text: 10 Sofía A. Cora et al. 1.5 0.0 Boselli et al (2014) SAG SAGβ1.3 1.0 z = 0 −0.5 lloogg((MM⋆⋆[[MM⊙⊙]]))∈∈[[91.07.,11,01.01.]5] log(M⋆[M⊙])∈[10.5,10.9] 0.5 −1.0 lloogg((MM⋆⋆[[MM⊙⊙]]))∈>[1110..39,11.3] Behroozietal. (2013c) 0.0 −1.5 −0.5 −2.0 −1.0 −2.5 −1.5 −3.0 8.5 9.0 9.5 10.0 10.5 11.0 11.5 0 1 2 3 4 5 6 log(M⋆[M⊙]) z Figure 4. Fraction of mass in cold gas as a function of stellar mass at Figure 5. Evolution of the cosmic star formation rate density for all galaxies 0 used as constraints to calibrate the sag model. Results obtained from (redthicksolidline)comparedwithobservationaldatacompiledbyBehroozi z = the calibrated model sag with the best-ﬁtting values given in Table 1 are et al. (2013c). Contribution from galaxies lying in diﬀerent mass ranges at representedbyredsolidlinesandreddishﬁlledcontours.Redthinsolidlines 0 are represented by thin black lines of diﬀerent style, as indicated in the z = denote the standard deviation around the mean (thick red solid line). The legend.Downsizinginstellarmassassemblybecomesevident.Intermediate- caodnotpotuinrglevels1ar3e,thasesshamowenasbythroesseuoltfsFfirgo.m3.tCheGsMagF is almmoosdteulnraeﬀpreecsteendtbeyd mexacsesssgainlatxhieesSlFoRgD(Ma(cid:63)t [M(cid:12)0].)T∈he[1g0l.o1b,a1l0S.F5]RaDreotbhteaimneadinfrroemspothnesicballeiborfatthede by a blueβda=she.d line and empty dotted contours. Thβis1.r3elation is compared model sag but adoptinzg= 1 3 (model sag ) is added for comparison, with observations by Boselli et al. (2014, ﬁlled circles with error bars). beingrepresentedbyabluβe=thic.kdashedline.Mβ1o.3delsag ischaracterized 1 3 by higher (lower) levels of SFR at high (low) redshiftsβw.ith respect to the calibrated model sag, in better agreement with observations. gas accretions onto BHs are needed to make the AGN feedback as eﬀective as possible. Thus, during the calibration process, the pa- the shaded contours show the distribution of model values. For high rvaamlueetehrigthhaetrcthoanntr0ols02thwehgircohwhtahsobfeseunpfeorumnadstsoivbeeBmHosr,efaBdHe,qtuaakteestoa stellar masses (log(M(cid:63)[M(cid:12)]) > 10.52), sag predicts lower fractions . of cold gas with respect to those expected from a simple extrapo- reproduce the BHB relation (see Table 1). Despite this strong eﬀect, lation of the trend found at lower masses. However, these values the model is not able to recover the high mass end of the SMF at are still contained within the range of cold gas fractions inferred 0. However, the AGN feedback together with additional mod- z = from the atomic and molecular gas fractions presented by Sain- iﬁcations introduced in the model, that is, the redshift dependence tonge et al. (2016, see their ﬁg. 3), obtained from data of the Arecibo of the reheated and ejected material and the scalings adopted for Legacy Fast ALFA (ALFALFA), the GALEX Arecibo SDSS Sur- the reincorporation of the latter (Equations 10, 12 and 13), allow vey (GASS) and the CO Legacy Database for GASS (COLD GASS) to recover downsizing in the star formation rates, as shown later in surveys. The same trend is obtained with model sag . Fig. 5. Hence, there is not an excess of SF in high mass galaxies β1.3 6 MODEL PREDICTIONS that could be responsible for the excess of the high mass end of the SMF at 0. Other physical processes must be revisited, like TS We present galaxy properties predicted by the model, that are z = and disruption of satellites galaxies, since dry mergers at (cid:46) 2 of not used as constraints to calibrate it. We ﬁrst focus on the evolution z low mass galaxies with massive ones seem to produce the excess of the cosmic star formation rate density (SFRD) and of the speciﬁc at the high mass end of the SMF at low redshift. This hypothesis star formation rate (sSFR). The former is given by the volume- is supported by a test made with the semi-analytic model Sage dis- averaged sum of SF of all galaxies at any given time, and the latter cussed in Knebe et al. (2017a), which shows that stars added to an is deﬁned as the ratio between the SFR and the stellar mass of the intra-cluster component as a result of tidal disruption of satellite galaxy. We connect the trends of the sSFR of satellites and centrals galaxies would enhance the massive end of the SMF if they would with the stellar mass content of galaxies populating DM haloes have merged with the central galaxy. Deeper analysis of the impact of diﬀerent mass through the stellar-to-halo mass relation (SHM). of galaxy mergers and TS in sag are presented in Vega-Martínez et According to the value of the sSFR, we classify galaxies as active al. (in prep.). and passive. We analyse the dependences of the fraction of passive 5.2.4 Cold gas mass fraction galaxies with stellar mass, halo mass and halo-centric distance. 6.1 Star formation rate density The fact that galaxy growth and star formation quenching are well captured by sag becomes evident in the cold gas content of The evolution of the cosmic SFRD predicted by sag is pre- model galaxies (Fig. 4). The CGMF obtained from sag also behaves sentedinFig.5(thicksolidline)andiscomparedwithdatacompiled as expected within the stellar mass range considered for the cali- byBehroozietal.(2013c).Althoughmodelresultsshowconsistency bration of the model (log(M(cid:63)[M(cid:12)]) ∈ [9.15,10.52]). Their mean with observations for z < 4, they underpredict the SFRD at higher values (thick solid line) are in good agreement with observational redshifts. Besides, the decline in the SFRD towards low redshifts is data from Boselli et al. (2014, ﬁlled circles with error bars); thin less pronounced in the model than in observations, thus resulting solid lines represent the standard deviation around the mean and in an excess of SFRD at 0. The peak and normalization of the z = MNRAS 000, 1–21 (2017) )⋆M/dlocM(gol )]3−cpM1−ry⊙M[ DRFS(gol ##LTFigure## ##LTFigure## ##LTFigure## ##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## 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text: 2 Sofía A. Cora et al. contribute to determine the role of environment in the evolution of quenched galaxies as a function of stellar mass, halo mass and the low-mass galaxies (Balogh et al. 2017). halo-centric distances, in good agreement with the trends inferred Thedependencesofthefractionofquenchedgalaxieswithstel- by Wetzel et al. (2012). Thus, the galaxy population generated by lar mass, halo mass and halo-centric distance are a consequence of applying this model, referred to as sag in Paper I, on the dark 1 3 the interplay between mass quenching and environmental quench- matter (DM) cosmological MultiDarkβsi.mulation MDPL2 is suit- ing along galaxy evolution and their dependences on stellar mass able to carry out the current study. The aim of this work is to for diﬀerent environments. From the stellar and halo mass depen- contribute to our understanding of the relevance of mass quenching dences of the quenched fractions, Wetzel et al. (2012) conclude and environmental quenching on currently passive satellite galaxies, that, at higher stellar mass, environmental eﬀects are less impor- and to determine the time-scales involved in the quenching process. tant in determining the passive fraction since more galaxies quench This paper is organized as follows. Section presents a brief as central galaxies before becoming satellites. This was later cor- description of the MDPL2 simulation and model sag , summa- 1 3 roborated by Wetzel et al. (2013, W13, hereafter) through a more rizing the main conclusions of Paper I. In Section ,βwe. present a detailed study. They ﬁnd that, for M (cid:46) 1 × 1010 M(cid:12), all currently detailed analysis of the dependences of the fraction of quenched quiescent satellites quenched as sate(cid:63)llites, and that half of satellites galaxies with the time of ﬁrst infall, focusing on the fraction of with M ∼ 1×1011 M(cid:12) that are quiescent at z = 0 quench their SF currently quenched satellites and those satellites already quenched as centr(cid:63)al galaxies prior to infall. Their results are obtained from a when being accreted. We discuss the implications of the relation galaxy catalogue constructed from a high-resolution cosmological between these fractions. Section includes estimations of the time- simulation by applying the subhalo abundance matching (SHAM) scales involved in the whole quenching process, which comprises a technique (Conroy et al. 2006; Vale & Ostriker 2006) to assign delay before the onset of SF quenching and the fading of the SFR. stellar mass to subhaloes at each redshift. We identify the physical processes associated to these two stages, Numerous works in the literature addressed these issues using evaluating the relative role of mass and environmental quenching. both hydrodynamical simulations (e.g. Bahé & McCarthy 2015; We present our conclusions in Section . Taylor et al. 2017; Davé et al. 2017) and semi-analytic models of 2 GALAXY FORMATION MODEL galaxy formation (SAMs; e.g. De Lucia et al. 2012; Henriques et al. The galaxy formation model combines our semi-analytic 2017; Stevens & Brown 2017). Although there is intense debate on model of galaxy formation and evolution sag and the cosmolog- the mechanisms driving SF quenching, it is being accepted that it ical DM MultiDark simulation MDPL2, which is part of the begins once the satellite hot gas halo has been completely removed CosmoSim database . MDPL2 simulation follows the evolution (cid:63) (Schawinski et al. 2014; Peng et al. 2015). Thus, the cold gas disc of 38403 particles within a box of side-length 1 −1 Gpc, with a iwsinthoinloangfeerwreGpylerngisihveindgbyplaccoeoltiongaﬂdoewcslinanindgitstwarillfobremcaotinosnumraetde mass resolution mp = 1.5 × 109 h−1 M(cid:12) per DM hparticle (Klypin et al. 2016). It is consistent with a ﬂat CDM model character- (SFR) until the galaxy becomes passive. This can be considered as Λ ized by Planck cosmological parameters: = 0.307, = 0.693, the second stage in the delay-then-rapid quenching scenario pro- = 0.048, = 0.96 and = 100 −Ω1mkm s−1 MpΩcΛ−1, where posed by W13, in which the time-scales for the onset of quenching ΩB ns H0 h = 0.678 (Planck Collaboration et al. 2014). The simulation pro- since time of ﬁrst infall vary from ≈ 4Gyr for low mass galaxies h vides the DM haloes and their corresponding merger trees required (≈ 1010 M(cid:12)) to ≈ 2Gyr for high mass ones (≈ 1011 M(cid:12)). Once the by sag to generate the galaxy population and track the evolution of SF quenching begins, it takes place in a short time-scale ((cid:46) 1Gyr). galaxy properties. DM haloes have been identiﬁed with the Rock- These results were later supported by Oman & Hudson (2016), who star halo ﬁnder (Behroozi et al. 2013a), and merger trees were use outputs of the Multidark Run 1 dark matter only cosmological constructed with ConsistentTrees (Behroozi et al. 2013b). DM simulation to construct subhalo orbit libraries in projected phase haloes detected over the background density are referred to as main space and estimate the probability distribution of possible orbits for host haloes, whereas those lying within another DM haloes are sub- a sample of observed Sloan Digital Sky Survey (SDSS) galaxies haloes. sag assigns one galaxy to each new detected halo in the for which they have their SFR. Short time-scales for the fading of simulation. Thus, each system of haloes contains a central galaxy SFR are also characteristic of high redshift galaxies as found from associated to the main host halo and satellite galaxies. Those galax- phase space modelling (Muzzin et al. 2014; Haines et al. 2015) and ies that are assigned to DM haloes that are no longer identiﬁed by the fraction of red, blue, and green galaxies in groups (Mok et al. the halo ﬁnder after the merger of two haloes are called orphan 2013). satellites. By identifying the mergers of haloes in the merger trees, The way in which the hot gas halo of accreted galaxies is we can also deﬁne the time of ﬁrst infall of a galaxy as the last time removed is crucial in determining the star formation history and a galaxy was identiﬁed as central. quenching time-scales of satellite galaxies. The updated version The model sag originates from the semi-analytic model pre- of our semi-analytic model of galaxy formation sag (acronym for sented in Springel et al. (2001) and was further modiﬁed as de- Semi-Analytic Galaxies) presented in Cora et al. (2018a, submit- scribed in Cora (2006), Lagos et al. (2008), Tecce et al. (2010), ted; Paper I, hereafter) considers the gradual removal of hot gas Orsi et al. (2014), Muñoz Arancibia et al. (2015) and Gargiulo of satellites by the action of RPS and TS. The former process also et al. (2015). sag includes the eﬀects of radiative cooling of hot aﬀects the cold gas disc, under certain conditions. The combination gas, star formation, feedback from SN explosions and AGN, and of this improved treatment of environmental eﬀects with a modi- starbursts triggered by disc instabilities and/or galaxy mergers, and ﬁed modelling of supernova (SN) feedback, in which the reheated features a chemical enrichment model that tracks several chemical and ejected mass depend explicitly on redshift, has been success- elements contributed by diﬀerent sources (stellar winds and super- ful in reproducing several galaxy properties at both low and high novae Type Ia and II) taking into account the lifetime of progenitors. redshifts. A variant of the model in which this redshift dependence This model has been improved by the implementation of a proper is mild, as the one suggested by the zoom FIRE (Feedback in Re- alistic Environments) hydrodynamical simulations (Muratov et al. 2015), allows to recover the expected behaviour of the fractions of (cid:63) https://www.cosmosim.org MNRAS 000, 1–11 (2017)

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text: detailed

text: analysis

text: of

text: the

text: dependences

text: of

text: the

text: fraction

text: of

text: quenched

text: quiescent

text: satellites

text: quenched

text: as

text: and

text: that

text: half

text: of

text: satellites

text: galaxies

text: with

text: the

text: time

text: of

text: ﬁrst

text: infall,

text: focusing

text: on

text: the

text: fraction

text: of

text: with

text: M

text: that

text: are

text: quiescent

text: at

text: z

text: quench

text: their

text: SF

text: currently

text: quenched

text: satellites

text: and

text: those

text: satellites

text: already

text: quenched

text: as

text: galaxies

text: prior

text: to

text: infall.

text: Their

text: results

text: are

text: obtained

text: from

text: a

text: when

text: being

text: accreted.

text: We

text: discuss

text: the

text: implications

text: of

text: the

text: relation

text: galaxy

text: catalogue

text: constructed

text: from

text: a

text: high-resolution

text: cosmological

text: between

text: these

text: fractions.

text: Section

text: includes

text: estimations

text: of

text: the

text: time-

text: simulation

text: by

text: applying

text: the

text: subhalo

text: abundance

text: matching

text: (SHAM)

text: scales

text: involved

text: in

text: the

text: whole

text: quenching

text: process,

text: which

text: comprises

text: a

text: technique

text: (Conroy

text: et

text: al.

text: Vale

text: Ostriker

text: to

text: assign

text: delay

text: before

text: the

text: onset

text: of

text: SF

text: quenching

text: and

text: the

text: fading

text: of

text: the

text: SFR.

text: stellar

text: mass

text: to

text: subhaloes

text: at

text: each

text: redshift.

text: We

text: identify

text: the

text: physical

text: processes

text: associated

text: to

text: these

text: two

text: stages,

text: Numerous

text: works

text: in

text: the

text: literature

text: addressed

text: these

text: issues

text: using

text: evaluating

text: the

text: relative

text: role

text: of

text: mass

text: and

text: environmental

text: quenching.

text: both

text: hydrodynamical

text: simulations

text: (e.g.

text: Bahé

text: McCarthy

text: We

text: present

text: our

text: conclusions

text: in

text: Section

text: Taylor

text: et

text: al.

text: Davé

text: et

text: al.

text: and

text: semi-analytic

text: models

text: of

text: GALAXY

text: FORMATION

text: MODEL

text: galaxy

text: formation

text: (SAMs;

text: e.g.

text: De

text: Lucia

text: et

text: al.

text: Henriques

text: et

text: al.

text: The

text: galaxy

text: formation

text: model

text: combines

text: our

text: semi-analytic

text: Stevens

text: Brown

text: Although

text: there

text: is

text: intense

text: debate

text: on

text: model

text: of

text: galaxy

text: formation

text: and

text: evolution

text: sag

text: and

text: the

text: cosmolog-

text: the

text: mechanisms

text: driving

text: SF

text: quenching,

text: it

text: is

text: being

text: accepted

text: that

text: it

text: ical

text: DM

text: MultiDark

text: simulation

text: which

text: is

text: part

text: of

text: the

text: begins

text: once

text: the

text: satellite

text: hot

text: gas

text: halo

text: has

text: been

text: completely

text: removed

text: CosmoSim

text: database

text: simulation

text: follows

text: the

text: evolution

text: (Schawinski

text: et

text: al.

text: Peng

text: et

text: al.

text: Thus,

text: the

text: cold

text: gas

text: disc

text: of

text: particles

text: within

text: a

text: box

text: of

text: side-length

text: −1

text: Gpc,

text: with

text: a

text: mass

text: resolution

text: mp

text: per

text: DM

text: hparticle

text: (Klypin

text: et

text: al.

text: It

text: is

text: consistent

text: with

text: a

text: ﬂat

text: CDM

text: model

text: character-

text: (SFR)

text: until

text: the

text: galaxy

text: becomes

text: passive.

text: This

text: can

text: be

text: considered

text: as

text: Λ

text: ized

text: by

text: Planck

text: cosmological

text: parameters:

text: the

text: second

text: stage

text: in

text: the

text: delay-then-rapid

text: quenching

text: scenario

text: pro-

text: and

text: where

text: posed

text: by

text: in

text: which

text: the

text: time-scales

text: for

text: the

text: onset

text: of

text: quenching

text: ΩB

text: ns

text: h

text: (Planck

text: Collaboration

text: et

text: al.

text: The

text: simulation

text: pro-

text: since

text: time

text: of

text: ﬁrst

text: infall

text: vary

text: from

text: for

text: low

text: mass

text: galaxies

text: h

text: vides

text: the

text: DM

text: haloes

text: and

text: their

text: corresponding

text: merger

text: trees

text: required

text: to

text: for

text: high

text: mass

text: ones

text: Once

text: the

text: by

text: sag

text: to

text: generate

text: the

text: galaxy

text: population

text: and

text: track

text: the

text: evolution

text: of

text: SF

text: quenching

text: begins,

text: it

text: takes

text: place

text: in

text: a

text: short

text: time-scale

text: galaxy

text: properties.

text: DM

text: haloes

text: have

text: been

text: identiﬁed

text: with

text: the

text: Rock-

text: These

text: results

text: were

text: later

text: supported

text: by

text: Oman

text: Hudson

text: who

text: star

text: halo

text: ﬁnder

text: (Behroozi

text: et

text: al.

text: and

text: merger

text: trees

text: were

text: use

text: outputs

text: of

text: the

text: Multidark

text: Run

text: dark

text: matter

text: only

text: cosmological

text: constructed

text: with

text: ConsistentTrees

text: (Behroozi

text: et

text: al.

text: DM

text: simulation

text: to

text: construct

text: subhalo

text: orbit

text: libraries

text: in

text: projected

text: phase

text: haloes

text: detected

text: over

text: the

text: background

text: density

text: are

text: referred

text: to

text: as

text: main

text: space

text: and

text: estimate

text: the

text: probability

text: distribution

text: of

text: possible

text: orbits

text: for

text: host

text: haloes,

text: whereas

text: those

text: lying

text: within

text: another

text: DM

text: haloes

text: are

text: sub-

text: a

text: sample

text: of

text: observed

text: Sloan

text: Digital

text: Sky

text: Survey

text: (SDSS)

text: galaxies

text: haloes.

text: sag

text: assigns

text: one

text: galaxy

text: to

text: each

text: new

text: detected

text: halo

text: in

text: the

text: for

text: which

text: they

text: have

text: their

text: SFR.

text: Short

text: time-scales

text: for

text: the

text: fading

text: of

text: simulation.

text: Thus,

text: each

text: system

text: of

text: haloes

text: contains

text: a

text: central

text: galaxy

text: SFR

text: are

text: also

text: characteristic

text: of

text: high

text: redshift

text: galaxies

text: as

text: found

text: from

text: associated

text: to

text: the

text: main

text: host

text: halo

text: and

text: satellite

text: galaxies.

text: Those

text: galax-

text: phase

text: space

text: modelling

text: (Muzzin

text: et

text: al.

text: Haines

text: et

text: al.

text: and

text: ies

text: that

text: are

text: assigned

text: to

text: DM

text: haloes

text: that

text: are

text: no

text: longer

text: identiﬁed

text: by

text: the

text: fraction

text: of

text: red,

text: blue,

text: and

text: green

text: galaxies

text: in

text: groups

text: (Mok

text: et

text: al.

text: the

text: halo

text: ﬁnder

text: after

text: the

text: merger

text: of

text: two

text: haloes

text: are

text: called

text: orphan

text: satellites.

text: By

text: identifying

text: the

text: mergers

text: of

text: haloes

text: in

text: the

text: merger

text: trees,

text: The

text: way

text: in

text: which

text: the

text: hot

text: gas

text: halo

text: of

text: accreted

text: galaxies

text: is

text: we

text: can

text: also

text: deﬁne

text: the

text: time

text: of

text: ﬁrst

text: infall

text: of

text: a

text: galaxy

text: as

text: the

text: last

text: time

text: removed

text: is

text: crucial

text: in

text: determining

text: the

text: star

text: formation

text: history

text: and

text: a

text: galaxy

text: was

text: identiﬁed

text: as

text: central.

text: quenching

text: time-scales

text: of

text: satellite

text: galaxies.

text: The

text: updated

text: version

text: The

text: model

text: sag

text: originates

text: from

text: the

text: semi-analytic

text: model

text: pre-

text: of

text: our

text: semi-analytic

text: model

text: of

text: galaxy

text: formation

text: sag

text: (acronym

text: for

text: sented

text: in

text: Springel

text: et

text: al.

text: and

text: was

text: further

text: modiﬁed

text: as

text: de-

text: Semi-Analytic

text: Galaxies)

text: presented

text: in

text: Cora

text: et

text: al.

text: submit-

text: scribed

text: in

text: Cora

text: Lagos

text: et

text: al.

text: Tecce

text: et

text: al.

text: ted;

text: Paper

text: I,

text: hereafter)

text: considers

text: the

text: gradual

text: removal

text: of

text: hot

text: gas

text: Orsi

text: et

text: al.

text: Muñoz

text: Arancibia

text: et

text: al.

text: and

text: Gargiulo

text: of

text: satellites

text: by

text: the

text: action

text: of

text: RPS

text: and

text: TS.

text: The

text: former

text: process

text: also

text: et

text: al.

text: sag

text: includes

text: the

text: eﬀects

text: of

text: radiative

text: cooling

text: of

text: hot

text: aﬀects

text: the

text: cold

text: gas

text: disc,

text: under

text: certain

text: conditions.

text: The

text: combination

text: gas,

text: star

text: formation,

text: feedback

text: from

text: SN

text: explosions

text: and

text: AGN,

text: and

text: of

text: this

text: improved

text: treatment

text: of

text: environmental

text: eﬀects

text: with

text: a

text: modi-

text: starbursts

text: triggered

text: by

text: disc

text: instabilities

text: and/or

text: galaxy

text: mergers,

text: and

text: ﬁed

text: modelling

text: of

text: supernova

text: (SN)

text: feedback,

text: in

text: which

text: the

text: reheated

text: features

text: a

text: chemical

text: enrichment

text: model

text: that

text: tracks

text: several

text: chemical

text: and

text: ejected

text: mass

text: depend

text: explicitly

text: on

text: redshift,

text: has

text: been

text: success-

text: elements

text: contributed

text: by

text: diﬀerent

text: sources

text: (stellar

text: winds

text: and

text: super-

text: ful

text: in

text: reproducing

text: several

text: galaxy

text: properties

text: at

text: both

text: low

text: and

text: high

text: novae

text: Type

text: Ia

text: and

text: II)

text: taking

text: into

text: account

text: the

text: lifetime

text: of

text: progenitors.

text: redshifts.

text: A

text: variant

text: of

text: the

text: model

text: in

text: which

text: this

text: redshift

text: dependence

text: This

text: model

text: has

text: been

text: improved

text: by

text: the

text: implementation

text: of

text: a

text: proper

text: is

text: mild,

text: as

text: the

text: one

text: suggested

text: by

text: the

text: zoom

text: FIRE

text: (Feedback

text: in

text: Re-

text: alistic

text: Environments)

text: hydrodynamical

text: simulations

text: (Muratov

text: et

text: al.

text: allows

text: to

text: recover

text: the

text: expected

text: behaviour

text: of

text: the

text: fractions

text: of

text: MNRAS


text: Semi-Analytic Galaxies II 3 treatment of environmental eﬀects on satellite galaxies. The stran- characterizing model sag , allows to achieve the expected be- 1 3 gulation scheme, in which the hot gas halo is removed instantly haviour of the fractions ofβqu.enched galaxies as a function of stellar when the galaxy becomes a satellite, is replaced by gradual starva- mass, halo mass and the halo-centric distances (see ﬁgs. 10 and 11 tion produced by combined action of RPS and TS. Cold gas disc in Paper I). The analysis of the galaxy catalogue generated by the is also subject to these processes when it is not longer shielded by model sag , allows us to conclude that RPS plays a dominant 1 3 the hot gas halo. Stellar mass loss as a result of TS is also taken role amongβth. e environmental processes considered in our model, into account. The advantage of our implementation with respect to being more eﬀective for lower mass galaxies and at lower redshifts, previous SAMs (Gonzalez-Perez et al. 2014; Henriques et al. 2017; and contributes to regulate adequately the mass of the hot gas halo Stevens & Brown 2017) resides in the integration of the orbits of and the cold gas disc. While the gradual starvation of the hot gas orphan galaxies according to the potential well of the host halo, reservoir is a key ingredient to determine the right SF history of taking into account mass loss by TS and dynamical friction eﬀects, satellite galaxies, the mild eﬀect suﬀered by the cold gas disc is and in the use of ﬁtting formulae to estimate RP as a function of halo necessary to achieve the correct atomic gas content (see ﬁg. 14 mass, halo-centric distance and redshift implemented in our model in Paper I). In view of these results, the current work is based on (Vega-Martínez et al., in prep.). The combination of these two im- the analysis of the properties of galaxies generated by the model provements allows to have values of RP consistent with the position sag . 1 3 and velocity of satellite galaxies. The latest version of sag also 3 βQU. ENCHED FRACTIONS: DEPENDENCES WITH considers a modiﬁcation in the SN feedback scheme that includes TIME AT FIRST INFALL an explicit redshift dependence of the reheated and ejected mass. A Both mass and environmental quenching can aﬀect the galaxy detailed description of the updated version of sag is presented in after being accreted but their relative role depends on both stellar Paper I. mass and host halo mass and cannot be estimated from the fraction The free parameters of this model were tuned using the Particle ofquenchedgalaxiesat 0.Weexplorethisissuemakingadeeper Swarm Optimization Technique (PSO) by Ruiz et al. (2015) adopt- z = analysis of further information provided by the model sag . This ing a set of ﬁve observational constraints deﬁned in Knebe et al. involves the dependences of quenched fractions and steβll1a.r3 mass (2017). Namely, the stellar mass functions (SMF) at 0 and 2 z = z = growth with the time of ﬁrst infall, deﬁned as the moment in which (compilation data used by Henriques et al. 2015), the star formation the galaxy becomes a satellite for the ﬁrst time; there is strong rate (SFR) distribution function (data presented by Gruppioni et al. evidence that environmental eﬀects produce SF quenching since 2015 for the redshift interval ∈ [0 0 0 3]), the fraction of mass in z . , . ﬁrst infall, regardless of the host halo mass (W13). We denote the cold gas as a function of stellar mass (data from Boselli et al. 2014), corresponding redshift as . and the relation between bulge mass and the mass of the central su- zinfall 3.1 Currently quenched fractions permassive black hole (combined datasets from McConnell & Ma 2013 and Kormendy & Ho 2013). During the calibration process, Fig. 1 shows the fraction of satellite galaxies that are quenched the power-law slope that characterizes the redshift dependence in- at 0, , as a function of . All galaxies identiﬁed as β z = f qz0 zinfall cluded in the modiﬁed SN feedback scheme was considered a free satellites within main host haloes of mass log(Mhalo[M(cid:12)]) > 12.5 parameter of the model. As discussed in Paper I, the constraining at the present are considered. Satellites are binned according to their power of the SMF at z = 2 makes the value of this parameter too present-day stellar mass as indicated in the legend. We can see that, high (β = 1.99) with respect to the one suggested by the zoom for a given stellar mass range, the trend of f qz0 is as expected, with FIRE hydrodynamical simulations of Muratov et al. (2015) from larger fractions achieved by satellite galaxies with higher values of which the redshift dependence of the mass-loading factor was taken , which means that galaxies have been satellites longer hav- zinfall (β = 1.3). As a consequence of this, the predicted SFRD becomes ing more time to suﬀer environmental quenching. Values of f qz0 too low with respect of observational data at high redshifts. A bet- are larger for larger stellar masses, at any . However, the de- zinfall ter agreement between model predictions and observational data is pendence of with is more pronounced for less massive f qz0 zinfall obtained by adopting β = 1.3, keeping the values of the rest of the satellites. This trend becomes gradually milder for more massive free parameters obtained from the calibration process. This variant galaxies. For a given stellar mass range, the average values of the of the model is referred to as the model sag in Paper I. The cost quenched fractions involving all possible values of are con- 1 3 zinfall of this choice is a higher number density oβf .low mass galaxies in sistent with the results of the stellar and halo mass dependence of the SMF at z = 2. f qz0 shown in ﬁg. 10 of Paper I. The relations f qz0 vs. zinfall are Galaxy star formation histories have a strong impact on the separated by a major gap between the two lowest and two high- fractionofpassiveorquenchedgalaxiesatdiﬀerentredshifts.Galax- est stellar mass bins which is produced by the underprediction of ies are classiﬁed as active or passive according to their speciﬁc star passive galaxies in the mass range log(M(cid:63)[M(cid:12)]) ∈ [10.1,10.5], formation rate density (sSFR, given by the ratio between their stel- discussed in Paper I. lar mass and SFR). Passive or quenched galaxies are those with Low mass satellites characterized by ≈ 0 are mostly ac- zinfall sSFR 10−10 7 yr−1, according to the cut that emerges from the tive galaxies ( ≈ 0 0 − 0 1). This is because SF quenching as < . f qz0 . . distribution of sSFR of our model galaxies (see ﬁg. 8 in Paper I). a result of environmental eﬀects is negligible for galaxies that have The underprediction of SFRD at high redshifts given by the cali- been recently accreted, and these galaxies are too small to suﬀer the brated model sag translates into less passive galaxies at 0, since consequences of mass quenching while they were centrals. How- z = the bulk of SF is shifted to lower redshifts and the galaxy population ever, more than 50 per cent of high mass satellites accreted at the has no enough time to quench their SF. Thus, the local fraction of present epoch are passive, which means that massive galaxies have quenched galaxies is lower than observed (see ﬁg. 9 in Paper I). Only suﬀered mass quenching processes as centrals prior to infall (e.g., tDhMe mhoosstt hmaalosssivweitghalmaxaisesses(lMogh(aMlo(cid:63)(cid:38)[M1(cid:12)0]1)4 M∈ (cid:12)[1,0a.c9h,i1e1v.e3]q)u,ewncithheidn AtieGsNthafeteedxbhaacuks,t sthtaerbcoulrdstsgatsrirgegseerrevdoibr)yams earlrgeeardsyanfodudnidskbyinvsatnabdielin- fractions similar to those presented by Wetzel et al. (2012). A milder Bosch et al. (2008). This eﬀect is more pronounced for more mas- redshift dependence of the reheated and ejected mass, as the one sive galaxies, in agreement with results from Lin et al. (2014) and MNRAS 000, 1–11 (2017)

text: Semi-Analytic

text: Galaxies

text: II

text: treatment

text: of

text: environmental

text: eﬀects

text: on

text: satellite

text: galaxies.

text: The

text: stran-

text: characterizing

text: model

text: sag

text: allows

text: to

text: achieve

text: the

text: expected

text: be-

text: gulation

text: scheme,

text: in

text: which

text: the

text: hot

text: gas

text: halo

text: is

text: removed

text: instantly

text: haviour

text: of

text: the

text: fractions

text: ofβqu.enched

text: galaxies

text: as

text: a

text: function

text: of

text: stellar

text: when

text: the

text: galaxy

text: becomes

text: a

text: satellite,

text: is

text: replaced

text: by

text: gradual

text: starva-

text: mass,

text: halo

text: mass

text: and

text: the

text: halo-centric

text: distances

text: (see

text: ﬁgs.

text: and

text: tion

text: produced

text: by

text: combined

text: action

text: of

text: RPS

text: and

text: TS.

text: Cold

text: gas

text: disc

text: in

text: Paper

text: I).

text: The

text: analysis

text: of

text: the

text: galaxy

text: catalogue

text: generated

text: by

text: the

text: is

text: also

text: subject

text: to

text: these

text: processes

text: when

text: it

text: is

text: not

text: longer

text: shielded

text: by

text: model

text: sag

text: allows

text: us

text: to

text: conclude

text: that

text: RPS

text: plays

text: a

text: dominant

text: the

text: hot

text: gas

text: halo.

text: Stellar

text: mass

text: loss

text: as

text: a

text: result

text: of

text: TS

text: is

text: also

text: taken

text: role

text: amongβth.

text: e

text: environmental

text: processes

text: considered

text: in

text: our

text: model,

text: into

text: account.

text: The

text: advantage

text: of

text: our

text: implementation

text: with

text: respect

text: to

text: being

text: more

text: eﬀective

text: for

text: lower

text: mass

text: galaxies

text: and

text: at

text: lower

text: redshifts,

text: previous

text: SAMs

text: (Gonzalez-Perez

text: et

text: al.

text: Henriques

text: et

text: al.

text: and

text: contributes

text: to

text: regulate

text: adequately

text: the

text: mass

text: of

text: the

text: hot

text: gas

text: halo

text: Stevens

text: Brown

text: resides

text: in

text: the

text: integration

text: of

text: the

text: orbits

text: of

text: and

text: the

text: cold

text: gas

text: disc.

text: While

text: the

text: gradual

text: starvation

text: of

text: the

text: hot

text: gas

text: orphan

text: galaxies

text: according

text: to

text: the

text: potential

text: well

text: of

text: the

text: host

text: halo,

text: reservoir

text: is

text: a

text: key

text: ingredient

text: to

text: determine

text: the

text: right

text: SF

text: history

text: of

text: taking

text: into

text: account

text: mass

text: loss

text: by

text: TS

text: and

text: dynamical

text: friction

text: eﬀects,

text: satellite

text: galaxies,

text: the

text: mild

text: eﬀect

text: suﬀered

text: by

text: the

text: cold

text: gas

text: disc

text: is

text: and

text: in

text: the

text: use

text: of

text: ﬁtting

text: formulae

text: to

text: estimate

text: RP

text: as

text: a

text: function

text: of

text: halo

text: necessary

text: to

text: achieve

text: the

text: correct

text: atomic

text: gas

text: content

text: (see

text: ﬁg.

text: mass,

text: halo-centric

text: distance

text: and

text: redshift

text: implemented

text: in

text: our

text: model

text: in

text: Paper

text: I).

text: In

text: view

text: of

text: these

text: results,

text: the

text: current

text: work

text: is

text: based

text: on

text: (Vega-Martínez

text: et

text: al.,

text: in

text: prep.).

text: The

text: combination

text: of

text: these

text: two

text: im-

text: the

text: analysis

text: of

text: the

text: properties

text: of

text: galaxies

text: generated

text: by

text: the

text: model

text: provements

text: allows

text: to

text: have

text: values

text: of

text: RP

text: consistent

text: with

text: the

text: position

text: sag

text: and

text: velocity

text: of

text: satellite

text: galaxies.

text: The

text: latest

text: version

text: of

text: sag

text: also

text: βQU.

text: ENCHED

text: FRACTIONS:

text: DEPENDENCES

text: WITH

text: considers

text: a

text: modiﬁcation

text: in

text: the

text: SN

text: feedback

text: scheme

text: that

text: includes

text: TIME

text: AT

text: FIRST

text: INFALL

text: an

text: explicit

text: redshift

text: dependence

text: of

text: the

text: reheated

text: and

text: ejected

text: mass.

text: A

text: Both

text: mass

text: and

text: environmental

text: quenching

text: can

text: aﬀect

text: the

text: galaxy

text: detailed

text: description

text: of

text: the

text: updated

text: version

text: of

text: sag

text: is

text: presented

text: in

text: after

text: being

text: accreted

text: but

text: their

text: relative

text: role

text: depends

text: on

text: both

text: stellar

text: Paper

text: I.

text: mass

text: and

text: host

text: halo

text: mass

text: and

text: cannot

text: be

text: estimated

text: from

text: the

text: fraction

text: The

text: free

text: parameters

text: of

text: this

text: model

text: were

text: tuned

text: using

text: the

text: Particle

text: ofquenchedgalaxiesat

text: Swarm

text: Optimization

text: Technique

text: (PSO)

text: by

text: Ruiz

text: et

text: al.

text: adopt-

text: z

text: analysis

text: of

text: further

text: information

text: provided

text: by

text: the

text: model

text: sag

text: This

text: ing

text: a

text: set

text: of

text: ﬁve

text: observational

text: constraints

text: deﬁned

text: in

text: Knebe

text: et

text: al.

text: involves

text: the

text: dependences

text: of

text: quenched

text: fractions

text: and

text: mass

text: Namely,

text: the

text: stellar

text: mass

text: functions

text: (SMF)

text: at

text: and

text: z

text: z

text: growth

text: with

text: the

text: time

text: of

text: ﬁrst

text: infall,

text: deﬁned

text: as

text: the

text: moment

text: in

text: which

text: (compilation

text: data

text: used

text: by

text: Henriques

text: et

text: al.

text: the

text: star

text: formation

text: the

text: galaxy

text: becomes

text: a

text: satellite

text: for

text: the

text: ﬁrst

text: time;

text: there

text: is

text: strong

text: rate

text: (SFR)

text: distribution

text: function

text: (data

text: presented

text: by

text: Gruppioni

text: et

text: al.

text: evidence

text: that

text: environmental

text: eﬀects

text: produce

text: SF

text: quenching

text: since

text: for

text: the

text: redshift

text: interval

text: the

text: fraction

text: of

text: mass

text: in

text: z

text: ﬁrst

text: infall,

text: regardless

text: of

text: the

text: host

text: halo

text: mass

text: We

text: denote

text: the

text: cold

text: gas

text: as

text: a

text: function

text: of

text: stellar

text: mass

text: (data

text: from

text: Boselli

text: et

text: al.

text: corresponding

text: redshift

text: as

text: and

text: the

text: relation

text: between

text: bulge

text: mass

text: and

text: the

text: mass

text: of

text: the

text: central

text: su-

text: zinfall

text: Currently

text: quenched

text: fractions

text: permassive

text: black

text: hole

text: (combined

text: datasets

text: from

text: McConnell

text: Ma

text: and

text: Kormendy

text: Ho

text: During

text: the

text: calibration

text: process,

text: Fig.

text: shows

text: the

text: fraction

text: of

text: satellite

text: galaxies

text: that

text: are

text: quenched

text: the

text: power-law

text: slope

text: that

text: characterizes

text: the

text: redshift

text: dependence

text: in-

text: at

text: as

text: a

text: function

text: of

text: All

text: galaxies

text: identiﬁed

text: as

text: β

text: z

text: f

text: zinfall

text: cluded

text: in

text: the

text: modiﬁed

text: SN

text: feedback

text: scheme

text: was

text: considered

text: a

text: free

text: satellites

text: within

text: main

text: host

text: haloes

text: of

text: mass

text: parameter

text: of

text: the

text: model.

text: As

text: discussed

text: in

text: Paper

text: I,

text: the

text: constraining

text: at

text: the

text: present

text: are

text: considered.

text: Satellites

text: are

text: binned

text: according

text: to

text: their

text: power

text: of

text: the

text: SMF

text: at

text: z

text: makes

text: the

text: value

text: of

text: this

text: parameter

text: too

text: present-day

text: stellar

text: mass

text: as

text: indicated

text: in

text: the

text: legend.

text: We

text: can

text: see

text: that,

text: high

text: (β

text: with

text: respect

text: to

text: the

text: one

text: suggested

text: by

text: the

text: zoom

text: for

text: a

text: given

text: stellar

text: mass

text: range,

text: the

text: trend

text: of

text: f

text: is

text: as

text: expected,

text: with

text: FIRE

text: hydrodynamical

text: simulations

text: of

text: Muratov

text: et

text: al.

text: from

text: larger

text: fractions

text: achieved

text: by

text: satellite

text: galaxies

text: with

text: higher

text: values

text: of

text: which

text: the

text: redshift

text: dependence

text: of

text: the

text: mass-loading

text: factor

text: was

text: taken

text: which

text: means

text: that

text: galaxies

text: have

text: been

text: satellites

text: longer

text: hav-

text: zinfall

text: (β

text: As

text: a

text: consequence

text: of

text: this,

text: the

text: predicted

text: SFRD

text: becomes

text: ing

text: more

text: time

text: to

text: suﬀer

text: environmental

text: quenching.

text: Values

text: of

text: f

text: too

text: low

text: with

text: respect

text: of

text: observational

text: data

text: at

text: high

text: redshifts.

text: A

text: bet-

text: are

text: larger

text: for

text: larger

text: stellar

text: masses,

text: at

text: any

text: However,

text: the

text: de-

text: zinfall

text: ter

text: agreement

text: between

text: model

text: predictions

text: and

text: observational

text: data

text: is

text: pendence

text: of

text: with

text: is

text: more

text: pronounced

text: for

text: less

text: massive

text: f

text: zinfall

text: obtained

text: by

text: adopting

text: β

text: keeping

text: the

text: values

text: of

text: the

text: rest

text: of

text: the

text: satellites.

text: This

text: trend

text: becomes

text: gradually

text: milder

text: for

text: more

text: massive

text: free

text: parameters

text: obtained

text: from

text: the

text: calibration

text: process.

text: This

text: variant

text: galaxies.

text: For

text: a

text: given

text: stellar

text: mass

text: range,

text: the

text: average

text: values

text: of

text: the

text: of

text: the

text: model

text: is

text: referred

text: to

text: as

text: the

text: model

text: sag

text: in

text: Paper

text: I.

text: The

text: cost

text: quenched

text: fractions

text: involving

text: all

text: possible

text: values

text: of

text: are

text: con-

text: zinfall

text: of

text: this

text: choice

text: is

text: a

text: higher

text: number

text: density

text: oβf

text: .low

text: mass

text: galaxies

text: in

text: sistent

text: with

text: the

text: results

text: of

text: the

text: stellar

text: and

text: halo

text: mass

text: dependence

text: of

text: the

text: SMF

text: at

text: z

text: f

text: shown

text: in

text: ﬁg.

text: of

text: Paper

text: I.

text: The

text: relations

text: f

text: vs.

text: zinfall

text: are

text: Galaxy

text: star

text: formation

text: histories

text: have

text: a

text: strong

text: impact

text: on

text: the

text: separated

text: by

text: a

text: major

text: gap

text: between

text: the

text: two

text: lowest

text: and

text: two

text: high-

text: est

text: stellar

text: mass

text: bins

text: which

text: is

text: produced

text: by

text: the

text: underprediction

text: of

text: ies

text: are

text: classiﬁed

text: as

text: active

text: or

text: passive

text: according

text: to

text: their

text: speciﬁc

text: star

text: passive

text: galaxies

text: in

text: the

text: mass

text: range

text: formation

text: rate

text: density

text: (sSFR,

text: given

text: by

text: the

text: ratio

text: between

text: their

text: stel-

text: discussed

text: in

text: Paper

text: I.

text: lar

text: mass

text: and

text: SFR).

text: Passive

text: or

text: quenched

text: galaxies

text: are

text: those

text: with

text: Low

text: mass

text: satellites

text: characterized

text: by

text: are

text: mostly

text: ac-

text: zinfall

text: sSFR

text: according

text: to

text: the

text: cut

text: that

text: emerges

text: from

text: the

text: tive

text: galaxies

text: This

text: is

text: because

text: SF

text: quenching

text: as

text: f

text: distribution

text: of

text: sSFR

text: of

text: our

text: model

text: galaxies

text: (see

text: ﬁg.

text: in

text: Paper

text: I).

text: a

text: result

text: of

text: environmental

text: eﬀects

text: is

text: negligible

text: for

text: galaxies

text: that

text: have

text: The

text: underprediction

text: of

text: SFRD

text: at

text: high

text: redshifts

text: given

text: by

text: the

text: cali-

text: been

text: recently

text: accreted,

text: and

text: these

text: galaxies

text: are

text: too

text: small

text: to

text: suﬀer

text: the

text: brated

text: model

text: sag

text: translates

text: into

text: less

text: passive

text: galaxies

text: at

text: since

text: consequences

text: of

text: mass

text: quenching

text: while

text: they

text: were

text: centrals.

text: How-

text: z

text: the

text: bulk

text: of

text: SF

text: is

text: shifted

text: to

text: lower

text: redshifts

text: and

text: the

text: galaxy

text: population

text: ever,

text: more

text: than

text: per

text: cent

text: of

text: high

text: mass

text: satellites

text: accreted

text: at

text: the

text: has

text: no

text: enough

text: time

text: to

text: quench

text: their

text: SF.

text: Thus,

text: the

text: local

text: fraction

text: of

text: present

text: epoch

text: are

text: passive,

text: which

text: means

text: that

text: massive

text: galaxies

text: have

text: quenched

text: galaxies

text: is

text: lower

text: than

text: observed

text: (see

text: ﬁg.

text: in

text: Paper

text: I).

text: Only

text: suﬀered

text: mass

text: quenching

text: processes

text: as

text: centrals

text: prior

text: to

text: infall

text: (e.g.,

text: tDhMe

text: mhoosstt

text: M∈

text: AtieGsNthafeteedxbhaacuks,t

text: fractions

text: similar

text: to

text: those

text: presented

text: by

text: Wetzel

text: et

text: al.

text: A

text: milder

text: Bosch

text: et

text: al.

text: This

text: eﬀect

text: is

text: more

text: pronounced

text: for

text: more

text: mas-

text: redshift

text: dependence

text: of

text: the

text: reheated

text: and

text: ejected

text: mass,

text: as

text: the

text: one

text: sive

text: galaxies,

text: in

text: agreement

text: with

text: results

text: from

text: Lin

text: et

text: al.

text: and

text: MNRAS


text: ALMOST EVERY SIMPLY TYPED λ-TERM HAS A LONG β-REDUCTION SEQUENCE 15 Proof. The proof is given by a straightforward induction on E. In the case E = (cid:74) (cid:75)n[E , . . . , E ], k 1 k we use the fact |C[E , . . . , E ]| = |C| + |E | + · · · + |E | (which can be shown by induction 1 k 1 k on C). • Now we deﬁne the decomposition function Φ . Let Σ be a ranked alphabet and m ≥ 1. (cid:44) m We deﬁne r max(Im (Σ)), which is also written as r for short. We always assume that Σ r ≥ 1, since it holds whenever #(L(G)) = ∞, which is an assumption of Theorem 2.11. We • • • ,Σ ,Σ shall deﬁne the decomposition function Φ (we omit Σ and just write Φ for Φ below) • m m m so that Φ (T ) = (E, P ) where (i) E is the second-order context, and (ii) P is a sequence of m (cid:5) aﬃne contexts, (iii) E(cid:74)P (cid:75) = T , and (iv) m ≤ |P i| ≤ r(m−1)+1 for each i ∈ {1, . . . , #(P )}. • The function Φ is deﬁned as follows, using an auxiliary decomposition function Φ m m given below. • (cid:44) Φ (T ) (U[E], P ) where (U, E, P ) = Φ (T ). m m The auxiliary decomposition function (just called “decomposition function” below) Φ m traverses a given tree T in a bottom-up manner, extracts a sequence P of subcontexts, and returns it along with a linear context U and a second-order context E; U and E together represent the “remainder” of extracting P from T . During the bottom-up traversal, the (cid:48) (cid:48) U-component for each subtree T represents a context containing the root of T ; whether it is extracted as (a part of) a subcontext or becomes a part of the remainder will be decided later based on the surrounding context. The E-component will stay as a part of the remainder (cid:48) during the decomposition of the whole tree (unless the current subtree T is too small, i.e., (cid:48) |T | < m). We deﬁne Φ by: m (cid:44) • If |T | < m, then Φ (T(cid:0)) ([ ], T, (cid:15)). (cid:1) m  • If |T | ≥ m, T =a(T1, . . . , TΣ(a)), and Φm(Ti) = (Ui, Ei, Pi) (for each i ≤ Σ(a)), then:    ([ ], a U [E ], . . . , U [E ] , P · · · P )  1 1 Σ(a) Σ(a) 1 Σ(a)    if there exist i, j ≤ Σ(a) such that i (cid:54)= j and |Ti|, |Tj| ≥ m  ([ ], (cid:74) (cid:75)n1[Ei], a(T1, . . . , Ui, . . . , TΣ(a)) · Pi) if |T | < m for every j (cid:54)= i, |T | ≥ m, and j i  (cid:44)  Φm(T ) (cid:44)  n |a(T1, . . . , Ui, . . . , TΣ(a))| ≥ m (3.2) (a(T1, . . . , Ui, . . . , TΣ(a)), Ei, Pi)    if |Tj| < m for every j (cid:54)= i, |Ti| ≥ m, and    |a(T1, . . . , Ui, . . . , TΣ(a))| < m ([ ], (cid:74) (cid:75)n, T ) 0 (cid:44) if |T | < m for every i ≤ Σ(a), and n |T | i As deﬁned above, the decomposition is carried out by case analysis on the size of a given tree. If T is not large enough (i.e., |T | < m), then Φ (T ) returns an empty sequence of m contexts, while keeping T in the E-component. If |T | ≥ m, then Φ (T ) returns a non-empty m sequence of contexts, by case analysis on the sizes of T ’s subtrees. If there are more than one subtree whose size is no less than m (the ﬁrst case above), then Φ (T ) concatenates m ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: ALMOST

text: EVERY

text: SIMPLY

text: TYPED

text: λ-TERM

text: HAS

text: A

text: LONG

text: β-REDUCTION

text: SEQUENCE

text: Proof.

text: The

text: proof

text: is

text: given

text: by

text: a

text: straightforward

text: induction

text: on

text: E.

text: In

text: the

text: case

text: E

text: E

text: k

text: k

text: we

text: use

text: the

text: fact

text: |C[E

text: E

text: |C|

text: |E

text: |E

text: (which

text: can

text: be

text: shown

text: by

text: induction

text: k

text: k

text: on

text: C).

text: Now

text: we

text: deﬁne

text: the

text: decomposition

text: function

text: Φ

text: Let

text: Σ

text: be

text: a

text: ranked

text: alphabet

text: and

text: m

text: m

text: We

text: deﬁne

text: r

text: max(Im

text: (Σ)),

text: which

text: is

text: also

text: written

text: as

text: r

text: for

text: short.

text: We

text: always

text: assume

text: that

text: Σ

text: r

text: since

text: it

text: holds

text: whenever

text: #(L(G))

text: which

text: is

text: an

text: assumption

text: of

text: Theorem

text: We

text: ,Σ

text: ,Σ

text: shall

text: deﬁne

text: the

text: decomposition

text: function

text: Φ

text: (we

text: omit

text: Σ

text: and

text: just

text: write

text: Φ

text: for

text: Φ

text: below)

text: m

text: m

text: m

text: so

text: that

text: Φ

text: (T

text: (E,

text: P

text: where

text: (i)

text: E

text: is

text: the

text: second-order

text: context,

text: and

text: (ii)

text: P

text: is

text: a

text: sequence

text: of

text: m

text: aﬃne

text: contexts,

text: (iii)

text: T

text: and

text: (iv)

text: m

text: |P

text: i|

text: for

text: each

text: i

text: #(P

text: The

text: function

text: Φ

text: is

text: deﬁned

text: as

text: follows,

text: using

text: an

text: auxiliary

text: decomposition

text: function

text: Φ

text: m

text: m

text: given

text: below.

text: Φ

text: (T

text: (U[E],

text: P

text: where

text: (U,

text: E,

text: P

text: Φ

text: (T

text: m

text: m

text: The

text: auxiliary

text: decomposition

text: function

text: (just

text: called

text: “decomposition

text: function”

text: below)

text: Φ

text: m

text: traverses

text: a

text: given

text: tree

text: T

text: in

text: a

text: bottom-up

text: manner,

text: extracts

text: a

text: sequence

text: P

text: of

text: subcontexts,

text: and

text: returns

text: it

text: along

text: with

text: a

text: linear

text: context

text: U

text: and

text: a

text: second-order

text: context

text: E;

text: U

text: and

text: E

text: together

text: represent

text: the

text: “remainder”

text: of

text: extracting

text: P

text: from

text: T

text: During

text: the

text: bottom-up

text: traversal,

text: the

text: U-component

text: for

text: each

text: subtree

text: T

text: represents

text: a

text: context

text: containing

text: the

text: root

text: of

text: T

text: whether

text: it

text: is

text: extracted

text: as

text: (a

text: part

text: of)

text: a

text: subcontext

text: or

text: becomes

text: a

text: part

text: of

text: the

text: remainder

text: will

text: be

text: decided

text: later

text: based

text: on

text: the

text: surrounding

text: context.

text: The

text: E-component

text: will

text: stay

text: as

text: a

text: part

text: of

text: the

text: remainder

text: during

text: the

text: decomposition

text: of

text: the

text: whole

text: tree

text: (unless

text: the

text: current

text: subtree

text: T

text: is

text: too

text: small,

text: i.e.,

text: |T

text: m).

text: We

text: deﬁne

text: Φ

text: by:

text: m

text: If

text: |T

text: m,

text: then

text: Φ

text: T,

text: m

text: If

text: |T

text: m,

text: T

text: TΣ(a)),

text: and

text: Φm(Ti)

text: (Ui,

text: Ei,

text: Pi)

text: (for

text: each

text: i

text: Σ(a)),

text: then:

text: a

text: U

text: [E

text: U

text: [E

text: P

text: P

text: Σ(a)

text: Σ(a)

text: Σ(a)

text: if

text: there

text: exist

text: i,

text: j

text: Σ(a)

text: such

text: that

text: i

text: j

text: and

text: |Ti|,

text: |Tj|

text: m

text: Ui,

text: TΣ(a))

text: Pi)

text: if

text: |T

text: m

text: for

text: every

text: j

text: i,

text: |T

text: m,

text: and

text: j

text: i

text: Φm(T

text: n

text: Ui,

text: TΣ(a))|

text: m

text: Ui,

text: TΣ(a)),

text: Ei,

text: Pi)

text: if

text: |Tj|

text: m

text: for

text: every

text: j

text: i,

text: |Ti|

text: m,

text: and

text: Ui,

text: TΣ(a))|

text: m

text: T

text: if

text: |T

text: m

text: for

text: every

text: i

text: Σ(a),

text: and

text: n

text: |T

text: i

text: As

text: deﬁned

text: above,

text: the

text: decomposition

text: is

text: carried

text: out

text: by

text: case

text: analysis

text: on

text: the

text: size

text: of

text: a

text: given

text: tree.

text: If

text: T

text: is

text: not

text: large

text: enough

text: (i.e.,

text: |T

text: m),

text: then

text: Φ

text: (T

text: returns

text: an

text: empty

text: sequence

text: of

text: m

text: contexts,

text: while

text: keeping

text: T

text: in

text: the

text: E-component.

text: If

text: |T

text: m,

text: then

text: Φ

text: (T

text: returns

text: a

text: non-empty

text: m

text: sequence

text: of

text: contexts,

text: by

text: case

text: analysis

text: on

text: the

text: sizes

text: of

text: T

text: ’s

text: subtrees.

text: If

text: there

text: are

text: more

text: than

text: one

text: subtree

text: whose

text: size

text: is

text: no

text: less

text: than

text: m

text: (the

text: ﬁrst

text: case

text: above),

text: then

text: Φ

text: (T

text: concatenates

text: m

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 16 K. ASADA, N. KOBAYASHI, R. SIN’YA, AND T. TSUKADA the sequences of contexts extracted from the subtrees, and returns the remainder as the second-order context. If only one of the subtrees, say T , is large enough (the second and i third cases), then it basically returns the sequence P extracted from T ; however, if the i i remaining part a(T , . . . , U , . . . , T ) is also large enough, then it is added to the sequence 1 i Σ(a) (the second case). If none of the subtrees is large enough (but T is large enough), then T is returned as the P -component (the last case). Example 3.5. Recall Figure 1. Let T be the tree on the left hand side. For some of the 0 subtrees of T , Φ can be calculated as follows. 0 3 Φ (b(c)) = ([ ], b(c), (cid:15)) 3 Φ (b(b(c))) = ([ ], (cid:74) (cid:75)3, b(b(c))) (by the last case of (3.2)) 3 0 Φ (a(b(c), b(c))) = ([ ], (cid:74) (cid:75)5, a(b(c), b(c))) (by the last case of (3.2)) 3 0 Φ (a(b(b(c)), b(· · · ))) = ([ ], a((cid:74) (cid:75)3, b((cid:74) (cid:75)5)), b(b(c)) · a(b(c), b(c))) 3 0 0 (by the ﬁrst case of (3.2)) Φ (T ) = (b([ ]), (cid:74) (cid:75)3[a((cid:74) (cid:75)3, b((cid:74) (cid:75)5))], a(b([ ]), c) · b(b(c)) · a(b(c), b(c))) 3 0 1 0 0 (cid:0) (by the third case of (3(cid:1).2)) From Φ (T ) above, we obtain: 3 0 • Φ (T ) = b((cid:74) (cid:75)3[a((cid:74) (cid:75)3, b((cid:74) (cid:75)5))]), a(b([ ]), c) · b(b(c)) · a(b(c), b(c)) . 3 0 1 0 0 • This subsection shows that the decompos(cid:97)itio(cid:89)n function Φ above provides a witness for m a bijection of the form (cid:96)i ∼ Tn(Σ) = Ui,j, ∈ i I j=1 where (cid:96) is suﬃciently large for each i. i We prepare some deﬁnitions to precisely state the bijection. We say that an aﬃne context U is good for m if |U| ≥ m and U is of the form a(U , . . . , U ) where |U | < m for each 1 Σ(a) i i ≤ Σ(a). For example, a(b([ ]), b(c)) is good for 3, but neither b(b([ ])) nor a(b([ ]), b(b(c))) is. (cid:74)(cid:75) We deﬁne Em and Um by: n n k (cid:44) • Em {E | (E, P ) = Φ (T ) for some T ∈ T (Σ) and P } n m n (cid:74)(cid:75) (cid:44) Um {U | U : (cid:74) (cid:75)n, U is good for m}. n k k • The goal of this subsection is to show that Φ provides a witness for: (cid:97) (cid:89)m shn(E) Tn(Σ) ∼= UEm(cid:5)j, (3.3) ∈E E m j=1 n and that shn(E) is suﬃciently large for every E ∈ Em. n The following are basic properties of the auxiliary decomposition function Φ . Lemma 3.6(3) • m below ensures that Φ (T ) extracts only good contexts from T . m Lemma 3.6. Let T be a tree. If Φ (T ) = (U, E, P ), then: m

text: K.

text: ASADA,

text: N.

text: KOBAYASHI,

text: R.

text: SIN’YA,

text: AND

text: T.

text: TSUKADA

text: the

text: sequences

text: of

text: contexts

text: extracted

text: from

text: the

text: subtrees,

text: and

text: returns

text: the

text: remainder

text: as

text: the

text: second-order

text: context.

text: If

text: only

text: one

text: of

text: the

text: subtrees,

text: say

text: T

text: is

text: large

text: enough

text: (the

text: second

text: and

text: i

text: third

text: cases),

text: then

text: it

text: basically

text: returns

text: the

text: sequence

text: P

text: extracted

text: from

text: T

text: however,

text: if

text: the

text: i

text: i

text: remaining

text: part

text: a(T

text: U

text: T

text: is

text: also

text: large

text: enough,

text: then

text: it

text: is

text: added

text: to

text: the

text: sequence

text: i

text: Σ(a)

text: (the

text: second

text: case).

text: If

text: of

text: the

text: subtrees

text: is

text: large

text: enough

text: (but

text: T

text: is

text: large

text: enough),

text: then

text: T

text: is

text: returned

text: as

text: the

text: P

text: -component

text: (the

text: last

text: case).

text: Example

text: Recall

text: Figure

text: Let

text: T

text: be

text: the

text: tree

text: on

text: the

text: left

text: hand

text: side.

text: For

text: some

text: of

text: the

text: subtrees

text: of

text: T

text: Φ

text: can

text: be

text: calculated

text: as

text: follows.

text: Φ

text: (b(c))

text: b(c),

text: Φ

text: (b(b(c)))

text: b(b(c)))

text: (by

text: the

text: last

text: case

text: of

text: Φ

text: (a(b(c),

text: b(c)))

text: a(b(c),

text: b(c)))

text: (by

text: the

text: last

text: case

text: of

text: Φ

text: (a(b(b(c)),

text: b(·

text: b(b(c))

text: a(b(c),

text: b(c)))

text: (by

text: the

text: ﬁrst

text: case

text: of

text: Φ

text: (T

text: (b([

text: a(b([

text: c)

text: b(b(c))

text: a(b(c),

text: b(c)))

text: (by

text: the

text: third

text: case

text: of

text: From

text: Φ

text: (T

text: above,

text: we

text: obtain:

text: Φ

text: (T

text: a(b([

text: c)

text: b(b(c))

text: a(b(c),

text: b(c))

text: This

text: subsection

text: shows

text: that

text: the

text: function

text: Φ

text: above

text: provides

text: a

text: witness

text: for

text: m

text: a

text: bijection

text: of

text: the

text: form

text: Tn(Σ)

text: Ui,j,

text: i

text: I

text: where

text: is

text: suﬃciently

text: large

text: for

text: each

text: i.

text: i

text: We

text: prepare

text: some

text: deﬁnitions

text: to

text: precisely

text: state

text: the

text: bijection.

text: We

text: say

text: that

text: an

text: aﬃne

text: context

text: U

text: is

text: good

text: for

text: m

text: if

text: |U|

text: m

text: and

text: U

text: is

text: of

text: the

text: form

text: a(U

text: U

text: where

text: |U

text: m

text: for

text: each

text: Σ(a)

text: i

text: i

text: Σ(a).

text: For

text: example,

text: a(b([

text: b(c))

text: is

text: good

text: for

text: but

text: neither

text: b(b([

text: nor

text: a(b([

text: b(b(c)))

text: is.

text: We

text: deﬁne

text: Em

text: and

text: Um

text: by:

text: n

text: n

text: k

text: Em

text: {E

text: (E,

text: P

text: Φ

text: (T

text: for

text: some

text: T

text: T

text: (Σ)

text: and

text: P

text: n

text: m

text: n

text: Um

text: {U

text: U

text: U

text: is

text: good

text: for

text: m}.

text: n

text: k

text: k

text: The

text: goal

text: of

text: this

text: subsection

text: is

text: to

text: show

text: that

text: Φ

text: provides

text: a

text: witness

text: for:

text: shn(E)

text: Tn(Σ)

text: ∈E

text: E

text: m

text: n

text: and

text: that

text: shn(E)

text: is

text: suﬃciently

text: large

text: for

text: every

text: E

text: Em.

text: n

text: The

text: following

text: are

text: basic

text: properties

text: of

text: the

text: auxiliary

text: decomposition

text: function

text: Φ

text: Lemma

text: m

text: below

text: ensures

text: that

text: Φ

text: (T

text: extracts

text: only

text: good

text: contexts

text: from

text: T

text: m

text: Lemma

text: Let

text: T

text: be

text: a

text: tree.

text: If

text: Φ

text: (T

text: (U,

text: E,

text: P

text: then:

text: m