Metrical Structure and Licensing: An Argument from Ukrainian

It has been debated in phonological literature whether word stress should be modeled using metrical grids or feet and how its directionality is assigned. In this article, we discuss new data from Ukrainian, which has a hybrid metrical system with unpredictable lexical stress and grammatical iterative secondary stress. We demonstrate that Ukrainian poses a challenge for current metrical theories relying on gradient alignment and propose an analysis based on categorical alignment coupled with a rhythmic licensing constraint mandating that lapses are located near the main stress (LAPSE-AT-PEAK). We argue that LAPSE-AT-PEAK is required regardless of the stress representations (feet or grids) assumed.


Introduction
This article discusses the mixed lexical-grammatical metrical system of Ukrainian, whose complexity poses an analytical challenge to current metrical theories. Two central issues are raised by interactions between free lexical stress and grammatically assigned rhythmic stresses in Ukrainian: • The directionality of stress assignment, given that rhythmic stresses are attracted concurrently by the left and the right edge of the word • Formal representation of stress, given that the system has no default trochaic or iambic foot parsing According to standard descriptions as well as recent acoustic studies (Nakone?nyj 1969, Łukaszewicz and Mołczanow 2018a,b), Ukrainian represents a complex bidirectional stress system with internal lapses, with secondary stresses iterating from the edges of the word toward the syllable carrying lexical stress: for example, i+ munii+ cypali} * tet 'municipality', } * lahodyi+ tymei+ te 'repair, 2PL'. In bidirectional stress systems reported in the literature, primary stress is typically assigned at (or near) one edge of the word, and secondary stresses iterate from the opposite end toward the syllable carrying primary stress (e.g., Kager 2001, Gordon 2002, Hyde 2002, Hermans 2011. Classic examples of such systems are Piro (Matteson 1965) and Polish (Rubach We are grateful to two LI reviewers for helpful suggestions and constructive criticism. The preliminary stage of the research was presented at Approaches to Phonology and Phonetics 2017, the 16th Conference on Laboratory Phonology, and the International Symposium on Monolingual and Bilingual Speech 2019. We would like to thank the audiences at these conferences for comments and questions. All errors are ours. Research was funded by the National Science Centre (Poland), grant 2017/27/B/HS2/00780. and Booij 1985, Kraska-Szlenk 2003, Łukaszewicz 2015, 2018, Łukaszewicz, Zajbt, and Krawczyk 2018, Łukaszewicz, Zajbt, and Mołczanow 2020, exhibiting a rightward iteration of secondary stresses, and Garawa (Furby 1974), which has a leftward stress iteration pattern. Characteristically, odd-parity words in these languages have lapses (i.e., sequences of unstressed syllables) adjacent to the peak, as schematized in (1).
(1) a. Rightward iteration of secondary stresses (Piro): i + i + } * b. Leftward iteration of secondary stresses (Garawa): } * i + i + Unlike bidirectional stress systems described previously in the literature, Ukrainian has free lexical stress and exhibits both the rightward and leftward iteration of secondary stress. The location of secondary stress depends on the position of lexical stress and the number of syllables separating lexical stress from the edges of the word, which we discuss in more detail in section 2. The iterative and bidirectional character of secondary stress can be inferred from examples in which lexical stress is separated by at least five syllables from the left or right edge of the word, as illustrated in (2a) and (2b). n is a shorthand for zero or more (maximally four) syllables (see also footnotes 1 and 2).
(2) a. Rightward iteration: i + i + } * ( n ) i + amei + rykani} * zovanyj 'Americanized' b. Leftward iteration: ( n ) } * i + i + } * vylikui + vatyi + sja 'heal, REFL' The basic generalization is that secondary stresses in Ukrainian are "repelled" from the syllable carrying primary stress toward the word's edges, as in the bidirectional systems in (1). Hence, the lapse is predicted to occur in the vicinity of primary stress and not between secondary stresses (e.g., [} + } + } * ( n )], not *[} + } + } * ( n )]). It is important to note that the hitherto described bidirectional systems have predictable main stress and either a rightward or a leftward stress iteration pattern. The point of interest of the Ukrainian data, which as yet has not been discussed in the generative literature, consists in the presence of unpredictable lexical stress that coexists with different patterns of grammatical stress. Although practically all contemporary metrical theories provide tools for describing bidirectional stress systems with internal lapses-rooted in grid-based, foot-based, or parameter-based mechanisms (e.g., Rubach and Booij 1985, McCarthy and Prince 1993, van der Hulst 1996, 2012, Kager 2001, Gordon 2002, Hyde 2002, 2016, Alber 2005)-the coexistence of the two opposite-edge-based secondary stress patterns within a single system poses a challenge for most of the hitherto proposed accounts. Most of these accounts are couched in the constraintbased framework of Optimality Theory (OT), regulating the positioning of stress in terms of alignment constraints, which require that the designated edge (left or right) of some prosodic domain coincide with the corresponding edge of some other prosodic or morphological domain (McCarthy and Prince 1993). Alignment constraints have been formulated as either gradient (McCarthy andPrince 1993, Alber 2005) or categorical (Kager 2001(Kager , 2005 constraints. As gradient constraints, they can be multiply violated, depending on the distance (the number of syllable units) from the designated edge. As categorical constraints, they can cause only single violations, as they operate on strictly local factors, such as immediate adjacency to the domain edge or peak. In crosslinguistic studies of metrical systems, the alignment mechanisms have proved notoriously insufficient to generate the asymmetries in the existing patterns and are supplemented by licensing, which admits rhythmically marked entities (lapses) in a limited set of prosodic positions (the peak and the right edge of the prosodic domain); see Kager 2001Kager , 2005. This article demonstrates that the Ukrainian pattern is problematic from the point of view of OT accounts of rhythmic stress appealing to gradient alignment (McCarthy andPrince 1993, Alber 2005). Instead, we suggest an analysis that is based on categorical alignment, which is coupled with a rhythmic licensing constraint mandating that lapses are located near the main stress.
The Ukrainian system, which is weight-insensitive and does not have default trochaic or iambic foot parsing, also raises a number of issues concerning the formal representation of stress. Beginning with the seminal work of Liberman and Prince (1977), generative metrical phonology has assumed grid-based representations, with grids reflecting relative syllabic prominence, as in (3). Rather than referring to grids alone (3a), most models of metrical structure have also assumed metrical footing (e.g., Liberman and Prince 1977, Hayes 1980, Halle and Vergnaud 1987, grouping syllables into a hierarchy of headed binary constituents and aligning the heads of feet with prominence marks on the grid (3b).
Secondary stress line 1 line 0 b. Grids cum feet: (x .) Primary stress line 2 (x .) (x .) Secondary stress line 1 line 0 The two aspects of metrical structure are largely redundant with respect to each other. Nevertheless, in most theoretical analyses of stress patterns, both feet and grids are used simultaneously as they are acknowledged to play a complementary role in accounting for various stress-related phenomena (e.g., Hayes 1995). While feet are mostly irrelevant from the point of view of stress distribution alone (see Hermans 2011), it has been argued that from the point of view of application of certain phonological processes, unstressed syllables embedded within feet are not structurally the same as unstressed syllables outside of foot structure. For example, vowel reduction in Dutch is more likely to occur in the weak syllable of a foot than in the weak syllable that remains unparsed (Booij 1995:133-134). Also, reference to foot structure remarkably clarifies and simplifies the description of certain otherwise complex and vague patterns; for example, the process of High Vowel Deletion in Old English (/hēafud‫ם‬es/ -[hēafdes], /word‫ם‬u/ -[word]) can be generalized to take place in the weak branch of the strong-weak Germanic foot (Dresher and Lahiri 1991: sec. 1). The structural differences captured straightforwardly in models assuming foot-sized constituents are left unexpressed in terms of grid-only models, which encode only the differences in degrees of stress. In the latter models, all unstressed syllables are structurally the same; for an insightful discussion of the role of feet cum grids and grid-only representations in generating grammatical stress systems, see Hermans 2011. "center," consisting of a pretonic and a tonic syllable (2-3), and a weak "periphery," comprising the remaining syllables (Kasatkina 1996). In contrast, Ukrainian rhythm is described, on the basis of impressionistic observations, by the formula 1-0.75-2-0.75-1.25 (see Toc'ka 2002:72), suggesting a wavelike pattern of alternating strong and weak syllables. Standard grammars of Ukrainian (e.g., Nakone?nyj 1969, Toc'ka 2002 explicitly state that secondary stresses are assigned at the word's edges as well as on every other syllable, forming a predictable alternating pattern. The existence of rhythmic prominence in Ukrainian has also been confirmed in a series of acoustic studies, which have demonstrated that rhythmic stress is cued by increased syllable duration (Łukaszewicz and Mołczanow 2018a,b,c). It is worthwhile pointing out that the phenomenon of rhythmic stress is not confined to one dialect or to a geographic region, because its presence has been reported both in the standard literary language (Nakone?nyj 1969), which is based on the southeastern dialectal group, and in the standard Ukrainian spoken in the west of the country (Łukaszewicz and Mołczanow 2018a,b,c).
Possible rhythmic configurations for words consisting of three to six syllables are shown in (5). 1 Illustrative data come from Nakone?nyj 1969 and Łukaszewicz and Mołczanow 2018a,b.
As can be observed in (5), lexical stress can appear on any syllable, and secondary stress is predictably located on the initial and final syllables, given that at least one syllable intervenes between lexical and rhythmic stress; compare } * i + horodyna 'vegetables' and i + } * robitnyctvo 'working class' in (5b). Additional rhythmic beats appear word-medially in words with lexical stress removed from the left or the right word edge by four or more syllables: for instance, i + i + } * peredovyky 'leaders' and } * i + i + vybuduvaty 'build, PERF' (5c). 2 The examples in (5d) show that whenever lexical stress is separated from the word edge by five syllables, there is a lapse close to lexical stress (i + i + } * municypalitet 'municipality', } * i + i + vylikuvatysja 'heal, REFL'), with the additional beats iterating from word edges toward main stress. This characteristic (i.e., rightward assignment of secondary stress in i + i + } * and leftward assignment in } * i + i + ) renders the metrical system of Ukrainian bidirectional.
As Ukrainian is a highly inflectional language with complex morphology, most words in (5) contain affixes: for example, zolot‫ם‬o 'gold', borot'‫ם‬b‫ם‬a 'struggle', vy‫ם‬bud‫ם‬uva‫ם‬ty 'build, PERF'. It is generally known that morphology can potentially interact with stress assignment. Welldocumented examples of such interactions come from English, where secondary stress placement is cyclic and thus dependent on morphological structure (Chomsky and Halle 1968, Hayes 1980, Halle and Vergnaud 1987. For instance, secondary stress is placed on the first syllable of the word i+ amica} * bility because this syllable carries main stress in the base form } * amicable. In contrast, secondary stress on the second syllable of Ai+ meri} * cana is inherited from the stress pattern of the stem A} * merica. No effects of this kind occur in Ukrainian. The fact that the assignment of secondary stress in this language proceeds regardless of lexical stress in related words is confirmed by several pieces of evidence. First, there is no correspondence between lexical stress in the base form and the grammatical stress in the derived form in word pairs such as a} * meryka 'America' -i+ amery} * kanec' 'an American, MASC SG', xa} * rakter 'character' -i+ xarakte} * rystyi+ ka 'characteristics', and many others. If morphological structure modulated the assignment of rhythmic stress, both i+ amery} * kanec' and i+ xarakte} * rystyi+ ka would have secondary stress on the second syllable: *ai+ mery-} * kanec' and *xai+ rakte} * rystyi+ ka. The second piece of evidence demonstrating that secondary stress is not lexicalized in Ukrainian comes from stress patterns of unaccented stems. 3 Such stems lack an underlying accent specification and do not carry primary stress in the surface representation, and yet they regularly receive secondary stress. For instance, words such as bu} * d‫ם‬yn‫ם‬ok 'building', i+ bud‫ם‬u} * va‫ם‬ty 'build', i+ bud‫ם‬i} * v‫ם‬el'‫ם‬nyk 'builder' have an unaccented stem bud-, with primary stress realized on the suffix (cf. the accented stem -rod-in na‫}ם‬ * rodÅ‫ם‬uva‫ם‬i+ ty 'give birth', na‫ם‬ } * rod‫ם‬nist' 'ethnicity', na‫}ם‬ * rod‫ם‬nyk 'patriot'). Regardless of its lack of lexical accent, budreceives rhythmic prominence in those words in which it is separated from primary stress by one syllable, which indicates that secondary stress assignment in Ukrainian is postlexical in nature. Finally, secondary stress is not marked in pronouncing dictionaries of Ukrainian (see äovtobrjux 1973), and native speakers are unaware of its existence. In this regard, Ukrainian is similar to grammatical stress systems such as Polish, which show the same surface-true rhythmic stress pattern in simple and complex words.
As mentioned previously, acoustic correlates of rhythmic stress in Ukrainian have been the subject of several experimental studies Mołczanow 2018a,b,c, Mołczanow, Łukaszewicz, andŁukaszewicz 2018). These studies have detected a rhythmic pattern in words with lexical stress located both at the right word edge (e.g., i+ veloi+ sype} * dyst 'cyclist') and at the left word edge (e.g., } * vyzoi+ lotyi+ ty 'gild'). However, whereas secondary stress was obligatorily present at both word edges (i.e., initially in i+ veloi+ sype} * dyst and finally in } * vyzoi+ lotyi+ ty ), an additional beat in word-medial position was optional in words with lexical stress at the left word edge (} * vyzoi+ lotyi+ ty ).
Also, phonetic studies have reported an increased duration of the syllable immediately preceding the syllable carrying lexical stress (Łukaszewicz and Mołczanow 2018b). Other East Slavic languages for which the existence of pretonic lengthening has been well-documented include Standard Russian and some East Slavic dialects (Bethin 2006). In these systems, increased duration is unconnected with rhythm and is argued to be caused by a high tone that is spread from the lexically stressed syllable (Bethin 2006, Mołczanow 2015, 2017. As mentioned above, Standard Russian as well as pretonic-length dialects do not have secondary stress; their rhythmic pattern is traditionally described using the formula 1-2-3-1, pretonic lengthening being marked with 2, which is meant to reflect the fact that the duration of this syllable is intermediate between that of the tonic syllable (marked with 3) and that of the unstressed syllable (marked with 1). Ukrainian rhythm is usually described by the formula 1-0.75-2-0.75-1.25, suggesting that the pretonic syllable has the same status as other unstressed syllables. In this respect, the detection of increased duration of the pretonic syllable in Ukrainian reported in Łukaszewicz and Mołczanow 2018b is unexpected. However, note that the formula provided above was stated on the basis of impressionistic observations, and was primarily meant to highlight the differences between the rhythmic structures of Ukrainian and Russian. Pretonic lengthening, in turn, is unconnected with rhythm; rather, it is related to the domain of lexical stress, which appears to extend beyond one syllable in East Slavic (for further discussion, see Łukaszewicz and Mołczanow 2018b). It should also be noted that pretonic lengthening is not as robust in Ukrainian as it is in the Russian dialects with pretonic length, where pretonic vowels often have the same duration as the vowels in stressed syllables. In contrast, Ukrainian pretonic vowels do not exceed two-thirds of the tonic syllable's duration. For these reasons, the presence of pretonic lengthening may have gone unnoticed in traditional descriptions of the Ukrainian metrical structure. Needless to say, the presence of increased duration that is related to lexical stress and unrelated to rhythm contributes to the complex- ity of the metrical system of Ukrainian. However, it also provides an argument that primary and secondary stresses have a different status within the metrical grammar, rather than representing two degrees of the same phenomenon.
To summarize, Ukrainian represents a hybrid system, combining primary lexical stress with secondary grammatical stress. The two types of stress are distinct phenomena; the former is lexically encoded and cumulative, whereas the latter is predictable and iterating. In what follows, we argue that the contemporary theories of rhythmic stress cannot adequately model the Ukrainian pattern.

Theoretical Implications
In sections 3.1-3.5, we demonstrate that the Ukrainian pattern is problematic from the point of view of OT accounts of rhythmic stress based on feet, regardless of whether the mechanism employed is gradient alignment (McCarthy and Prince 1993, Alber 2005 or categorical alignment combined with licensing (Kager 2001(Kager , 2005, and regardless of the type of feet assumed (e.g., Hayes 1980, Kager 2001, Hyde 2002, 2016, Pruitt 2010, Martínez-Paricio and Kager 2015.

Approaches Postulating Binary Feet
The gradient alignment approach fails to generate both the concurrent secondary stresses in initial and final positions and the internal lapses adjacent to the peak that occur on both sides of the peak. This is because, in essence, it requires that all feet (or stresses in the pure-grid approach) are pushed either toward the left or toward the right edge of the word. Depending on whether ALIGN-FT-L ("The left edge of every foot is aligned with the left edge of a prosodic word"; McCarthy and Prince 1993) or ALIGN-FT-R ("The right edge of every foot is aligned with the right edge of a prosodic word"; McCarthy and Prince 1993) is the dominating constraint in the hierarchy, the rightward or the leftward pattern can be generated, but never both. In the following analyses, we focus on two examples, i+ munii+ cypali} * tet 'municipality' and } * lahodyi+ tymei+ te 'repair, 2PL', representing the rightward and leftward pattern, respectively. As shown earlier in (5), the interactions between lexical and rhythmic stress result in a spectrum of possible stress patterns depending on the position of lexical stress. The patterns exemplified by i+ munii+ cypali} * tet and } * lahodyi+ tymei+ te are patently the best to look at because they represent the limits of this spectrum: with lexical stress positioned at one extreme point (the right word edge) in i+ munii+ cypali} * tet and at the other extreme point (the left word edge) in } * lahodyi+ tymei+ te, these examples pose the superset of problems with respect to those posed by the remaining lexical-rhythmic stress interactions exemplified in (5). This is so because they involve (a) placing rhythmic stress at the edge of the word, (b) iteration of rhythmic stress on the third syllable counting from the edge, and (c) a lapse in the vicinity of lexical stress. The evaluation of candidates is shown in (6). It is important to bear in mind that the position of lexical stress is determined at the underlying level and does not follow from output well-formedness conditions. Thus, in this and subsequent analyses, only candidates with correctly marked lexical stress are considered; the violation marks referring to the syllable carrying lexical stress are given in parentheses as irrelevant to the point of discussion.
In (6a), the correct output (6ai) requires the ranking ALIGN-FT-L ϾϾ ALIGN-FT-R. The contradictory ranking is required for (6bi) to win: ALIGN-FT-R ϾϾ ALIGN-FT-L. However, this setting would produce an incorrect result for words with main stress on the final syllable: Let us point out that we arbitrarily assume trochaic feet here because it is unclear whether any default foot parsing exists in Ukrainian. In this respect, Ukrainian is similar to Russian, where 4 A similar problem arises in the approach of Hyde (2002), who departs from the standard assumptions of proper bracketing and one-to-one correspondence between stresses and feet, instead proposing intersected feet. A detailed presentation of this approach would be beyond the scope of this article (see Martínez-Paricio and Kager 2020 for discussion and a critical assessment). However, let us point to the aspects of Hyde's model that could potentially be employed in the analysis of Ukrainian. Hyde (2002:316) proposes the following intersected feet (vertical association lines indicate prosodic heads): Both the foot type and the footing directionality are derived by the alignment constraints HDS-L ("The left edge of every foot-head is aligned with the left edge of some prosodic word") and HDS-R ("The right edge of every foot-head is aligned with the right edge of some prosodic word") (Hyde 2002:319). HDS-L favors trochaic feet (ia), drawing them to the left word edge, while HDS-R gives preference to iambs (ib) located as close as possible to the right word edge. As demonstrated in (ii), the mirror-image metrical patterns in } * lahodyi+ tymei+ te and i+ munii+ cypali} * tet require both an intersecting trochee (iia) and an intersecting iamb (iib). The former can be generated by the ranking HDS-L ϾϾ HDS-R, whereas the latter requires the opposite ranking, HDS-R ϾϾ HDS-L. Hence, this model encounters the same problem as the approach assuming binary feet presented above, in that it cannot generate both patterns employing one constraint ranking.
(ii) a. x no empirical evidence for foot parsing has been found and for which, as a result, both trochaic and iambic feet have been assumed on purely theoretical grounds. 5 However, analyses assuming iambic feet would also yield contradictory results for the rightward and leftward patterns. That is, for the stresses in (6a), feet would have to be aligned with the left word edge in the pattern (i + )(i + )(} * ), so the ranking generating the rightward stress iteration would be ALIGN-FT-L ϾϾ ALIGN-FT-R. The pattern with the leftward stress iteration, (} * )(i + )(i + ), mimicking the arrangement of prominences in (6b), would align feet with the right word edge; hence, the opposite ranking would be required: ALIGN-FT-R ϾϾ ALIGN-FT-L. Aside from the contradictory direction effects and the iambic-trochaic controversy, lurking beneath these analyses is the question of how to ensure that candidate (6bi), where secondary stress resides in both binary and unary feet, can ever win in the competition with candidate (6biii), where secondary stress resides in uniform binary feet. (Candidates with both trochaic and iambic feet are excluded by GEN, so * is not a viable solution to this puzzle.) This question is also pertinent to the licensing-based account that we discuss below. The Ukrainian pattern is also problematic from the perspective of the licensing approach proposed by Kager (2001Kager ( , 2005. This approach accommodates a sequence of two unstressed syllables in the vicinity of main stress, both to the right and to the left, either via LAPSE-AT-PEAK ("Lapse must be adjacent to the peak"; Kager 2001) or via *LAPSE-IN-TROUGH ("No sequences of two unstressed syllables between secondary stresses"; Kager 2005), which must dominate the general *LAPSE constraint ("No two adjacent unstressed syllables"; Selkirk 1984). Yet it offers no mechanism for placing secondary stresses at the opposite word edges simultaneously; see (7). The point is that the nongradient ALIGN-WD-L and ALIGN-WD-R ("Every prosodic word starts/ ends with a foot"; Kager 2001) operate in terms of foot units. While this solution is feasible for systems that exhibit clear iambic or trochaic patterns, it is problematic for Ukrainian, whose foot structure (iambic or trochaic) is far from obvious. In hexasyllabic words, this approach correctly generates initial secondary stress in words with main stress at the right edge of the word, such as i+ munii+ cypali} * tet (7a). However, in words with main stress at the left edge of the word, such as } * lahodyi+ tymei+ te (7b), the fully parsed and incorrect output *[(} * laho)(i + dyty)(i + mete)] (7biii) violates a subset of well-formedness conditions relative to the correct output with final stress [(} * laho)dy (i + tyme)(i + te)] (7bi). K Note also that while the lapse on the third syllable in the correct output [} * lahodyi + tymei + te] can be modeled without reference to foot structure (in terms of LAPSE-AT-PEAK or *LAPSE-IN-TROUGH), the parsing of the word into feet raises several questions (as anticipated in the gradient alignment analysis above). Assuming that parsing is trochaic in Ukrainian, is there a unary foot at the end of the word, as in candidates (7bi) and (7bii)? If so, under what conditions are unary feet allowed in Ukrainian? Would the candidate [(} * la)ho(dyi + ty)(mei + te)] with leftward iambic parsing also be a viable output? What mechanisms would ensure unambiguous trochaic or iambic interpretation of the string, and on what grounds could they be posited? Further, putting aside the iambic/trochaic controversy, it seems that the incorrect parsing *[(} * laho)(i + dyty)(i + mete)] (7biii) would emerge as suboptimal only if one assumed priority of rhythmic beats over foot structure. Apparently, if foot parsing were imposed in such cases, it would have to be generated in accordance with the "optimal grid" requirements: a high-ranked constraint demanding that a stress mark be aligned with the edge of the word.

Internally Layered Feet
An attractive alternative seems to be offered by Martínez-Paricio and Kager's (2015) approach employing internally layered feet with minimal recursion. This model postulates both binary and ternary feet, which may coexist within one system. Systems with strictly binary or strictly ternary feet are at the end of the rhythmic continuum; the interior of the continuum is occupied by mixed systems, in which the coexistence of binary and ternary (and sometimes also unary) feet is determined by a restricted set of principles. Apart from regular trochees (} * ), iambs (} * ), and unary feet (} * ), four types of internally layered feet (ILF) are permitted by GEN: ((} * )), ((} * )), ((} * )), ((} * )). Alignment constraints refer to minimal/maximal foot projections as loci of violation, penalizing their separation (in terms of some prosodic category, such as the foot) from the left/right edge of some prosodic domain. In the analysis of bidirectional stress systems such as Piro and Garawa, ternary feet combine with binary feet in odd-parity words, as shown in (8) (see Martínez-Paricio and Kager 2015:479).
Both Piro and Garawa are classified as mixed binary/ternary systems. In these systems, binary parsing is the default option, but a single ternary foot emerges in odd-parity words (Martínez-Paricio and Kager 2015:472). Thus, allowing ILF makes it possible to replace the former mechanisms for generating unparsed syllables in odd-numbered words by exhaustive parsing. The appearance of ternary feet is controlled by the constraint ALIGN-L/R nonmin ("For every nonminimal foot Ft nonmin , assign a violation mark if some foot intervenes between Ft nonmin and the left/right edge of its containing "). Apart from minimizing recursive (ternary) feet, ALIGN-L/R nonmin ensures that the ternary foot-that is, the foot accommodating a lapse-is located at the left/right edge of the word. Ternarity, on the other hand, is promoted by the constraint ALIGN-L/R min , which disfavors binary feet that are both maximal and minimal ("For every foot that is minimal and maximal ([Ft min ] ), assign a violation mark if some foot intervenes between [Ft min ] and the left/right edge of its containing ").
In Piro and Garawa, ALIGN-L/R nonmin must be ranked above ALIGN-L/R min , to ensure that the default foot parsing is binary; for a more exhaustive analysis, see Martínez-Paricio and Kager 2015:481. The directionality of stress iteration depends on the location of the ternary feet vis-àvis the word edge (right-aligned in Piro and left-aligned in Garawa) and the structure of ternary feet (left-branching ((} * )) in Piro and right-branching ((} * )) in Garawa). A consistent iambic or trochaic layout of recursive feet in a given metrical system is ensured by the constraints TROCHEE nonmin /IAMB nonmin ("For every minimal foot Ft min , assign a violation mark if some footed syllable intervenes between Ft nonmin and the left/right edge of its containing Ft"). Furthermore, exhaustive parsing in binary/ternary systems is generated by the high-ranked constraint CHAIN-L/R ("For every unfooted syllable () , assign a violation mark if some foot intervenes between () and the left/right edge of its containing "). The constraint ranking of Garawa is provided in (9)  The mirror-image system of Piro is derived by locating the ILF at the right word edge, through the reranking of ALIGN-L/R nonmin as ALIGN-R nonmin ϾϾ ALIGN-L nonmin , and by changing the position of the foot head from the leftmost to the rightmost, which is achieved by the ranking IAMB, IAMB nonmin ϾϾ TROCHEE, TROCHEE nonmin . As discussed previously, both rightward and leftward iteration of secondary stresses are found in Ukrainian in words with lexical stress separated from either the left or the right edge of the word by at least five syllables, as in i+ munii+ cypali} * tet and } * lahodyi+ tymei+ te. From the point of view of the distribution of secondary stresses, Ukrainian meets the definition of a bidirectional system. Therefore, it is expected that the rightward iteration pattern could be generated using the ranking of constraints established for languages with rightward stress iteration (as in Piro), while the leftward iteration pattern would require the ranking generating systems with leftward stress iteration (as in Garawa). The two rankings are incompatible; therefore, as in other foot-based approaches discussed above, an approach assuming ILF is not capable of modeling both rightward and leftward stress iteration with one constraint ranking. This is illustrated in (10), where, on the basis of the ranking established for Garawa, the test items i+ munii+ cypali} * tet and } * lahodyi+ tymei+ te are evaluated. To simplify the presentation, we do not consider candidates with unfooted syllables (which are prohibited by the high-ranked CHAIN-L/R). Note also the presence of unary feet in (10). Martínez-Paricio and Kager (2015) exclude unary feet from bidirectional stress systems via the high-ranked constraint ALIGN-L/R unary . We argue later in this section that Ukrainian must allow unary feet. Here, we note that eliminating unary feet from the candidate set in (10) would not lead to generating correct outputs. iii. iv.

J A N I N A M O Ł C Z A N O W A N D B E A T A Ł U K A S Z E W I C Z
choosing candidate (10bii), where the lapse occurs between secondary stresses. For candidate (10bi) to win, ALIGN-R nonmin must outrank ALIGN-L nonmin . However, this ranking would not generate a correct output in (10a). Besides the problem connected with the coexistence of leftward and rightward rhythmic stress iterations, a complication arises in the case of trisyllabic words. While the interweaving of binary and ternary feet ensures exhaustive parsing in classic bidirectional systems, such as Piro and Garawa, secondary stress in Ukrainian trisyllabic words such as } * zoloi+ to 'gold' and i+ borot'} * ba 'struggle' suggests that this system must also admit unary feet: (} * zolo)(i+ to) or (} * zo)(loi+ to), (i+ borot') (} * ba) or (i+ bo)(rot'} * ba). These forms also indicate that the default foot parsing in Ukrainian is binary, not ternary. This would make Ukrainian analogous to the binary/unary systems analyzed by Martínez-Paricio and Kager (2015:478-479).
However, other examples are not in concert with assuming the primacy of binary parsing: lahodyty 'repair' is parsed [} * i + ], not *[} * i + ]. From the point of view of such examples, Ukrainian would have to be classified as a mixed ternary/binary/unary system. This is a contradiction in terms, because within the ILF theory, systems that have exhaustive parsing group threesyllable strings either into a single recursive foot (compare binary/ternary systems with ternary/ binary/unary systems; Martínez-Paricio and Kager 2015:479, 481, respectively) or into a sequence of a binary and unary foot (binary/unary systems), never both. 6 The theory does overgenerate in the case of three-syllable strings; however, the only inconsistent parsing of strings that cannot be parsed by a canonical foot is "three-syllable exceptionality"-a "pathological" scenario whereby three-syllable strings are parsed by a recursive foot, while longer odd-parity strings are parsed by a series of binary feet plus a single unary foot located at the edge; Martínez-Paricio and Kager 2015:488). Thus, the Ukrainian system does not fill the unattested gap; instead, it presents an unpredicted scenario.
The failed evaluation of trisyllabic words with primary stress on the final and initial syllables is shown in (11), where we employ the Garawa ranking (see (9) and tableau (10)). For reasons of space, we do not show ALIGN-L/R nonmin because none of the candidates violates this constraint. Also, we do not consider candidates that exhibit reversal of lexical and rhythmic stress (e.g., *(} * borot')(i+ ba), *((} * borot')ba)), because they fatally violate a high-ranked constraint demanding preservation of lexical stress.

Grid-Based Approach
Another way of modeling metrical bidirectionality is offered by approaches that do not appeal to the metrical foot but refer directly to the grid (Gordon 2002). In Gordon's (2002) approach, attraction of stress to word edges is straightforwardly ensured by ALIGN EDGES, which requires that the edges of level 0 of a prosodic word-that is, both the initial and the final syllable-be aligned with a grid mark on level 1, corresponding to secondary stress (p. 497). However, the family of *LAPSE constraints expanded along the lines that Gordon suggests does not include any equivalent of LAPSE-AT-PEAK or *LAPSE-IN-TROUGH. Iteration of rhythmic stress is modeled in terms of gradient alignment, reformulated in terms of the metrical grid: ALIGN (X 1 , L) and ALIGN (X 1 , R) require that every level 1 grid mark be aligned with the left/right edge of level 0 of grid marks in a prosodic word. As a result, as in (6)-(7) and (9)-(10), the predicted pattern is compatible with either the left or the right side of the rhythmic mirror attested in Ukrainian, but not with both. In (12), ALIGN EDGES needs to be ranked highest to ensure the occurrence of polar beats at the left/right edge of the prosodic word. The rightward iteration in (12a) is effected by the

J A N I N A M O Ł C Z A N O W A N D B E A T A Ł U K A S Z E W I C Z
ranking ALIGN (X 1 , L) ϾϾ ALIGN (X 1 , R); the leftward iteration in (12b) can only be modeled in terms of the opposite ranking of the two constraints. Note that positioning primary stresses at word edges in the two words cannot be enforced by the activity of ALIGN EDGES, as primary stress is lexical in Ukrainian. ⇐ ii. laho dytyme te **, ***** (7) *** (3) laho dyty mete **, **** (6) *, *** (4) iii.

Combining the Grid-Based Approach and Licensing
In the light of the above discussion, it is clear that the Ukrainian mirror-image secondary stress pattern can be generated if and only if (a) foot-based constraints are replaced by grid-based constraints, and (b) gradient alignment pushing all secondary stresses toward the left or the right edge of the word is abandoned. In technical terms, this is achieved by the grid-based ALIGN EDGES accompanied by the licensing LAPSE-AT-PEAK. 7 The two constraints are defined in (13).

ALIGN EDGES
The edges of level 0 of a prosodic word, i.e., both the initial and the final syllable, are aligned with a level 1 grid mark. (Gordon 2002:497) 7 The ranking ALIGN EDGES ϾϾ LAPSE-AT-PEAK will also generate the bidirectional systems illustrated in (8). However, in order to generate penultimate stress in languages such as Piro, ALIGN EDGES would have to be outranked by NONFINALITY (Prince andSmolensky 2004, Walker 1996), formulated by Gordon (2002:501) as "A final syllable does not have a grid mark on level 1." In Gordon's proposal, the placement of primary stress on the penult is regulated not in terms of ALIGN EDGES but by a gradient alignment constraint referring to level 2 of the metrical grid, ALIGN (x2, R). In the account we propose here, we could instead use a grid-based version of the familiar End Rule-R (McCarthy 2003: 111). Note that this issue is not relevant to Ukrainian, where primary stress is lexically marked. 567 LAPSE-AT-PEAK ensures that rhythmic stress iterates from word edges so that the lapse is located in the vicinity of the lexical stress in even-parity words such as i+ munii+ cypali} * tet and } * lahodyi+ tymei+ te. ALIGN EDGES, in turn, derives edge-based secondary stress. This constraint is needed to account for edge-based stress patterns in a number of languages, including Canadian French, Armenian, and Udihe (see Gordon 2002:500 for discussion and references). These systems are different from Ukrainian in that they involve primary stress at one word edge and secondary stress at the opposite word edge. Thus, the point of interest of the Ukrainian pattern lies in the fact that ALIGN EDGES enforces secondary stresses at both word edges in this language. The analysis is shown in (14). iii.

⇒
The Ukrainian data are also consistent with the parameter-based approach developed by van der Hulst (1996Hulst ( , 2012Hulst ( , 2014, who assumes that rhythmic stress is derived without reference to foot structure by two independent mechanisms. First, a polar beat is placed at the edge opposite to the primary stress (on the initial and/or final syllable), shown in (15a). Next, rhythmic beats are added in between the two stresses (15b).
(16) a. Echo rhythm: In this approach, Ukrainian can be analyzed as having obligatory polar beats (15a) and optional rhythmic beats (15b). When rhythmic beats are present, they spread from the polar beats (16b), producing a bidirectional stress system with internal lapses near the primary stress. There are obvious parallels between this approach and the OT analysis suggested in (14). Polar beats correspond to ALIGN EDGES and polar rhythm achieves the same result as LAPSE-AT-568 J A N I N A M O Ł C Z A N O W A N D B E A T A Ł U K A S Z E W I C Z PEAK: the former attracts rhythmic stress to word edges while the latter repels it from the primary stress. The difference between van der Hulst's approach and the analysis suggested in (14) is that the latter does not predict the existence of patterns such as (16a). In our approach, only polar rhythm is possible in systems with polar beats. Thus, lapses are possible only at the peak, as in (16b). This prediction is consistent with the fact that bidirectional systems with secondary stress at word edges and a lapse not adjacent to main stress are unattested among the languages of the world (Kager 2001).
Further research is needed to predict the consequences of the hybrid approach that we propose here for the factorial typology of stress patterns. Also, the role of LAPSE-AT-PEAK needs to be further investigated. Intuitively, however, LAPSE-AT-PEAK offers a direct expression of the symmetrical distribution of secondary stress around main stress in Ukrainian.

Licensing Approach Based on Moraic Feet
An alternative foot-based analysis has been suggested by an anonymous LI reviewer, who points out that the increased duration of stressed syllables, reported in experimental research, may be encoded phonologically. Assuming that feet are maximally binary at the level of the mora, and that all heads of feet are heavy, moraic feet can be constructed at both word edges. As in the grid-based licensing model proposed above, the iterative stresses can be derived using LAPSE-AT-PEAK. The derivation of the two test items, i+ munii+ cypali} * tet and } * lahodyi+ tymei+ te, is shown in (18). The constraint enforcing bimoraic foot heads, which we provisionally call STRESS-TO-WEIGHT ("If stressed, then heavy"), is the reversal of the familiar Weight-to-Stress principle ("If heavy, then stressed"). 8 As in the grid-based approach laid out in section 3.4, the position of iterative stresses is derived by the rhythmic licensing constraint LAPSE-AT-PEAK, defined in (13), repeated here for convenience.
( la)ho( dy)tyme( te) ( la)ho( dyty)( mete) ⇒ *! * *! *** As demonstrated by the tableau above, the foot-based model combined with licensing constraints can successfully derive the Ukrainian pattern. The assumption that foot heads are bimoraic allows for the correct placement of secondary stresses at word edges, which is technically achieved by means of the constraints ALIGN-WD-L and ALIGN-WD-R. LAPSE-AT-PEAK, in turn, accounts for the bidirectional iteration of stresses, locating them in positions removed by two syllables from lexical stress.
As we have shown, both the grid-based and the foot-based licensing models are able to account for the distribution of stress in Ukrainian. Unlike the grid-based analysis, the foot-based analysis retains the foot constituent, which demonstrably plays an important role in many phonological systems. However, though the foot-based model can account for the distribution of secondary prominence in Ukrainian, it is based on a theoretical assumption that does not dovetail with the rest of the Ukrainian phonological system. In particular, the bimoraicity of foot heads makes wrong predictions concerning the behavior of geminates and glides in Ukrainian, as demonstrated in the remainder of this section. Ukrainian has true geminates, which contrast with single consonants in various contexts, including morpheme-medial positions: for instance, } * synju 'blue, FEM ACC SG ADJ' -} * synnju 'blue, INSTR SG NOUN', su} * di 'court, LOC SG' -su} * ddi 'judge, GEN SG' (Toc'ka 1969, Bethin 1992, Loboda 2009). 9 It has been standardly assumed in the literature that single consonants are distinguished from geminates by the presence of a mora in the latter (Hayes 1989:257 (20a) and (20b). 10 The stressed vowel [é] in (20a) is bimoraic due to STRESS-TO-WEIGHT. It remains monomoraic in (20b), because the stressed syllable contains two moras (the second of which comes from the consonant) and thus fulfills the bimoraicity requirement. The representation of the word } * syl'nu [} * sél j nu] 'strong, ACC SG ADJ', with a stressed syllable closed by a coda consonant, is shown in (20c). Since the stressed syllable is followed by a nonmoraic consonant in } * syl'nu (Weight-by-Position does not apply in Ukrainian), the stressed vowel [é] is bimoraic, as in (20a). l j u n s This model predicts increased duration of the bimoraic vowels in (20a) and (20c), but not of the initial monomoraic vowel in (20b). 11 While the vowel in a closed syllable, as in (20c) and (20b), is expected to be phonetically shorter than a vowel in an open syllable, as in (20a), to the best 9 Ukrainian true geminates arose historically at morpheme boundaries where coronals were followed by a sequence of a short vowel and a glide (mazz'u 'ointment, INSTR SG' Ͻ *maz‫ם‬ǐıˆiu). True geminates are different from sequences of identical consonants that are present at morpheme boundaries-for example, vid‫ם‬dil 'department', son‫ם‬n‫ם‬yj 'sleepy'-in that they do not allow epenthesis. Though there exist minimal pairs, there is no agreement in the literature about whether Ukrainian geminates are phonemic (see Bethin 1992Bethin , 1998 for detailed discussion).
10 For an alternative interpretation, see Bethin 1998:232. Bethin suggests that geminates share the mora with the preceding vowel. As a result, geminates are moraic, but they do not add to the syllabic mora count. 11 It has been noted in the literature that Hayes's (1989) theory of geminates predicts the existence of systems in which syllables closed by geminates act as heavy (similarly to syllables with long vowels), and syllables closed by nonmoraic consonants act as light (on a par with syllables containing short vowels); for a detailed discussion, see Davis 2011. Such a system is in fact found in Seto (Southeastern Estonian), where vowel lengthening occurs in syllables closed by coda consonants but not in syllables closed by geminates (Kiparsky 2008, discussed in Davis 2011. of our knowledge the vowels in the stressed syllables of (20b) and (20c) do not exhibit durational differences. 12 Another problem with the moraic-feet analysis arises when we consider the alternation of high vowels with their nonsyllabic counterparts, which is attested both at morpheme and at word boundaries in Ukrainian (Toc'ka 2002 It is a generally accepted view that in languages such as Ukrainian, high vowels and the corresponding glides are derived from one underlying source. Gliding, which consists in the deletion of a mora, is employed as a strategy to avoid vowel hiatus. Underlyingly, then, the sequence [i j] in lycari jdut' [} * léts Ë ai + r j i j} * dut j ] 'knights go' contains two moras. Assuming that stressed vowels are bimoraic on the surface, the vowel /i/ in the rhythmically stressed final syllable of the word } * lycai+ ri could acquire the second mora from the gliding of the word-initial /i/ of the word idut', producing the output *[} * léts Ë ai + r j i: } * dut j ]. The evaluation of the relevant sequence is presented in (22). 12 Toc'ka (1969:82-83), who provides measurements of vowel duration in different consonantal contexts, does not mention any differences in vocalic duration before clusters and before geminates. An anonymous LI reviewer rightly points out that the fact that Toc'ka does not mention the relevant difference may as well indicate that no measurements were conducted that would systematically compare vocalic duration in the two contexts. At present, we are not in a position to resolve this issue, and thus it must await future instrumental study. 13 In some contexts, the process applies optionally; for a detailed discussion, see Toc'ka 2002:60. Another caveat is that gliding occurs across word boundaries, while in word-medial positions hiatus is resolved by glide insertion, as in stojik [} * st:jik] 'Stoic', kokajin [k:ka} * jin] 'cocaine' (Rubach 2002  The intended winner (22i) attaches the second mora to the initial vowel, fatally violating IDENT, which mandates that the mora of the input should be preserved on the same output vowel. Candidate (22ii) wins because it violates a subset of constraints violated by candidate (22i). Thus, the model assuming bimoraicity makes the wrong prediction that the underlying sequence /i i/ could be realized as a long vowel. 14 In sum, though the foot-based model presented in this section can account for the distribution of secondary prominence in Ukrainian, it is based on a theoretical assumption that has certain undesirable consequences for the rest of the Ukrainian phonological system. In particular, the bimoraicity of foot heads makes wrong predictions concerning the behavior of geminates and glides in Ukrainian.

Conclusions
In this article, we have discussed new data from Ukrainian, which has a hybrid metrical system with unpredictable lexical stress and grammatical iterative secondary stress. We have argued that the Ukrainian metrical system poses a challenge for current theories relying on foot structure construction and/or employing the mechanism of gradient alignment. We have suggested an alternative account based on licensing theories that appeal directly to the grid. However, it is important to point out that we do not posit that feet can be entirely dispensed with in phonological analysis. The considerable body of evidence for foot structure amassed by research on metrical phonology allows us to predict that that would be an undesirable move. Rather, the Ukrainian data suggest that languages differ in which aspects of metrical structure-feet or grids-they highlight as primary, and which they mark off as redundant. From a typological perspective, there is abundant evidence for the interaction between metrical and segmental phenomena in Germanic languages (vowel reduction in Dutch (Booij 1995), flapping in English (Kiparsky 1979), and glottal stop insertion in German (Wiese 1996), to mention just a few), but there is no stressindependent evidence for foot structure in Slavic languages. Regardless of whether stress is lexical (as in Russian) or grammatical (as in Polish), feet have been posited as theoretical artifacts, rather than independently motivated structures (see also footnote 5).
The metrical system of Ukrainian sheds new light on the nature of bidirectional stress systems and the relationship between primary and secondary stress. Although the initial intuition was that lapses in such systems are adjacent to the peak (Kager 2001), the technical apparatus evolved to regulate this characteristic by fixing feet with respect to word edges, rather than with respect to the peak. This resulted in making the distinction between primary stress (the peak) and the rhythmic beats irrelevant (see, e.g., the analyses in Hyde 2002, 2016and Martínez-Paricio and Kager 2015. In our analysis, we restore the view that the window in which secondary stresses apply is delimited by primary stress at one end and the word edge at the other end. In grammatical systems, primary stress is necessarily fixed with respect to one of the word's edges, which invites technical solutions aimed at orienting all metrical structure left or right-that is, with respect to edges of the word. In such systems, it is impossible to disentangle the potential demarcating effect of primary stress from that of the corresponding word edge because they coincide. The confound does not exist in Ukrainian, which has free lexical stress. Directionality paradoxes in Ukrainian reveal that it is the primary stress that acts as the landmark toward which secondary stresses radiate from both edges (as in polar rhythm systems predicted in van der Hulst 1996Hulst , 2012Hulst , 2014; the exclusively left-or exclusively right-orientation of secondary stress alternation is not borne out. Including LAPSE-AT-PEAK among the licensing mechanisms correctly predicts that secondary stresses are repelled from primary stress, rather than being pushed to the left edge or to the right edge. This mechanism has been criticized (Alber 2005, referred to in Kager 2005 for its potential to generate the unattested metrical pattern whereby the peak stays in initial position in even-parity words, [(} * )(i + )(i + )], while shifting toward the end of the word in order to license a lapse in odd-parity words: *[(i + )(i + )(} * )]. The assumed ranking for such a purported system was ALIGN-L ϾϾ LAPSE-AT-END ϾϾ LAPSE-AT-PEAK ϾϾ LEFTMOST (Alber 2005:533); LEFTMOST/ RIGHTMOST derive from the EDGEMOST constraint family of Prince and Smolensky (2004) imposing the requirement that the head foot of the prosodic word be placed in initial/final position. The above criticism works on the premise that a language with default primary stress on the initial syllable is generated with LEFTMOST ranked at the bottom of the constraint hierarchy. One solution to the problem would be to posit that either LEFTMOST or RIGHTMOST must be ranked at the top of the constraint hierarchy in default stress systems (see Kager 2004Kager , 2005. Another solution would be to use End Rule-L/R ("The head foot is not preceded/followed by another foot within the same prosodic word"; McCarthy 2003:111).
Though it is technically possible to prevent generating an unattested pattern in which primary and secondary stress swap positions, it is important to bear in mind that the problem arises because main stress is treated on a par with rhythmic stress and the model lacks a mechanism that would give priority to main stress assignment. In the light of our analysis, by referring to the peak as the licensor, LAPSE-AT-PEAK correctly brings to the fore the issue of the difference in status between the primary stress and secondary stresses. Notably, this difference is also highlighted in the parameter-based approach of van der Hulst (1996Hulst ( , 2012Hulst ( , 2014. In Ukrainian, it is self-evident that lexical stress and rhythmic stress must be different phenomena, as they are located in different components, lexical as opposed to grammatical. But also in grammatical stress systems such as Polish, the status of the two phenomena is different: primary stress is insensitive to the presence of rhythmic stress, and hence to its position, while the application of secondary stress is restricted within a domain demarcated by the primary stress and the edge of the word. Two-and three-syllable words have primary stress, but no secondary stresses; obviously, there are no words with secondary but not primary stress in Polish to reify the hypothetically reversed scenario of primary stress assignment conditioned by the distribution of rhythmic beats. This makes the situation in Polish and Ukrainian fully comparable: secondary stresses apply only when the distance between the primary stress and the edge of the word is sufficiently big to accommodate these stresses. Finally, it should be pointed out that theories of lexical stress (see, e.g., Alderete 1999, Revithiadou 1999 do not provide tools to model a possible interaction of lexical stress with rhythmic stress. At the same time, lexical stress systems fall outside the purview of the models of bidirectional stress systems, which have been designed to account exclusively for grammatical stress systems (see, e.g., Martínez-Paricio and Kager 2015:487). The question then arises whether grammatical stress theories should be capable of modeling grammatical stress in systems with primary lexical stress or whether these theories can only provide technical apparatus to describe systems in which both primary and secondary stress are predictable. If the latter is the case, then we need another model, which would target grammatical stress in languages with primary lexical stress. This, however, would be an undesirable move, as introducing two technical devices to account for what is essentially the same phenomenon (rhythmic stress) would amount to restating descriptive generalizations and, in effect, call into question the basic rationale of metrical theory. We hope that the complexity of the Ukrainian metrical system, which has not been previously discussed in the theoretical literature, will point to new research directions and contribute to advancing the theory of stress.