1. INTRODUCTION
The present paper is part of a larger study of Object Shift (OS) in Scandinavian (see Engels & Vikner Reference Engels, Vikner, Broekhuis and Vogel2013a, Reference Engels and Viknerb); it focuses on differences between Danish and Swedish as to the applicability of OS.
Object Shift is a movement operation that moves an unfocused object from its canonical position to the right of the main verb to a position to the left of a sentential adverbial. In Danish and Swedish, OS is restricted to weak pronouns while full DPs cannot undergo OS, as shown by the contrast between (1)/(3) and (2)/(4).Footnote 1 In addition note that OS of a weak pronoun is obligatory in Danish (Da) whereas it is optional in Swedish (Sw).Footnote 2
(1)
(2)
(3)
(4)
Object Shift presupposes movement of the main verb; as shown in (5), it cannot cross a verb in situ.
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This observation, i.e. that the object only moves if the main verb has moved, forms the basis of Holmberg's generalisation (Holmberg Reference Holmberg1986:165, Reference Holmberg1997:208). Holmberg's (Reference Holmberg1997) formulation is given in (6), where ‘within VP’ has to mean that only elements ‘properly inside’ VP (i.e. not adverbials or other elements adjoined to VP) may block OS.
(6) Holmberg's generalisation
Object Shift is blocked by any phonologically visible category preceding/c-commanding the object position within VP.
However, the main verb does not have to undergo head movement (V°-to-I°-to-C° movement) as in (2) and (4) in order to license OS. Object Shift is also possible in clauses with a non-finite main verb if the verb occurs in clause-initial position, as illustrated in (7).
(7)
Likewise, OS cannot take place if it would cross other visible non-adverbial material such as the indirect object in (8), but can apply if the indirect object is moved out of the way (e.g. by OS or wh-movement), as shown in (9).
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(9)
This prohibition against OS across intervening non-adverbial material gives rise to a contrast between Danish and Swedish as to OS in particle verb constructions and ‘let’-constructions. In Danish, where the object precedes the particle as well as the infinitive embedded under ‘let’, as in (14) and (16), OS is possible, see (15) and (17).
(14)
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In contrast in Swedish, where the object follows the particle as well as the infinitive embedded under ‘let’, as in (18) and (20), OS is ungrammatical, see (19) and (21).
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(19)
(20)
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As we will show in Section 3, the mirror image arises if the particle verb or ‘let’ occurs in clause-initial position: In this case, OS becomes possible in Swedish but not in Danish.
Section 2 sets out the basics of our analysis which is couched in an Optimality-Theoretic framework. Section 3 discusses the differences between Danish and Swedish particle verb constructions (Section 3.1), and between Danish and Swedish causative constructions with ‘let’ (Section 3.2), and it is shown that while OS is possible in V2 contexts in Danish but not in Swedish, OS is possible in remnant VP-topicalisation contexts in Swedish but not in Danish. Section 4 summarises the main results.
2. AN OPTIMALITY THEORY APPROACH TO OBJECT SHIFT
In our Optimality Theory (OT) approach, we take OS to be motivated by the constraint Shift, which is based on the idea that the ‘purpose’ of OS is to move non-focused (or [–focus]) elements out of the VP; see Diesing (Reference Diesing1992) and Diesing & Jelinek (Reference Diesing and Jelinek1993).Footnote 3 The constraint Shift outranks the constraint Stay that prohibits movement. Shift is satisfied if the pronoun is adjoined to the top VP, see e.g. the syntactic tree in (24).
(22) Shift: A [–focus] constituent is left-adjoined to an extended VP that contains all VP-adjoined adverbials.
(23) Stay: Don't move. (Grimshaw Reference Grimshaw1997:374)
(24)
Recall that weak pronouns and full DPs differ as to the applicability of OS in Danish and Swedish: OS is restricted to weak pronouns, full DPs cannot undergo OS, see the examples in (1)–(4) above. We therefore assume that the constraint Stay is differentiated as to syntactic complexity. In addition to the general constraint Stay, there exists a more specific constraint that prohibits movement of full DPs (see also the appendix).
(25) StayBranch: Don't move a constituent that contains a branching node.
As illustrated in Tableau 1 and Tableau 2, the ranking StayBranch >> Shift >> Stay permits only weak pronouns but not full DPs to undergo OS. In addition, recall that Danish and Swedish contrast as to the obligatoriness of OS: While pronominal OS is obligatory in Danish, it is optional in Swedish. Optionality can be accounted for in OT by a constraint tie, Stay <> Shift, which means that both relative rankings of the two constraints co-exist. Depending on the actual ranking, OS is required (Shift >> Stay) or prohibited (Stay >> Shift). Constraint ties are marked by a dotted line in the tableaux, between the constraints that are tied. (In terms of Müller's (Reference Müller, Cheng and Sybesma2001b) classification of constraint ties, these would be ordered global ties.)Footnote 4
No full DP shift, obligatory pronominal OS (Danish).
No full DP shift, optional pronominal OS (Swedish).
In these and the following tableaux, only Stay-violations induced by OS are listed; Stay-violations induced by e.g. V°-to-I°-to-C° movement or VP-topicalisation are left out because they do not vary between competing candidates. The same holds for the violations of the constraint Order Preservation, which we will turn to now.
Following Fox & Pesetsky (Reference Fox and Pesetsky2005a, Reference Fox and Pesetskyb), we will assume here that Holmberg's generalisation results from a high ranking condition on order preservation (see also Déprez Reference Déprez, Corver and van Riemsdijk1994, Müller Reference Müller, Grimshaw, Legendre and Vikner2001a, Sells Reference Sells2001, Williams Reference Williams2003, Fox & Pesetsky Reference Fox and Pesetsky2005a, Koeneman Reference Koeneman, Hartmann and Molnárfi2006).
(26) Order Preservation (OrdPres): An independently moved constituent α must not precede a non-adverbial constituent β if the canonical position of α (or parts of α) follows the canonical position of β.Footnote 5
Some remarks on the formulation of OrdPres are in order here. First, ‘independently moved’ is relevant for cases where a complex constituent is moved, as e.g. in the VP-topicalisations discussed in Section 3 below. Crucially, movement of a complex constituent induces a violation of OrdPres for each non-adverbial constituent crossed, independently of how many overt elements the moved constituent contains. As a result, topicalisation of an entire VP and topicalisation of a remnant VP give rise to the same number of violations of OrdPres.
Secondly, we take the ‘canonical position’ of an element to be the lowest position where all case requirements are satisfied. This is crucial in cases where the base-generated position of an element differs from the position in which the element is assigned case or assigns case itself, as e.g. in particle verb constructions (Section 3.1), ‘let’-constructions (Section 3.2) and double object constructions (see Engels & Vikner Reference Engels, Vikner, Broekhuis and Vogel2013a, Reference Engels and Viknerb). If an element does not assign or is not assigned case, the canonical position is its base-generated position. If an element assigns case or is assigned case in a position different from its base-generated position, then the canonical position is the position where case assignment takes place (compare Section 3). If an element assigns case in more than one position, the canonical position is the highest of these positions (in terms of c-command).Footnote 6 This restriction to canonical position means that on one hand, movement into a case-assigning or case-assigned position cannot possibly violate OrdPres and that, on the other hand, OrdPres is evaluated with regard to the case position of an element, and not with regard to its base-generated position (if the two should differ).
Dominance of OrdPres over Shift predicts that OS is only possible if it maintains the canonical order of certain constituents.Footnote 7 If the main verb stays in situ, OS gives rise to a fatal violation of OrdPres (as the object would have to move across the verb) and is thus excluded; the object must remain in situ to the right of the main verb, as shown by the optimal candidate in Tableau 3.
No OS with in situ verb.
For OS to be possible, the main verb must occur in a position to the left of the target position of OS, such that the relative order between verb and object is preserved. This can be guaranteed e.g. by V°-to-I°-to-C° movement illustrated in Tableau 4. (The restriction of OrdPres to non-adverbial constituents is necessary to permit OS across clause-medial adverbials.)
OS with V°-to-I°-to-C° movement.
However, OS does not presuppose V°-to-I°-to-C° movement of the main verb. As shown by the example in (7) above (repeated here as (27)), OS can also take place if a non-finite main verb occurs in clause-initial position. In fact, OS has to take place in this case even in Swedish, see (28).
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As we have argued in Engels & Vikner (Reference Engels, Vikner, Broekhuis and Vogel2013a:195–196) (see also Engels & Vikner Reference Engels and Vikner2013b), we take the construction in (27) not to involve V°-topicalisation as in (29), contra Holmberg (Reference Holmberg1999). Rather, we follow Fox & Pesetsky (Reference Fox and Pesetsky2005a, Reference Fox and Pesetskyb) in assuming that they involve remnant VP-topicalisation as in (30). (For theoretical and empirical problems with the approaches by both Holmberg (Reference Holmberg1999) and Fox & Pesetsky (Reference Fox and Pesetsky2005a, Reference Fox and Pesetskyb), see Broekhuis (Reference Broekhuis2008: 254–260), Engels & Vikner (Reference Engels, Vikner, Broekhuis and Vogel2013a: 195–202; Reference Engels and Vikner2013b).)
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Tableau 5 shows how our OT analysis predicts that OS (i.e. placement of the object to the left of a clause-medial adverbial) is obligatory when a non-finite verb occurs in clause-initial position, even in Swedish where OS is otherwise optional (compare Tableau 2). Candidate a in Tableau 5, where the object is moved to a position below the clause-medial adverbial, violates both Stay and Shift; it is thus suboptimal to candidate b in Tableau 5, where the object occurs in OS position (compare the definition of Shift in (22) above).
OS with verb in SpecCP.
In Holmberg's (Reference Holmberg1997, Reference Holmberg1999) approach, remnant VP-topicalisation is ruled out by the assumption that Holmberg's generalisation is derivational, i.e. that it cannot be violated at any point in the derivation. The OT constraint OrdPres, by contrast, is representational: Constraint violations are computed on the basis of the final structure of the candidates. Hence, although the individual steps of OS might violate OrdPres, this is of no consequence as long as the verb is subsequently placed to the left of the shifted object so that their original precedence relation is re-established, satisfying OrdPres on the surface.
The following section focuses on two particular differences between Danish and Swedish related to OS and remnant VP-topicalisation, which support the OT approach presented here, namely between Danish and Swedish particle verb constructions (Section 3.1), and between Danish and Swedish causative constructions with ‘let’ (Section 3.2).
3. TWO DANISH/SWEDISH DIFFERENCES CONCERNING OBJECT SHIFT
3.1 Particle verb constructions
That OS must be order preserving can also be observed in particle verb constructions. The Scandinavian languages differ as to the ordering of particle and object.Footnote 8 In Danish, the object precedes the verb particle, as in (31), whereas in Swedish, the object follows the particle, as in (32), repeated from, respectively, (14) and (18) above. This is completely independent of whether the object is a full DP or a weak pronoun.
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Vikner (Reference Vikner, Allan and Barnes1987:263) and Haegeman & Guéron (Reference Haegeman and Guéron1999:257–258), among many others, suggest that particle constructions have a structure parallel to prepositional phrases, i.e. that the particle (Prt°) is the head of a particle phrase (PrtP), and that Prt° may be followed by a complement DP, as in (33), to which it assigns a thematic role.
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Haegeman & Guéron (Reference Haegeman and Guéron1999:257–258) further suggest that particles do not assign case to their complement DPs, and that in English, there are two ways out of this predicament: DP-movement or particle adjunction.
Vikner (Reference Vikner, Allan and Barnes1987:263–265, 269; Reference Vikner2009:5–6) implements these two as follows: One option is that the DP moves to the specifier of the PrtP where it can be assigned case by the verb (much like exceptional case marking, ECM), see (34a) and candidate b in Tableau 6 and Tableau 7 below. The other option is that the particle adjoins to the verb, which allows the case assignment properties of the verb to be shared with the particle, so that the DP may now be assigned case by the trace of the particle, see (34b) and candidate a in Tableau 6 and Tableau 7. Danish only allows DP-movement, see (31) above and Tableau 6, and Swedish only allows particle adjunction, see (32) above and Tableau 7.
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We suggest that the difference between the Obj–Prt sequence in Danish and the Prt–Obj one in Swedish can be accounted for by means of three constraints: Case, which penalises DPs that are not assigned case, No X°-Adjunction, which penalises particle adjunction as a means of achieving case assignment, and No Exceptional Case Marking, which penalises the DP moving to SpecPrtP in order to be assigned case from the verb.
(35) Case: DPs must be case-marked. (Grimshaw Reference Grimshaw1997:374)
(36) No X°-Adjunction (*X°-Adj): A head must not adjoin to a lexical verb.
(37) No Exceptional Case Marking (*ECM): A head must not assign case to the specifier of its complement.
Particle verb construction (with DP-movement; Danish).
Particle verb construction (with particle adjunction; Swedish).
As we do not actually think that there are languages in which DPs are not assigned case, we assume that Case is an inviolable constraint in the generator GEN (rather than a violable one in the evaluator EVAL). Thus, only candidates that satisfy Case can be generated. This also guarantees that every canonical position of a case-assigning or case-assigned constituent contains at least a trace of this constituent.
The difference between Danish and Swedish in the ordering of verb particle and object can then be captured by a difference in the ranking of the two constraints: *X°-Adj and *ECM.Footnote 9 In Danish, *X°-Adjis ranked higher than *ECM,necessitating movement of the object to SpecPrtP, whereas in Swedish, it is the opposite, *ECM overrides *X°-Adj, such that the particle must adjoin to the verb to make case assignment possible. This is shown in Tableau 6 and Tableau 7. The object occurs in its canonical position in the optimal candidate in Tableau 6 and Tableau 7, respectively. It is thus these orders which will be relevant for computing of OrdPres.
Consider now the interaction between OS and particle verbs. If the particle verb itself is finite and thus moves to C° because of V2, a pronominal object of a particle verb has to undergo OS in Danish, as is shown in (38), but it cannot do so in Swedish, see (39).
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This contrast is expected in the present analysis because of OrdPres. As shown in Tableau 8 and Tableau 9, OrdPres plays no part in the choice between the two non-adjoining candidates, b and d, as neither candidate violates it (the particle also follows the object in the canonical order). The crucial constraint in the Danish Tableau 8 then becomes Shift, which favours the candidate with OS. OrdPres does play a part, however, in the choice between the two adjoining candidates, a and c (see the Swedish Tableau 9), as it is fatally violated by c, where the particle precedes the object in the canonical order.
OS with moved particle verb (Danish).
No OS with moved particle verb (Swedish).
Thus, the dominance of OrdPres over Shift predicts that OS can take place in particle verb constructions with V2 movement of the particle verb in Danish but not in Swedish, where the object would have to cross the particle. However, note that OS is possible in Swedish particle verb constructions if the particle occurs in SpecCP (compare the verb-topicalisation construction in (7) above). This is expected as the relative order of particle and object is maintained in this case, satisfying OrdPres.Footnote 10
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Whereas the definition of Holmberg's generalisation in (6) above only makes predictions as to the relative order of the object and elements to its left, the present analysis, with OrdPres ranked higher than Shift, forces maintenance of the order of the object relative to all non-adverbial elements, both to the left and to the right of the object. It is thus expected that OS during remnant topicalisation of a particle verb including the particle is possible in Swedish, where the object is right-peripheral in VP, but not in Danish, where the object precedes the particle inside VP. This expectation is borne out, as illustrated in (41) and (42).
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Because the particle in Tableau 10 and Tableau 11 is part of the remnant VP that moves to SpecCP, OrdPres does not play a part in the choice between the two adjoining candidates, a and c, as the order remains V–Prt–Obj. The crucial constraints in the Swedish Tableau 11 then become Shift and Stay, which are tied, and thus allow both the candidate with OS, see (42c) above, and a candidate with full VP-topicalisation, see (42a).Footnote 11OrdPres does play a part, however, in the choice between the two non-adjoining candidates, b and d (see the Danish Tableau 10), as it is fatally violated by d, where the particle precedes the object at the surface, but follows it in the canonical order.
No OS with remnant VP-topicalisation of particle verb (Danish).
OS with remnant VP-topicalisation of particle verb (Swedish).
Thus, Danish and Swedish display mirror images as regards OS in particle verb constructions. The canonical order in Danish, Obj–Prt, permits OS in case the particle verb undergoes V°-to-I°-to-C° movement but prohibits OS in case particle verb and particle undergo remnant VP-topicalisation. In contrast, Swedish, which has Prt–Obj order, does not permit OS if the particle verb undergoes V°-to-I°-to-C° movement, but OS is possible in remnant VP-topicalisation constructions. This follows from the fact that OS in both languages has to preserve the canonical order (OrdPres >> Shift).
3.2 Causative constructions with ‘let’
The situation concerning ‘let’-constructions is parallel to the one concerning particle verb constructions. In Danish, the object of a subjectless infinitive under the causative verb lade ‘let’ precedes the infinitival verb, as in (43), whereas it follows the infinitive (låta) in Swedish, as in (44), repeated from (16) and (20) above; see Vikner (Reference Vikner, Allan and Barnes1987:262–266) and many others.
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In the present analysis, this contrast again follows from the differences in the relative ranking of *X°-Adj and *ECM. The infinitive embedded under ‘let’ may or may not assign the external theta-role. If it does not assign its external theta-role as in (43) and (44), it also does not assign accusative case to its object (Burzio's generalisation); see Vikner (Reference Vikner, Allan and Barnes1987). In Danish, the object undergoes movement to SpecVP as in (46a) to be assigned case (*X°-Adj >> *ECM) while the infinitival verb adjoins to ‘let’ in Swedish, as shown in (46b), to make case assignment possible (*ECM >> *X°-Adj). This is thus similar to Tableau 6 and Tableau 7 above.
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As they reflect the canonical order, the structures in (46a) and (46b) are relevant for computing OrdPres. If ‘let’ now undergoes finite verb movement, OS into the ‘let’-clause becomes possible in Danish, as is shown in (47), but not in Swedish, see (48).
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This is expected by OrdPres. Object Shift preserves the relative ordering of object and infinitive in Danish but not in Swedish, where the object would have to move across the infinitival verb. This is thus similar to the situation in Tableau 8 and Tableau 9 above.
In contrast, if the infinitive undergoes VP-topicalisation together with non-finite ‘let’, stranding of the object is impossible in Danish, as shown in (49), whereas it is acceptable in Swedish, see (50). Again, this is predicted by OrdPres: Stranding of the object maintains the relative ordering of object and infinitive in Swedish but not in Danish, where the object is non-peripheral within VP. This situation is thus the same as the one in particle verb constructions analysed in Tableau 10 and Tableau 11 above:
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Summing up, contrasts as to the applicability of OS in ‘let’-constructions in Danish and Swedish are expected due to the differences in the canonical order of object and infinitival verb (i.e. the relative ranking of *X°-Adj and *ECM). Object Shift is only possible as long as it is order preserving, as accounted for by the ranking OrdPres >> Shift.
In addition, notice that if the infinitive has an overt subject, the object follows the infinitival verb in both Danish and Swedish, see (51) and (52).Footnote 12
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Given the presence of an infinitival subject, we assume the existence of an infinitival clause, namely the IP in the syntactic tree in (53). As the infinitive assigns its external theta-role, it is also able to assign accusative to its object.
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If ‘let’ is finite and thus undergoes V°-to-I°-to-C° movement, OS of the infinitival subject into the ‘let’-clause is possible while OS of the infinitival object is prohibited in both languages, as expected by OrdPres >> Shift: OS of the infinitival subject maintains the canonical order while OS of the infinitival object does not as the object would have to cross the infinitive. This is illustrated in Tableau 12.
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OS of infinitival subject but not of infinitival object in ‘let’-constructions.
However, if the infinitive undergoes VP-topicalisation together with non-finite ‘let’, neither the infinitival subject nor the infinitival object may be stranded; see (56) and (57).
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That OS of the infinitival subject is not possible under remnant VP-topicalisation is expected in the present analysis since the infinitival subject occurs in a non-peripheral position. In other words, stranding of the infinitival subject in OS position violates OrdPres. However, the infinitival object cannot be stranded either, even though it is right-peripheral within VP and consequently maintains the canonical ordering relations (satisfying OrdPres). The ungrammaticality of (56c) and (57c) thus cannot have to do with order preservation. Rather, the above data suggest that OS is clause-bound: In the ungrammatical sentences in (56c) and (57c), the infinitival object is stranded in the OS position of a higher clause, namely the ‘let’-clause. Hence, a constituent may apparently not be moved out of its own clause by OS. This may be derived by a constraint Clause-Boundedness (ClauseBound), which prohibits an element from moving out of its own clause. Dominance of ClauseBound over Shift then rules out stranding of an object in OS position of a higher clause.Footnote 13, Footnote 14
Note that the subject of the infinitive belongs to the ‘let’-VP by virtue of being assigned case by ‘let’. As it does not violate ClauseBound, OS of the subject of the infinitive is thus expected to be possible as long as it is order preserving; compare (54b)/(55b) with (56b)/(57b). Similarly, the object of a subjectless infinitive belongs to the extended ‘let’-VP: In Danish, it is assigned case by ‘let’ (in the specifier of its complement), in Swedish, the infinitive adjoins to ‘let’ and thus forms an extended VP with ‘let’; compare (46) above. Thus, OS to the extended ‘let’-VP complies with ClauseBound and is licit as long as OrdPres is satisfied; see the contrast between (47b) and (48b) as well as between (49c) and (50c) above.
Summing up, the contrasts between Danish and Swedish as to OS in particle verb constructions and ‘let’-constructions support the order preservation approach suggested here. Due to differences in case assignment captured by the relative ranking of *X°-Adj and *ECM, an object precedes the particle and the subjectless infinitival verb in Danish but follows them in Swedish. As a consequence, the two languages display mirror images with regard to OS in these constructions. The ranking OrdPres >> Shift predicts that OS is only possible if it retains the canonical order of the VP-internal elements. Finite verb movement of the particle verb/’let’ paves the way for OS in Danish but not in Swedish, where the object would have to cross the particle/infinitival verb. In contrast, stranding of the object in OS position during remnant VP-topicalisation is possible in Swedish particle verb constructions and subjectless ‘let’-constructions but not in Danish ones as the object is right-peripheral within VP in the former but not in the latter. Moreover, although the object is right-peripheral in ‘let’-constructions with an infinitival subject, stranding it during remnant VP-topicalisation is ruled out due to the fact that OS is clause-bound (ClauseBound >> Shift). ‘Let’ is not involved in assigning case to the object in this case; the object thus does not belong to the extended ‘let’-VP and it may not undergo OS into the ‘let’-clause.
4. CONCLUSION
On the basis of a set of less commonly discussed data concerning remnant VP-topicalisation, the present OT approach suggests that Holmberg's generalisation should be accounted for in terms of order preservation, as formulated in the violable constraint OrdPres.
This order preservation approach to OS finds support in the contrasts discussed between Danish and Swedish particle verb constructions and ‘let’-constructions. Differences in the VP-internal object position (V–DP–X in Danish and V–X–DP in Swedish) were taken to result from differences in case assignment, and this was shown to account for the mirror image sequences with regard to OS in clauses with finite verb movement of the matrix main verb (OS possible in Danish but not in Swedish) and clauses with remnant VP-topicalisation (OS possible in Swedish but not in Danish).
ACKNOWLEDGEMENTS
We are grateful for helpful comments and suggestions to Maia Andréasson, Theresa Biberauer, Ken Ramshøj Christensen, Elisabet Engdahl, Gunlög Josefsson, Shin-Sook Kim, Henrik Jørgensen, Gereon Müller, Vieri Samek-Lodovici, Peter Sells, Michelle Sheehan, Johanna Wood, Bjarne Ørsnes, three anonymous reviewers, as well as to audiences at the universities of Aarhus, Berlin (ZAS), Cambridge, Gothenburg, Leipzig, London (UCL), Lund, Newcastle and Stuttgart. This work was carried out as part of the projects Object Positions – Comparative Syntax in a Cross-Theoretical Perspective, and Similarities and Differences between Clauses and Nominals, both financed by Forskningsrådet for Kultur og Kommunikation (Danish Research Council for Culture and Communication).
APPENDIX. SYNTACTIC COMPLEXITY OF PRONOUNS
The examples in (1)–(4) repeated below as (A1)–(A4) have shown that in Danish and Swedish, OS is restricted to weak pronouns.
(A1)
(A2)
(A3)
(A4)
In contrast, also full DPs may optionally undergo OS in Icelandic (see also note 1).
(A5)
(A6)
In this connection note that not only a full DP like den her bog ‘this book’, as in (A1), but also syntactically complex pronouns, i.e. modified or conjoined ones as in (A7) and (A8), are excluded from OS in Danish and Swedish. In Icelandic, in contrast, they can undergo OS, shown in (A9) and (A10).
(A7)
(A8)
(A9)
(A10)
The difference between simple pronouns and all other DPs is that the former are DPs that do not contain a branching node whereas the latter are DPs that contain a branching node (compare (A11a) with (A11b, c) and (A12a–c) below). This insight forms the basis for the constraint StayBranch in (25) above.Footnote 15
(A11)
(A12)
