all

text: S

text: with

text: S

text: ﬁnite

text: and

text: S

text: there

text: exist

text: positive

text: integers

text: a

text: a

text: and

text: n

text: b

text: b

text: with

text: a

text: a

text: and

text: b

text: b

text: such

text: that

text: m

text: m

text: n

text: m

text: S+

text: a[xi],

text: S

text: b

text: )[yj],

text: and

text: S

text: S+

text: S

text: i

text: j

text: where

text: x

text: and

text: y

text: are

text: positive

text: integers

text: for

text: all

text: i

text: n]

text: and

text: j

text: m].

text: i

text: j

text: In

text: his

text: work

text: on

text: Diophantine

text: linear

text: equations,

text: Lambert

text: proved

text: the

text: following

text: theorem.

text: Z

text: Theorem

text: (Lambert

text: Let

text: S

text: be

text: a

text: minimal

text: zero-sum

text: sequence

text: over

text: with

text: S

text: ﬁnite

text: and

text: S

text: If

text: S

text: is

text: as

text: in

text: then

text: S+

text: S

text: b

text: and

text: S

text: S+

text: a

text: m

text: n

text: This

text: was

text: reformulated

text: and

text: reproved

text: in

text: the

text: language

text: of

text: sequences

text: by

text: Baginski

text: et

text: al.

text: Perhaps

text: due

text: to

text: inconsistent

text: notation

text: across

text: various

text: areas,

text: Theorem

text: has

text: been

text: indepen-

text: dently

text: rediscovered

text: by

text: Diaconis

text: et

text: al.

text: and

text: Sahs

text: et

text: al.

text: Currently,

text: the

text: best

text: bounds

text: for

text: S+

text: and

text: S

text: are

text: due

text: to

text: Henk-Weismantel

text: They

text: proved

text: the

text: following

text: theorem

text: for

text: which

text: Theorem

text: is

text: a

text: special

text: case

text: upon

text: setting

text: m

text: and

text: k

text: n.

text: Z

text: Theorem

text: (Henk-Weismantel

text: Let

text: S

text: be

text: a

text: minimal

text: zero-sum

text: sequence

text: over

text: with

text: S

text: ﬁnite

text: and

text: S

text: If

text: is

text: then

text: (J

text: S+

text: b

text: bj

text: y

text: m

text: bj

text: y

text: for

text: all

text: m],

text: an

text: j

text: j

text: (I

text: S

text: a

text: k

text: ak

text: ai

text: x

text: n

text: ai

text: ak

text: x

text: for

text: all

text: k

text: n].

text: k

text: k

text: bm

text: i

text: i

text: In

text: this

text: paper,

text: we

text: improve

text: on

text: Theorem

text: by

text: proving

text: the

text: following

text: theorem.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


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text: E.

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text: Hlozek,

text: [astro-ph.CO].


text: set of angular cross-spectra derived from multiple detector combination between the 100, 143, and 217 GHz frequency channels, marginalizing over power-spectrum foreground templates. In order to break the well-known parameter degeneracy between the reionization optical depth τ and the scalar spectral index n , the Planck team adopts the low-l WMAP polarization s likelihood (WP). As stated above in this paper we are also interested in the X-ray cluster data [22–24]. The CCCP project measures cluster mass function by using a high-redshift (0.4 < z < 0.9) subsample of the 400 square degree survey, 37 objects, and low-redshift (z < 0.2) subsample of the all-sky survey, 49 brightest clusters. The methodologies of likelihood construction follow the standard derivation of the Poisson distribution of cluster mass [25]. The likelihood function implicitly depends on the cosmological parameters through the model of cluster mass function (reﬂecting the growth, normalization, and shape of the density perturbation power spectrum), through the cosmological volume-redshift relation which determines the survey volume, and through the distance-redshift as well as the masses-temperature relation. The details of likelihood construction and systematic uncertainty control are mentioned in [22]. Furthermore, in order to break the parameter degeneracies we also use some other external data sets, including BAO measurements from the 6dFGS [5], SDSS DR7 [6] and BOSS DR9 (cid:80) [7] surveys, and HST Key project [3] H measurement. 0 Contrary to the Planck constraints on the summed neutrino mass, [11] found that adding X-ray cluster could gives the non-zero detection of the active or sterile neutrino mass ( m ν or m ) with great statistical signiﬁcance for various 8-parameter models, including the ac- s tive/sterile neutrino mass as well as eﬀective neutrino number N . Moreover, adding X-ray eﬀ data set could also lead to a signiﬁcant deviation in σ , namely the matter power spectrum − 8 amplitude on the 8h 1Mpc scale, from the Planck result [2]. In details, without X-ray clus- ter, the Planck result favors a larger value of σ , while with them the joint analysis give a 8 lower value. Therefore, in this paper we explore the tension between Planck and CCCP X-ray cluster data sets with several 8-parameter models, including eﬀective neutrino number N , eﬀ constant dark energy equation of state w, present spatial curvature Ω and lensing ampli- K (cid:80) tude A . Particularly, we here focus on the neutrino mass constraint, so that in our baseline L model the summed neutrino mass is always allowed to vary freely. We restrict ourselves to one-parameter extensions to the baseline model of ΛCDM + m , as listed in Tab.1. ν We compute the CMB angular and matter power spectra by using the public Einstein- Boltzmann solver CAMB [26] and explore the cosmological parameter space with a Markov Chain Monte Carlo sampler, namely CosmoMC [27]. For Planck we use the Planck Likelihood Code (PLC/clik) [28] which are available at the Planck Legacy Archive [29], and for CCCP our analysis is based on the likelihood grids presented in [22], which can be download from the website [30]. 3 Results In this section we will dig the information hidden in the CCCP X-ray cluster data [22– 24]. Hereafter, we denote CL − for this data. We will ﬁrst investigate the constraints on X ray the summed neutrino mass from the Planck+WP+BAO+HST data and CL − . Hereafter, X ray (cid:80) we dub Planck+WP+BAO+HST as PWBH. Then, we will turn to reveal some tensions in some cosmological parameters between these two data sets. 3.1 m results ν – 3 –

text: set

text: of

text: angular

text: cross-spectra

text: derived

text: from

text: multiple

text: detector

text: combination

text: between

text: the

text: and

text: GHz

text: frequency

text: channels,

text: marginalizing

text: over

text: power-spectrum

text: foreground

text: templates.

text: In

text: order

text: to

text: break

text: the

text: well-known

text: parameter

text: degeneracy

text: between

text: the

text: reionization

text: optical

text: depth

text: τ

text: and

text: the

text: scalar

text: spectral

text: index

text: n

text: the

text: Planck

text: team

text: adopts

text: the

text: low-l

text: WMAP

text: polarization

text: s

text: likelihood

text: (WP).

text: As

text: stated

text: above

text: in

text: this

text: paper

text: we

text: are

text: also

text: interested

text: in

text: the

text: X-ray

text: cluster

text: data

text: The

text: CCCP

text: project

text: measures

text: cluster

text: mass

text: function

text: by

text: using

text: a

text: high-redshift

text: z

text: subsample

text: of

text: the

text: square

text: degree

text: survey,

text: objects,

text: and

text: low-redshift

text: (z

text: subsample

text: of

text: the

text: all-sky

text: survey,

text: brightest

text: clusters.

text: The

text: methodologies

text: of

text: likelihood

text: construction

text: follow

text: the

text: standard

text: derivation

text: of

text: the

text: Poisson

text: distribution

text: of

text: cluster

text: mass

text: The

text: likelihood

text: function

text: implicitly

text: depends

text: on

text: the

text: cosmological

text: parameters

text: through

text: the

text: model

text: of

text: cluster

text: mass

text: function

text: (reﬂecting

text: the

text: growth,

text: normalization,

text: and

text: shape

text: of

text: the

text: density

text: perturbation

text: power

text: spectrum),

text: through

text: the

text: cosmological

text: volume-redshift

text: relation

text: which

text: determines

text: the

text: survey

text: volume,

text: and

text: through

text: the

text: distance-redshift

text: as

text: well

text: as

text: the

text: masses-temperature

text: relation.

text: The

text: details

text: of

text: likelihood

text: construction

text: and

text: systematic

text: uncertainty

text: control

text: are

text: mentioned

text: in

text: Furthermore,

text: in

text: order

text: to

text: break

text: the

text: parameter

text: degeneracies

text: we

text: also

text: use

text: some

text: other

text: external

text: data

text: sets,

text: including

text: BAO

text: measurements

text: from

text: the

text: SDSS

text: and

text: BOSS

text: surveys,

text: and

text: HST

text: Key

text: project

text: H

text: measurement.

text: Contrary

text: to

text: the

text: Planck

text: constraints

text: on

text: the

text: summed

text: neutrino

text: mass,

text: found

text: that

text: adding

text: X-ray

text: cluster

text: could

text: gives

text: the

text: non-zero

text: detection

text: of

text: the

text: active

text: or

text: sterile

text: neutrino

text: mass

text: m

text: ν

text: or

text: m

text: with

text: great

text: statistical

text: signiﬁcance

text: for

text: various

text: models,

text: including

text: the

text: ac-

text: s

text: tive/sterile

text: neutrino

text: mass

text: as

text: well

text: as

text: eﬀective

text: neutrino

text: number

text: N

text: Moreover,

text: adding

text: X-ray

text: eﬀ

text: data

text: set

text: could

text: also

text: lead

text: to

text: a

text: signiﬁcant

text: deviation

text: in

text: σ

text: namely

text: the

text: matter

text: power

text: spectrum

text: amplitude

text: on

text: the

text: scale,

text: from

text: the

text: Planck

text: result

text: In

text: details,

text: without

text: X-ray

text: clus-

text: ter,

text: the

text: Planck

text: result

text: favors

text: a

text: larger

text: value

text: of

text: σ

text: while

text: with

text: them

text: the

text: joint

text: analysis

text: give

text: a

text: lower

text: value.

text: Therefore,

text: in

text: this

text: paper

text: we

text: explore

text: the

text: tension

text: between

text: Planck

text: and

text: CCCP

text: X-ray

text: cluster

text: data

text: sets

text: with

text: several

text: models,

text: including

text: eﬀective

text: neutrino

text: number

text: N

text: eﬀ

text: constant

text: dark

text: energy

text: equation

text: of

text: state

text: w,

text: present

text: spatial

text: curvature

text: Ω

text: and

text: lensing

text: ampli-

text: K

text: tude

text: A

text: Particularly,

text: we

text: here

text: focus

text: on

text: the

text: neutrino

text: mass

text: constraint,

text: so

text: that

text: in

text: our

text: baseline

text: L

text: model

text: the

text: summed

text: neutrino

text: mass

text: is

text: always

text: allowed

text: to

text: vary

text: freely.

text: We

text: restrict

text: ourselves

text: to

text: one-parameter

text: extensions

text: to

text: the

text: baseline

text: model

text: of

text: ΛCDM

text: m

text: as

text: listed

text: in

text: ν

text: We

text: compute

text: the

text: CMB

text: angular

text: and

text: matter

text: power

text: spectra

text: by

text: using

text: the

text: public

text: Einstein-

text: Boltzmann

text: solver

text: CAMB

text: and

text: explore

text: the

text: cosmological

text: parameter

text: space

text: with

text: a

text: Markov

text: Chain

text: Monte

text: Carlo

text: sampler,

text: namely

text: CosmoMC

text: For

text: Planck

text: we

text: use

text: the

text: Planck

text: Likelihood

text: Code

text: (PLC/clik)

text: which

text: are

text: available

text: at

text: the

text: Planck

text: Legacy

text: Archive

text: and

text: for

text: CCCP

text: our

text: analysis

text: is

text: based

text: on

text: the

text: likelihood

text: grids

text: presented

text: in

text: which

text: can

text: be

text: download

text: from

text: the

text: website

text: Results

text: In

text: this

text: section

text: we

text: will

text: dig

text: the

text: information

text: hidden

text: in

text: the

text: CCCP

text: X-ray

text: cluster

text: data

text: Hereafter,

text: we

text: denote

text: CL

text: for

text: this

text: data.

text: We

text: will

text: ﬁrst

text: investigate

text: the

text: constraints

text: on

text: X

text: ray

text: the

text: summed

text: neutrino

text: mass

text: from

text: the

text: Planck+WP+BAO+HST

text: data

text: and

text: CL

text: Hereafter,

text: X

text: ray

text: we

text: dub

text: Planck+WP+BAO+HST

text: as

text: PWBH.

text: Then,

text: we

text: will

text: turn

text: to

text: reveal

text: some

text: tensions

text: in

text: some

text: cosmological

text: parameters

text: between

text: these

text: two

text: data

text: sets.

text: m

text: results

text: ν


text: Figure 3. Left: Marginalized likelihoods of σ . Right: 2D likelihood contours between the lensing 8 amplitude A and rms amplitude of linear ﬂuctuation σ . L 8 (cid:88) ± σ = 0.7894 0.0070 (68% ; ΛCDM + A : PWBH + CL) , (3.8) 8 L ± σ = 0.7550 0.0139 (68% ; ΛCDM + m + A : PWBH + CL) . (3.9) 8 ν L ± In 6 parameter ΛCDM model, Planck collaboration [35] gives σ (Ω /0.27)0.3 = 0.784 8 m 0.027 by using Planck+WP+BAO+BBN+CL − , which is consistent with the results re- X ray ported here. We notice that in Fig.3 the results with CL − (3.9) (blue curve) are in a 2σ X ray tension with those in absence of CL − data (3.10) (red curve) X ray ± σ = 0.8162 0.0119 (68% ; ΛCDM + A : PWB) . (3.10) 8 L Similar to A , with a 9% cluster mass correction we could also reconcile the tension with L Planck (cid:88) ± σ = 0.7639 0.0198 (68% ; wCDM + m + A : PWBH + CL + 9%Mass) . (3.11) 8 ν L 3.3 H and w results 0 In this subsection, we study two background parameters, Hubble constant H and Dark 0 (cid:80) Energy (DE) Equation of State (EoS) w. First of all, we can see from Tab.2 that without 9% mass correction to CL − data X ray sets, a larger H value is favored, e.g. for wCDM+ m model: 0 ν ± H = 74.0 2.1 (68% ; +w : PWBH + CL) . (3.12) 0 – 8 – ##LTFigure## ##LTFigure##

text: Figure

text: Left:

label: caption

text: Marginalized

text: likelihoods

text: of

text: σ

text: Right:

text: likelihood

text: contours

text: between

text: the

text: lensing

text: amplitude

text: A

text: and

text: rms

text: amplitude

text: of

text: linear

text: ﬂuctuation

text: σ

text: L

label: equation

text: σ

text: ΛCDM

text: A

text: PWBH

text: CL)

text: L

text: σ

text: ΛCDM

text: m

text: A

text: PWBH

text: CL)

text: ν

text: L

text: In

text: parameter

text: ΛCDM

text: model,

text: Planck

text: collaboration

text: gives

text: σ

text: (Ω

text: m

text: by

text: using

text: Planck+WP+BAO+BBN+CL

text: which

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text: consistent

text: with

text: the

text: results

text: re-

text: X

text: ray

text: ported

text: here.

text: We

text: notice

text: that

text: in

text: the

text: results

text: with

text: CL

text: (blue

text: curve)

text: are

text: in

text: a

text: 2σ

text: X

text: ray

text: tension

text: with

text: those

text: in

text: absence

text: of

text: CL

text: data

text: (red

text: curve)

text: X

text: ray

text: σ

text: ΛCDM

text: A

text: PWB)

text: L

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text: to

text: A

text: with

text: a

text: cluster

text: mass

text: correction

text: we

text: could

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text: reconcile

text: the

text: tension

text: with

text: L

text: Planck

text: σ

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text: m

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text: CL

text: ν

text: L

text: H

text: and

text: w

text: results

text: In

text: this

text: subsection,

text: we

text: study

text: two

text: background

text: parameters,

text: Hubble

text: constant

text: H

text: and

text: Dark

text: Energy

text: (DE)

text: Equation

text: of

text: State

text: (EoS)

text: w.

text: First

text: of

text: all,

text: we

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text: that

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text: correction

text: to

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text: sets,

text: a

text: larger

text: H

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text: e.g.

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text: wCDM+

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text: ##LTFigure##

text: ##LTFigure##


text: 660 km), and 0.1% (perovskite, below 660 km). Com- draw down suﬃcient CO to cool the planet out of the 2 bining these estimates with the pressure-dependence moist greenhouse state, leaving a partially water-covered of olivine water content from Hauri et al. (2006) puts planet (Abbot et al. 2012). Detailed atmospheric sim- Earth’s mantle water capacity at a dozen times the ulations of such hot planets, however, indicate that the current surface reservoir. Measurements of electrical tendency of carbon dioxide to cool the stratosphere im- conductivity (Dai & Karato 2009) suggest that Earth’s pedes the loss of water (Wordsworth & Pierrehumbert mantle contains 1–2 oceans worth of water at present- 2013). day, but other estimates have diﬀered by a factor of a In an alternate hypothesis, the Rayleigh number of few in either direction (e.g., Huang et al. 2005; Smyth & convecting water in hydrothermal systems at mid-ocean Jacobsen 2006; Khan & Shankland 2012). ridges exhibits a sharp peak when water becomes su- 1.2. The Deep Water Cycle percritical, which occurs at nearly the pressures and temperatures in hydrothermal systems today (Kasting There is a two-way ﬂux of water between ocean and & Holm 1992). If Earth’s water started in the mantle mantle on Earth. Ocean crust forms at mid-ocean ridges and gradually degassed to the surface, then this pressure- by depressurization melting of mantle, releasing volatiles dependence could explain why the oceans stabilized at into the ocean. Plate tectonics then drives the ocean their current depths. A planet starting with very deep crust toward subduction zones. Hydrothermal alteration, oceans (Korenaga 2008), however, would not be able principally serpentinization, produces ocean crust that is to eﬀectively subduct water and would remain a water- 5–10% water by mass. During subduction, much of the world. water is baked out of the subducting slab, producing ex- In order to solve the problem of regassing water into plosive volcanism like Mount St. Helens. Some of the the mantle, Holm (1996) suggested that the reduced ef- water, however, is subducted deep into the mantle, clos- ﬁciency of hydrothermal heat exchange beneath deep ing the loop (for a review of the relevant geochemistry, oceans would result in higher mantle temperatures, see Arndt 2013). greater plate velocities and therefore eﬃcient subduction The current H O degassing at mid-ocean ridges is approximately 2 ×2 1011 kg/yr (Hirschmann & Kohlst- of hydrated crust. This mechanism may not work quan- titatively, however, because hydrothermal heat transfer edt 2012), while the regassing ﬂux at subduction zones × only accounts for 30–50% of heat ﬂux through oceanic is estimated to be 0.7–2.9 1012 kg/yr (Jarrard 2003; crust (Stein et al. 1995). There are also qualitative prob- Schmidt & Poli 1998). Given the order-of-magnitude lems with this regassing argument: ﬁrst of all, the same agreement of these ﬁgures, and the large associated un- logic should apply to planets with very shallow oceans, certainties, Earth’s current ocean volume is typically potentially negating the supercritical circulation hypoth- assumed to be in a steady state (McGovern & Schubert esis. Moreover, hydrating the mantle lowers its viscosity, 1989). Moreover, realistic parameterizations of mantle further increasing plate velocities and making this mech- convection predict water ﬂuxes of 1011–1013 kg/yr in anism a destabilizing feedback. Lastly, although rapid Earth’s geological past, implying ocean-cycling times of subduction may aid regassing, a hotter mantle almost 108 yrs (McGovern & Schubert 1989). A signiﬁcant ﬂux certainly hinders it (Bounama et al. 2001). In short, it is imbalance would have long ago desiccated or submerged not clear that reduced heat transport through the ocean the planetary surface. crust leads to net regassing of water. 1.3. Previous Work Whatever geophysical processes govern Earth’s deep A planet entirely covered in water may develop exposed water cycle presumably operate on other terrestrial plan- continents by losing water to space, hydrating the crust ets with plate tectonics. The partitioning of water on and mantle, or reshaping continents and ocean basins. Earth is not only an outstanding problem in geophysics FoDrreaxfatmvpelres,ioannMoacyea2n6-,c2o0v2e0red planet with a hot strato- (Fyfe 1994; Langmuir & Broecker 2012), but one with spPhreerperinctoutylpdesleotsuesiwngatLAeTrEXtostsypleaecmeuulantteialpcj ovn. 8t/in13e/n1t0s are ex- important implications for the surface conditions and cli- posed. The resulting vigorous silicate weathering might mate of terrestrial exoplanets. WATER CYCLING BETWEEN OCEAN AND MANTLE: SUPER-EARTHS NEED NOT BE WATERWORLDS Nicolas B. Cowan1 & Dorian S. Abbot2 Subject headings: planets and satellites: composition — planets and satellites: interiors — planets and satellites: oceans — planets and satellites: physical evolution — planets and satellites: surfaces — planets and satellites: tectonics 1 Center for Interdisciplinary Exploration and Research in As- trophysics (CIERA), Department of Earth & Planetary Sciences, Department of Physics & Astronomy, Northwestern University, 2145 Sheridan Road, Evanston, IL, 60208, USA n-cowan@northwestern.edu 2 Department of the Geophysical Sciences, University of Chicago, 5734 South Ellis Avenue, Chicago, IL, 60637, USA

text: km),

text: and

text: (perovskite,

text: below

text: km).

text: Com-

text: draw

text: down

text: suﬃcient

text: CO

text: to

text: cool

text: the

text: planet

text: out

text: of

text: the

text: bining

text: these

text: estimates

text: with

text: the

text: pressure-dependence

text: moist

text: greenhouse

text: state,

text: leaving

text: a

text: partially

text: water-covered

text: of

text: olivine

text: water

text: content

text: from

text: Hauri

text: et

text: al.

text: puts

text: planet

text: (Abbot

text: et

text: al.

text: Detailed

text: atmospheric

text: sim-

text: Earth’s

text: mantle

text: water

text: capacity

text: at

text: a

text: dozen

text: times

text: the

text: ulations

text: of

text: such

text: hot

text: planets,

text: however,

text: indicate

text: that

text: the

text: current

text: surface

text: reservoir.

text: Measurements

text: of

text: electrical

text: tendency

text: of

text: carbon

text: dioxide

text: to

text: cool

text: the

text: stratosphere

text: im-

text: conductivity

text: (Dai

text: Karato

text: suggest

text: that

text: Earth’s

text: pedes

text: the

text: loss

text: of

text: water

text: (Wordsworth

text: Pierrehumbert

text: mantle

text: contains

text: oceans

text: worth

text: of

text: water

text: at

text: present-

text: day,

text: but

text: other

text: estimates

text: have

text: diﬀered

text: by

text: a

text: factor

text: of

text: a

text: In

text: an

text: alternate

text: hypothesis,

text: the

text: Rayleigh

text: number

text: of

text: few

text: in

text: either

text: direction

text: (e.g.,

text: Huang

text: et

text: al.

text: Smyth

text: convecting

text: water

text: in

text: hydrothermal

text: systems

text: at

text: mid-ocean

text: Jacobsen

text: Khan

text: Shankland

text: ridges

text: exhibits

text: a

text: sharp

text: peak

text: when

text: water

text: becomes

text: su-

text: The

text: Deep

text: Water

text: Cycle

text: percritical,

text: which

text: occurs

text: at

text: nearly

text: the

text: pressures

text: and

text: temperatures

text: in

text: hydrothermal

text: systems

text: today

text: (Kasting

text: There

text: is

text: a

text: two-way

text: ﬂux

text: of

text: water

text: between

text: ocean

text: and

text: Holm

text: If

text: Earth’s

text: water

text: started

text: in

text: the

text: mantle

text: mantle

text: on

text: Earth.

text: Ocean

text: crust

text: forms

text: at

text: mid-ocean

text: ridges

text: and

text: gradually

text: degassed

text: to

text: the

text: surface,

text: then

text: this

text: pressure-

text: by

text: depressurization

text: melting

text: of

text: mantle,

text: releasing

text: volatiles

text: dependence

text: could

text: explain

text: why

text: the

text: oceans

text: stabilized

text: at

text: into

text: the

text: ocean.

text: Plate

text: tectonics

text: then

text: drives

text: the

text: ocean

text: their

text: current

text: depths.

text: A

text: planet

text: starting

text: with

text: very

text: deep

text: crust

text: toward

text: subduction

text: zones.

text: Hydrothermal

text: alteration,

text: oceans

text: (Korenaga

text: however,

text: would

text: not

text: be

text: able

text: principally

text: serpentinization,

text: produces

text: ocean

text: crust

text: that

text: is

text: to

text: eﬀectively

text: subduct

text: water

text: and

text: would

text: remain

text: a

text: water-

text: water

text: by

text: mass.

text: During

text: subduction,

text: much

text: of

text: the

text: world.

text: water

text: is

text: baked

text: out

text: of

text: the

text: subducting

text: slab,

text: producing

text: ex-

text: In

text: order

text: to

text: solve

text: the

text: problem

text: of

text: regassing

text: water

text: into

text: plosive

text: volcanism

text: like

text: Mount

text: St.

text: Helens.

text: Some

text: of

text: the

text: the

text: mantle,

text: Holm

text: suggested

text: that

text: the

text: reduced

text: ef-

text: water,

text: however,

text: is

text: subducted

text: deep

text: into

text: the

text: mantle,

text: clos-

text: ﬁciency

text: of

text: hydrothermal

text: heat

text: exchange

text: beneath

text: deep

text: ing

text: the

text: loop

text: (for

text: a

text: review

text: of

text: the

text: relevant

text: geochemistry,

text: oceans

text: would

text: result

text: in

text: higher

text: mantle

text: temperatures,

text: see

text: Arndt

text: greater

text: plate

text: velocities

text: and

text: therefore

text: eﬃcient

text: subduction

text: The

text: current

text: H

text: O

text: degassing

text: at

text: mid-ocean

text: ridges

text: is

text: approximately

text: ×2

text: kg/yr

text: (Hirschmann

text: Kohlst-

text: of

text: hydrated

text: crust.

text: This

text: mechanism

text: may

text: not

text: work

text: quan-

text: titatively,

text: however,

text: because

text: hydrothermal

text: heat

text: transfer

text: edt

text: while

text: the

text: regassing

text: ﬂux

text: at

text: subduction

text: zones

text: only

text: accounts

text: for

text: of

text: heat

text: ﬂux

text: through

text: oceanic

text: is

text: estimated

text: to

text: be

text: kg/yr

text: (Jarrard

text: crust

text: (Stein

text: et

text: al.

text: There

text: are

text: also

text: qualitative

text: prob-

text: Schmidt

text: Poli

text: Given

text: the

text: order-of-magnitude

text: lems

text: with

text: this

text: regassing

text: argument:

text: ﬁrst

text: of

text: all,

text: the

text: same

text: agreement

text: of

text: these

text: ﬁgures,

text: and

text: the

text: large

text: associated

text: un-

text: logic

text: should

text: apply

text: to

text: planets

text: with

text: very

text: shallow

text: oceans,

text: certainties,

text: Earth’s

text: current

text: ocean

text: volume

text: is

text: typically

text: potentially

text: negating

text: the

text: supercritical

text: circulation

text: hypoth-

text: assumed

text: to

text: be

text: in

text: a

text: steady

text: state

text: (McGovern

text: Schubert

text: esis.

text: Moreover,

text: hydrating

text: the

text: mantle

text: lowers

text: its

text: viscosity,

text: Moreover,

text: realistic

text: parameterizations

text: of

text: mantle

text: further

text: increasing

text: plate

text: velocities

text: and

text: making

text: this

text: mech-

text: convection

text: predict

text: water

text: ﬂuxes

text: of

text: kg/yr

text: in

text: anism

text: a

text: destabilizing

text: feedback.

text: Lastly,

text: although

text: rapid

text: Earth’s

text: geological

text: past,

text: implying

text: ocean-cycling

text: times

text: of

text: subduction

text: may

text: aid

text: regassing,

text: a

text: hotter

text: mantle

text: almost

text: yrs

text: (McGovern

text: Schubert

text: A

text: signiﬁcant

text: ﬂux

text: certainly

text: hinders

text: it

text: (Bounama

text: et

text: al.

text: In

text: short,

text: it

text: is

text: imbalance

text: would

text: have

text: long

text: ago

text: desiccated

text: or

text: submerged

text: not

text: clear

text: that

text: reduced

text: heat

text: transport

text: through

text: the

text: ocean

text: the

text: planetary

text: surface.

text: crust

text: leads

text: to

text: net

text: regassing

text: of

text: water.

text: Previous

text: Work

text: Whatever

text: geophysical

text: processes

text: govern

text: Earth’s

text: deep

text: A

text: planet

text: entirely

text: covered

text: in

text: water

text: may

text: develop

text: exposed

text: water

text: cycle

text: presumably

text: operate

text: on

text: other

text: terrestrial

text: plan-

text: continents

text: by

text: losing

text: water

text: to

text: space,

text: hydrating

text: the

text: crust

text: ets

text: with

text: plate

text: tectonics.

text: The

text: partitioning

text: of

text: water

text: on

text: and

text: mantle,

text: or

text: reshaping

text: continents

text: and

text: ocean

text: basins.

text: Earth

text: is

text: not

text: only

text: an

text: outstanding

text: problem

text: in

text: geophysics

text: planet

text: with

text: a

text: hot

text: strato-

text: (Fyfe

text: Langmuir

text: Broecker

text: but

text: one

text: with

text: ovn.

text: are

text: ex-

text: important

text: implications

text: for

text: the

text: surface

text: conditions

text: and

text: cli-

text: posed.

text: The

text: resulting

text: vigorous

text: silicate

text: weathering

text: might

text: mate

text: of

text: terrestrial

text: exoplanets.

text: WATER

text: CYCLING

text: BETWEEN

text: OCEAN

text: AND

text: MANTLE:

text: SUPER-EARTHS

text: NEED

text: NOT

text: BE

text: WATERWORLDS

text: Nicolas

text: B.

text: Dorian

text: S.

text: Subject

text: headings:

text: planets

text: and

text: satellites:

text: composition

text: planets

text: and

text: satellites:

text: interiors

text: planets

text: and

text: satellites:

text: oceans

text: planets

text: and

text: satellites:

text: physical

text: evolution

text: planets

text: and

text: satellites:

text: surfaces

text: planets

text: and

text: satellites:

text: tectonics

text: Center

text: for

text: Interdisciplinary

text: Exploration

text: and

text: Research

text: in

text: As-

text: trophysics

text: (CIERA),

text: Department

text: of

text: Earth

text: Planetary

text: Sciences,

text: Department

text: of

text: Physics

text: Astronomy,

text: Northwestern

text: University,

text: Sheridan

text: Road,

text: Evanston,

text: IL,

text: USA

text: n-cowan@northwestern.edu

text: Department

text: of

text: the

text: Geophysical

text: Sciences,

text: University

text: of

text: Chicago,

text: South

text: Ellis

text: Avenue,

text: Chicago,

text: IL,

text: USA


text: Fig. 2.— Waterworld boundary as a function of water mass fraction, ω, and normalized surface gravity, g˜ = g/g⊕: planets in the upper-right corner are waterworlds, while those in the lower-left maintain exposed continents. The solid black line shows the waterworld boundary for our nominal parameters, including the negative feedbacks associated with seaﬂoor pressure. The dotted black line shows the waterworld boundary if one a−ssumes that water can only be stored in a mantle’s transition zone. The solid red line shows the waterworld boundary if one applies the g 1 dependence to Earth’s current hypsometry and pre§sumes that all of a planet’s water resides at the surfac⊕e. The dashed red line accounts for the eﬀects of erosion and isostatic adjustment ( ) but not the deep-water cycle. The black symbol, , denotes Earth if one only considers its surface water reservoir; the arrow indicates Earth’s probable location if one also accounts for water present in the mantle. The gray lines indicate the analytic waterworld boundary for nominal parameters (solid), as well as Earth mantle × − × − water content values of x⊕ = 5.8 10 3(dashed) and x⊕ = 5.8 10 5 (dotted). The blue lines show the waterworld boundary for seaﬂoor pressure-dependencies of φ = 3 (dashed) and φ = 1 (dotted). A terrestrial planet with surface gravity g˜ = 3 corresponds to a 10M⊕ super-Earth (Valencia et al. 2007b). In the high-gravity limit, the mantle contains the en- the higher gravity of super-Earths biases the deep water tirety of the planet’s water, x = ω/f . In practice this cycle in favor of mantle sequestration. Water inventory m caDnnraofttovcecrusriofnorMwaayt2e6r-,r2ic0h20plane≤ts because of the ﬁnite and mantle capacity are both proportional to planetary waPtreerprcinatptaypcietsyet ousfintghLeATmEXansttyllee e(mxulatexampjavx.).8/1N3/o1n0etheless, mass, producing a weak mass-dependence to the water- world boundary. WATER CYCLING BETWEEN OCEAN AND MANTLE: SUPER-EARTHS NEED NOT BE WATERWORLDS Nicolas B. Cowan1 & Dorian S. Abbot2 Subject headings: planets and satellites: composition — planets and satellites: interiors — planets and satellites: oceans — planets and satellites: physical evolution — planets and satellites: surfaces — planets and satellites: tectonics 1 Center for Interdisciplinary Exploration and Research in As- trophysics (CIERA), Department of Earth & Planetary Sciences, ##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## ##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## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Fig.

text: Waterworld

text: boundary

text: as

text: a

text: function

text: of

text: water

text: mass

text: fraction,

text: ω,

text: and

text: normalized

text: surface

text: gravity,

text: g˜

text: g/g⊕:

text: planets

text: in

text: the

text: upper-right

text: corner

text: are

text: waterworlds,

text: while

text: those

text: in

text: the

text: lower-left

text: maintain

text: exposed

text: continents.

text: The

text: solid

text: black

text: line

text: shows

text: the

text: waterworld

text: boundary

text: for

text: our

text: nominal

text: parameters,

text: including

text: the

text: negative

text: feedbacks

text: associated

text: with

text: seaﬂoor

text: pressure.

text: The

text: dotted

text: black

text: line

text: shows

text: the

text: waterworld

text: boundary

text: if

text: one

text: a−ssumes

text: that

text: water

text: can

text: only

text: be

text: stored

text: in

text: a

text: mantle’s

text: transition

text: zone.

text: The

text: solid

text: red

text: line

text: shows

text: the

text: waterworld

text: boundary

text: if

text: one

text: applies

text: the

text: g

text: dependence

text: to

text: Earth’s

text: current

text: hypsometry

text: and

text: pre§sumes

text: that

text: all

text: of

text: a

text: planet’s

text: water

text: resides

text: at

text: the

text: surfac⊕e.

text: The

text: dashed

text: red

text: line

text: accounts

text: for

text: the

text: eﬀects

text: of

text: erosion

text: and

text: isostatic

text: adjustment

text: but

text: not

text: the

text: deep-water

text: cycle.

text: The

text: black

text: symbol,

text: denotes

text: Earth

text: if

text: one

text: only

text: considers

text: its

text: surface

text: water

text: reservoir;

text: the

text: arrow

text: indicates

text: Earth’s

text: probable

text: location

text: if

text: one

text: also

text: accounts

text: for

text: water

text: present

text: in

text: the

text: mantle.

text: The

text: gray

text: lines

text: indicate

text: the

text: analytic

text: waterworld

text: boundary

text: for

text: nominal

text: parameters

text: (solid),

text: as

text: well

text: as

text: Earth

text: mantle

text: water

text: content

text: values

text: of

text: x⊕

text: and

text: x⊕

text: (dotted).

text: The

text: blue

text: lines

text: show

text: the

text: waterworld

text: boundary

text: for

text: seaﬂoor

text: pressure-dependencies

text: of

text: φ

text: (dashed)

text: and

text: φ

text: (dotted).

text: A

text: terrestrial

text: planet

text: with

text: surface

text: gravity

text: g˜

text: corresponds

text: to

text: a

text: super-Earth

text: (Valencia

text: et

text: al.

text: In

text: the

text: high-gravity

text: limit,

text: the

text: mantle

text: contains

text: the

text: en-

text: the

text: higher

text: gravity

text: of

text: super-Earths

text: biases

text: the

text: deep

text: water

text: tirety

text: of

text: the

text: planet’s

text: water,

text: x

text: ω/f

text: In

text: practice

text: this

text: cycle

text: in

text: favor

text: of

text: mantle

text: sequestration.

text: Water

text: inventory

text: m

text: because

text: of

text: the

text: ﬁnite

text: and

text: mantle

text: capacity

text: are

text: both

text: proportional

text: to

text: planetary

text: waPtreerprcinatptaypcietsyet

text: ousfintghLeATmEXansttyllee

text: mass,

text: producing

text: a

text: weak

text: mass-dependence

text: to

text: the

text: water-

text: world

text: boundary.

text: WATER

text: CYCLING

text: BETWEEN

text: OCEAN

text: AND

text: MANTLE:

text: SUPER-EARTHS

text: NEED

text: NOT

text: BE

text: WATERWORLDS

text: Nicolas

text: B.

text: Dorian

text: S.

text: Subject

text: headings:

text: planets

text: and

text: satellites:

text: composition

text: planets

text: and

text: satellites:

text: interiors

text: planets

text: and

text: satellites:

text: oceans

text: planets

text: and

text: satellites:

text: physical

text: evolution

text: planets

text: and

text: satellites:

text: surfaces

text: planets

text: and

text: satellites:

text: tectonics

text: Center

text: for

text: Interdisciplinary

text: Exploration

text: and

text: Research

text: in

text: As-

text: trophysics

text: (CIERA),

text: Department

text: of

text: Earth

text: Planetary

text: Sciences,

text: ##LTFigure##

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