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<Paper uid="C94-2163">
  <Title>Phonological Derivation in Optimality Theory*</Title>
  <Section position="3" start_page="0" end_page="1007" type="metho">
    <SectionTitle>
OPTIMALITY THEORY
</SectionTitle>
    <Paragraph position="0"> Optimality Theory tOT) is a constraint-bascd theory of phonoh)gy, developed by Prince and Smolensky (1993) (hereafter, this work will be referred to as P&amp;S) and is now being used by a growing numher of i~honologists (ito and Mester 1993, McCarthy and Prince 1993, McCarthy 1993).</Paragraph>
    <Paragraph position="1"> It difli~rs from declarative phonology (Bird 1994, Scobbie 199 I, Bird and l'31ison 1994) in that its constraints arc vio-lable and can conllict, with the contlicts resolved by an * This research was funded Ily the U.K. Science and F.ngineeling Re search Council, i.llldcl- grallt GP./G-22084 (.'t.'mputalional Pho.ology: A Co~lsH'ainl-BasedAl.,prrmch. I am grateful to Stcven Bird, 1&amp;quot;,wml Klein alld Jitll Scobbic for their COIIIIIlelI\[S Oll fin etlrliel- Version of this \[)aper. ordered system of defaults I . l)eclarative phonology evaluates candidate l'ornls ~ on a binary scale: whether they are accepted by a constraint system or not. In contrast, OT assigns a ranking to all of the candidale realisations of a word, calling the scale a tlleasta'e o1' harmony. ALl of the candidates which show the maximal amount of harmony are accepted by the constraint system, and others are rejected. A derivation in OT consists of an original candidate set produced by a \[unction called GI,;N, and the subsequent application of constraints to reduce tile candidate set, eliminating non-optimal candidates and I)reserving those with Ihe greatest harmony. At no stage can a constraint elimillale all candidates.</Paragraph>
    <Paragraph position="2"> Each constraint assigns to each candidate a list of iJlarks.</Paragraph>
    <Paragraph position="3"> These lllilrks illay, \['el instance, tag segntents as reguhtr or exceptional. The lllarks are wdues on the harnlony scale, and are totally ordered: for any two marks a and b, either a is more harmonic than b (symbolically, a ~- b) or tile reverse. In the list assigned to a candidate, however +, the Sallle lllal'k Inay occttl' inaay times, To COllll)arc the hal'illOIly Of twt) candidates with regard to a given constraint, their respective lists of marks are sorted into increasing order of harmony 3. The lists are then compared lirstqolast componentwise. The more harmonic candidate has Ihe more hammnie vahte at tim tirst point where the lists differ. The empty list always has the same harmony its lhe most hamlonic mark on the harmony scale, common to all consmtints, which we will call the zero mark, and write as (34 (2restraints which only use two different marks arc called binary constraints. For binary constrainls, the ewlluation of harmony is a sitnple affair. The candidate with the fewest non-zero marks is preferred.</Paragraph>
    <Paragraph position="4"> Consider, lbr exatnple, tile hinary constraint ONS(P&amp;S:25). This constraint discourages nuclei without onsets when selecting hetween diffc, rent syllabifications.</Paragraph>
    <Paragraph position="5"> Two syllabilications of the Arabic segntental sequence alqahmm are shown in (I), with syllables demarcated by parentheses. The nuclei are always the w)wels, The dishart l'31ison (1994) offers at I~armal amdysis of Ihe use of defaults in Optimalily Theory.</Paragraph>
    <Paragraph position="6"> 2 hi COllSll'aint-llased theories, consll'ailllS ilIipose \[illlits Oli possible rcalisations of objects, such as words or sentences. A caadidate is a tellialive realisalioll which is yet to be Icslcd against the eonstraillts. aEarly in their techaical report, P&amp;S illlroduce OllC COllStrgtiat, \[\[NUC, which requires stlrtiag into Ihe reverse order, l .alel&amp;quot; in the SalllO work Ihey replace this eollSll'aillt with a lltllnber o1' billary COliStl'aillls wilh I\[le tlSlllll ordering.</Paragraph>
    <Paragraph position="7">  monic mark L indicates on onsetless nucleus, the harmonic (zero) mark 9) is used tor other segments.</Paragraph>
    <Paragraph position="9"> In this example, the sorted lists of marks differ in the second position with the first candidate, (al)(qa)(ia)(mu), being the more harmonic of the two.</Paragraph>
    <Paragraph position="10"> When there is more than one constraint, we must consider not only the orderiug of marks assigned by one constraint, but the ordering of marks from diffcrent constraints. In OT, constraints are placed in a total order (6'1 :,--&gt;- C2), and all non-zero marks of higher-ranked constraints (C1) are less harmonic than all non-zero marks of lower-ranked constraints (C2). in effect, this means that higher-ranked constraints have priority in eliminating candidates. For all constraints, however, the zero mark has the same, maximally harmonic, value.</Paragraph>
    <Paragraph position="11"> binarity So far we have considered a general class of constraints including non-binary constraints. As it happens, non-binary constraints can often be replaced by binary constraints.</Paragraph>
    <Paragraph position="12"> Binary constraints are those which only assign two marks: the zero nmrk, and one other.</Paragraph>
    <Paragraph position="13"> In the simplest case, restating a constraint in a logically equivalent form can transform a non-binary constraint into a binary constraint. The constraint fanfily EDGEMOST is delined by P&amp;S(p35) as (2).</Paragraph>
    <Paragraph position="14"> (2) EDGEMOST((-'; E; D).</Paragraph>
    <Paragraph position="15"> The item 4) is situated at the edge E of domain D.</Paragraph>
    <Paragraph position="16"> This delinition covers a family of constraints depending on the instantiations of the arguments: E is either left (L) or right (R), domain \]nay be syllable, foot or word, and ~b can be any phonological object, such as stress or an affix.</Paragraph>
    <Paragraph position="17"> According to P&amp;S, constraints of this form are nonbinary, returning as their marks the distance of their objects from the designated edge of domain. The greater the distance, the less harmonic the mark. Constraints of this kind can, however, be replaced by logically equivalent binary constraints (3).</Paragraph>
    <Paragraph position="18">  (3) NOINTERVENING(O; E; D).</Paragraph>
    <Paragraph position="19">  There is no material intervening between 4) and edge E of domain D.</Paragraph>
    <Paragraph position="20"> This form of constraint assigns a disharmony mark to each item intervening between 4) and edge E. The more material lying between 4) and E, the greater the nmnber of marks and so the lower the harmony value.</Paragraph>
    <Paragraph position="21"> Other types of non-binary constraints can be converted into hierarchies (ordered sequences) of binary constraints. Suppose a constraint C produces N different kinds of marks. Applied to a candidate form c, this constraint produces a list C(e) of anal'ks. Now detine a function f which takes a list of marks, 1, and a mark type m, and replaces all marks in 1 which are different from m by thc zero mark 0, and then re-sorts the list. So with the marks</Paragraph>
    <Paragraph position="23"> If the marks generated by C are 0= ,q &gt;- r~2 &gt;-- .. &gt;- rt*N, then C can be replaced by consmfints Ci,i=l. N_ 1 such that Ci (c) = f(C(c), i) sut!ject to the ordering Ci&gt;-&gt;-Ci if i&gt;j.</Paragraph>
    <Paragraph position="24"> To see the equivalence of the single nm&gt;binary constraint with the family of binary constraints, let us look at the comparison of some candidate forms. Using the three-valued constraint of the earlier example, st, ppose candidates M, N and P are assigned mark lists 102, 219)12 and 9)122 respectively. Sorted, these lists become 2J0, 22119) and 2210. Comparing these lists, we arrive at the harmony ordering M &gt;- P &gt;- N.</Paragraph>
    <Paragraph position="25"> Now, let us apply the corresponding binary constraints.</Paragraph>
    <Paragraph position="26"> The first and dominant constraint preserves only 2s in the mark list, the second preserves only the mark 1. The two lists of marks lor M, N and P are 209) and 19)0, 22000 and 119)9)0, and 2209) and 19)9)9), respectively. By lhe ordcringof the coustraints, we know that 2 --&lt; I still, and so merging the two lists of marks for each candidate gives 219)9)9)0, 221.100009)9) and 22100000. Apart from the trailing 0s, these arc identical to the marks assigned by the single constraint, and so lead to the same ordering: M &gt; P &gt;- N.</Paragraph>
    <Paragraph position="27"> So all constraints which use a finite alphabet of marks, and some which do not, such as EDGEMOST constraints, can be t,'anslated into binary constraints or a linite sequence of binary constraints. Consequently, formalising binary constraints and their interaction wilt be enough to capture the bulk of constraints in OT.</Paragraph>
  </Section>
  <Section position="4" start_page="1007" end_page="7009" type="metho">
    <SectionTitle>
FORMALISATION
</SectionTitle>
    <Paragraph position="0"> The formalisation of OT developed here makes uses three idcalising assmnptions (4).</Paragraph>
    <Paragraph position="1">  (4) 1. All constraints are binary. 2. The output of GEN is a regular set. 3. All constraints are regular.</Paragraph>
    <Paragraph position="2">  We have aheady seen that most non-binmy constraints can be recast as binary constraints or families of binary constraints. Unlortnnately, P&amp;S are not explicit about whether there are other unbot, nded non-binary constraints (like EDGEMOST) -- there may he some which cannot be recast as binary constraints. Assumption I is, therefore, an idealisation imposing a slight linfitation on the theory. regular gen The second assumption requires that tim ontpnt of GEN be regnlar. Recall that GEN is the function which produces the initial set of candidate forms which is reduced by the constraints. In other words, the set of candidates must be i,fitialised to a set which can be dclincd by a regular expression, or, equivalently, by a \[inite-state autonmton (FSA).</Paragraph>
    <Paragraph position="3"> As an example, (5) shows a regular expression giving a subset of the candidate syllabifications of alqalamu accorcling to the syllabification rules of P&amp;S(p25). The set does  not include all candidates; for clarity I have omitted partial syllabilications in which segments have not been assigned a syllabic role, and completely empty syllahles. The set does include syllabic slots which do not correspond to segments. In snch slot-segmen! pairs, tile empty segment is written as 0.</Paragraph>
    <Paragraph position="4"> L01J..1 ,i. \[L0lJl I/rl. oJJJ :{(,:: },o ,;, (,;}} ,:{;} The hrackets cover disjunctions of lerms separaled by vertical bar I, while concatenation is expressed by juxtaposition. The vertical pairs of symbols are tile complex labels used (Ill alcs ill tile corresponding atttolnaton. The three syllabic slot types arc onset (O), nucleus (N) and coda (C). As a reguhn&amp;quot; expression, (5) captures 64 different possible syllabiIications of tile sequence ahlalanm, l:or example, Ihe syllahilicntion (al)(qal)(am)(u)is accepted by the (5), while (aiq)(al)(am)(u) is not.</Paragraph>
    <Paragraph position="5"> regular constraints The third assumption imposes regularity on conslraints. A constraint is regular if there is iF linite-slale tlansducer 5 (I&amp;quot;ST) which assigns tim same list of lllarks to a candidate form that the constraint does. Since we are only dealing with hinary constraints, the transducer will associate with each component of tile caudidale oue ,51&amp;quot; the two harmonic vahles ( ~ (/). Such transdtlcers Call be expressed its regular expressions over pairs of phonological material and nll.nks. P&amp;S (1/25) ase two constraints, lVIl,l~ (6) and ONS (7), to account for the limits tm cpenlhesis in Arabic. Epenlhetic material arises when syllabic slots which are not occultied by segments are realised, llerc the nlarks are given on the right hand side of the cohm in each pair. llere ~ is the disharmonic nlark, alld 0 the more harmonic zero illat'k.  (6) I;II.L. Sylhthte positions arc lilled with segnlenlal materM.</Paragraph>
    <Paragraph position="6"> (7) ONS. Every syllable has an onscL  These two constraints can be readily translated into regular expressions, using the ahbrcviatory notations: N for onset or coda, 0 for segmental nmterial and *, for anything. The transducers for 171I.I. alld ONS are defined hy tile regtdar expressions in (8) and (9) respectively.</Paragraph>
    <Paragraph position="7"> This trausduccr marks with c every syllabic slot associated</Paragraph>
    <Paragraph position="9"> SA lilfitc slate II'illlSdUct3r is all FSA which is labelled with pairs of values. In this case, the pairs will combilm phonological in forlnatioll with COIIStl'tthl\[ lllalks.</Paragraph>
    <Paragraph position="10"> This transducer is non-deterministic, producing more than one sequence of nlarks Ior a given input. All nttclei preceded by an onset are marked with 0 and wilh ~. All other segnmnls segments are marked as 0. The multiple evahmtions el:candidates is not a problem: candklates will survive so long its their best evaluation is its good as the hest of any other candidate.</Paragraph>
    <Paragraph position="11"> linearity The reader may be concerned that tim regularity consmtint in\]poses tmdue restrictions (51' linearity (m the candidate forms, and, in doing so, vitiates the phonological advantages of non-linear representations. This is not tile case. Bird and Eltison ( 1992,1994) have shown that it is possible to capture the semantics of autosegmental rules and representations using FSAs. The oulpttt of GEN, therefore, may correspond to a set of partially specilied autosegmental representations, attd still be i lUerprelcd as a regular set. candidate comparison For single binary constraints, tile harnlony of candidates is compared as sorted lists over the alphabet containing c and 0, whe,'e 0 has the higher harmony, the same, in fact, its tile empty list. Consequently, the results o1' comparing lists of these marks is identical with comparing #&lt;h(&lt;) where #e is the ntunber o1' times c occurs il5 tile list, and h(e) is the constant quantity of harnlony assigned to f.</Paragraph>
    <Paragraph position="12"> As ~ has Ihe same harmony as the empty list, h((/)) musl be zero. As e -X O, comparison is preserved il' h(e) &lt; 0, so we set h(~) =: -1. II' the arcs in file h'ansducer arc labellcd with -1 and 0 instead o1' e and (/), then the harmony of a candidate can be evaluated by just adding the numbers along the corresponding path in the constraint transducer. 'File greater tile (always non-positive) result, Ihe more hartnonic the candidate.</Paragraph>
    <Paragraph position="13"> Just its we Catl tneasure hartilony relative to a single COltstraint with a single integer, we can ineasure tile Imnnony relative to an ordered hierarchy of constraints wilh an ordered list of integers. The list of integers corresponds one-to-one to tile constraints in decreasing order (51: dominance. \[!ach integer maintains information about the number of e values o1: the corresponding constraint in lhc evaluation of the candidate. A candklate wilh tile list (--2, -I ) violates the first conshaint twice and tile second once: the corresponding sorted list of harnlony marks is 22 I.</Paragraph>
    <Paragraph position="14"> Lists of this t'ornt citn be compared just like lists of lmrmony marks. The first integer is tile most signilicant and tile last tile least. The greater of two lists is tile ode with the higher value at the most signiiicant point of difference.</Paragraph>
    <Paragraph position="15"> l;oi' example (- 10, --3 \[, -50) is nlOl'e harmonic than (&gt;) (-10,-34,-t2). I,ists of integers can be accumulated like single integers using componentwise addition.</Paragraph>
    <Paragraph position="16"> We can generalise transducers l)'om denoting single constraints to detloting hierarchies of constraints: frotIt translating candidates into sequences o1: {0, e} or { 0, - 1 } lllarks to transducers from candidates to sequences of lists of integers, each integer drawn fronl {0, - 1 }. Sunnning the lists  along a path gives a harmonic evaluation of the corresponding candidate.</Paragraph>
    <Paragraph position="17"> Let us call the length of the integer list the degree of tile transdt, cer. The output of GEN is an automaton -- a trailsducer without marks -- and so corresponds to a transducer of degree 0. The transducer for a single constraint needs only a single binary distinction for its marks, so a degree 1 transducer suflices. In general, the number o1' binary conslraints that a transducer encodes will equal its degree.</Paragraph>
    <Paragraph position="18"> The next section looks at how transducers of single constraints or small hierarchies can be combined into single transducers for larger hierarchies.</Paragraph>
  </Section>
  <Section position="5" start_page="7009" end_page="7009" type="metho">
    <SectionTitle>
ALGORITHMS
</SectionTitle>
    <Paragraph position="0"> product We have seen how a single constraint can tie regarded as a transducer fl'om candidate segments into a singleton list of integers, and further that multiple constraints can be cvaluated using longer lists of integers. Combining these two notions into tin extended version of the automaton product operation allows us to build tip transducers capturing a hierarchy of constraints fiom single constraint transducers. The product operation is easier to describe when trailsducers are thought (if in terms o1' automata rather than reguhlr expressions. For brevity, then, the algorithms will be phrased in terms of the states and arcs of an automa- ton, while, for clarity, regular expressions will be used to present the inputs and outputs of examples.</Paragraph>
    <Paragraph position="1"> Tile pseudocode for the standard automaton product operation appears in (10). As the initial states of any automaton can be identified with each other withont affecting the language recognised, and similarly the final states, we will assume that there is only a single initial state (I) and linal state (F) in each automaton. In this pseudocodc, semicolons are followed by colnments.</Paragraph>
    <Paragraph position="2">  The pseudocode in (10) applies to two automata A and/3, over the same alphabet, and constructs their product el x 17, an automaton which accepts only those strings accepted hy both ,,4 and/3. Each combination of arcs, one from ,,4 and one from/3, which could be traversed while reading tile same input, that is, an input in the intersection .Ad fI./V&amp;quot; of the labels of the two arcs, defines an arc in the product autonlaton.</Paragraph>
    <Paragraph position="3"> &amp;quot;Ib make the product mimic tile combination of constraints in OT, we need to introduce an asymmetric operation on the lists o1' marks: concatenation. Each arc in each automata passed to this prodnct operation is labelled not only with a set of possible phonological segments, but also a list of harmony marks. When two arcs are combined, these lists tire concatenated. The pseudocode for this augmented product operation appears in (I 1).</Paragraph>
    <Paragraph position="4">  augmented product does not commute: A x/3 does not assign the same marks to candidate forms as /3xel. The difference in interpretation is that A x/3 regards all constraints in el as higher priority than all constraints in/3, whereas/3 x el i nstantiates the reverse ordering.</Paragraph>
    <Paragraph position="5"> Tile augmented product operation provides a way of combining two constraints into a single transducer. As an example, (12) is the product of the transducers corresponding to the constraints ONS (9) and FmL (8) in that</Paragraph>
    <Paragraph position="7"> The prodnct is the crucial operation for implementing OT. The product of the regular expression or automaton produced by GEN with all of the constraints in order products a transducer encoding the harmony evaluations of all candidates. Let us call it the su#;i?we transducer. To evaluate the harmony of any fully specified candidate, we need only follow the corresponding paths in the sur\[ace transducer accumuhiting the integer lists associated with each arc. Tile path with the greatest total harmony is the crucial one lbr deciding whether the candidate is optimal or not, The surface transduccr which is the product of the candidate syllabilications of alqalanm with the constraints ONS and FILL, in that order, is shown in (13). tr ) }oooo.o</Paragraph>
    <Paragraph position="9"> harmony of suhstrings In OT, only tile candidates with maxinml harmony survive to the surface; non-optimal candidates are eliminated. To  implement this part of the derivation, we need to remove all paths fi'om the sttrface transducer which do not acctnntllate optinml values of harmony. The algorithms in this section and the next are designed to achieve this task, and will be proven to do so.</Paragraph>
    <Paragraph position="10"> The lirst algorithm (14) assigns Io every slate N the harmony value of the optimal path to it from the initial state, storing this wflue in the liekl harmony(N). Since there is only a sin gle linal state, F, harmony(F) will c{}ntain the harmony evalttation of all {}ptinml candidates.</Paragraph>
    <Paragraph position="11">  The algorithm sets the harmony of the inital slate to zero, and places Ihe inilial state itt an otherwise empty list. Tile most optinml member ill the list is expanded (lines 6-15) and relnoved from the list. When a state is expanded, all of the arcs from it are examined in turn. If ally of them point to states with undelined Imrtnony values, tile harlnony of the state being expanded, and o1' tile arc, are used to calculate the lmtumny value of tile {}tiler state and it is added to the list. If the arc points to a state with a delined harmony value, the hamlony value of the better palh is retained by that state, and its position ill tile sorted list adjusted appropriately.</Paragraph>
    <Paragraph position="12"> If the list is kept sorted, inserting each new state ill oMer of the value o1' its harmo,ly lield then, in the worst case, o(Iog \[sZatc,s D comparis(ms of harm{my wdues will need to he done fol+ an insertion into the list where ,+tcttc.s is the set of states in the transducer and arcs the set o\[ arcs. As each stale is expanded only once, each arc is examined only once. So lare.s I forms an upper bound on lhe number {)f insertions that need to he done. The single comparis{m on line 9 is insigniIicant in relation to tim coml}arisons used in insertion. So an upper hound on order of the worst case execution of tiffs algorithm is o(I,vc.q lo~ I.,zazc.H) COlltpari8011S.</Paragraph>
    <Paragraph position="13"> It is not obvious that this algorithm will, in fact, label each state with the lmnnony of the optimal 1}ath to it, so a l}roof follows.</Paragraph>
    <Paragraph position="14"> (15) I+emma. When state M is being expanded (lines 615), tile tree harlnony value of Ihe optimal path to M, namely h(M), and the computed value, harll}ony(M), arc equal, il'the same is true lbr all previously eXl}anded stales.</Paragraph>
    <Paragraph position="15"> Proof. Case .&lt;. Suppesc the lemma is false, and that h(M)&gt;harnlony(M). Then there is an el}tined path p.a.q where p is a (possibly null) path, (t is an arc from an already expanded state P, to an unexpanded state S and q is another (possihly null) path. There will always be such a path as M is reachable Dom the initial state, and the initial state is tile litst one expanded. This path is ot}timal, so h(M) = h.(R) I- h(a) -I-h(q) which in turn is less than or equal to h(P,) q. h(a) as h. is always non-positive. F'utting this inequality together with the SUl}position of the lemma that lmnnony and h. match for all expanded nodes, gives tile following inequality:</Paragraph>
    <Paragraph position="17"> A lower I}ound for harmony(S) was set when P, was expanded. As R is ah'eady expanded l~(P,)=harnaony(l~,), and consequently harn/ony(S)&gt;harmony(M) which contradicts the ntinimality o1' the choice of M (line 6 of algorithm (14.)).</Paragraph>
    <Paragraph position="18"> Thus h(M)&lt;harnaony(M), Case &gt;A. lf M is in list, then harmony(M) must he defined and set at a wdue _&lt; h(M).</Paragraph>
    <Paragraph position="19"> Tiros the equality o f harmony(M) and h.(M), and the lemma. When M is the initial state I, the result follows immediately from line 3 which sets hartnony(I) to zero. u (16) Theorem. After the al}plication o\[LAImI,NOI)ES , for all states N on which harmony(N) is deiined, harmony(N) is the harmony of the optimal path to N.</Paragraph>
    <Paragraph position="20"> Proof. By the lemma and induction on the sequence of expansion of stales, uI We call mimic the hthelling of nodes ill the transducer with harmonic evaluations by labelling disjuncts in reguhu expressions with harmonic values. The value of a whole disjunction is most Ilannonic value amongst the disjuncts. As before, the harmonic cvahlations arc added during concatenation. The evahtations lor the surlhce transducer (13) of alqalamu are shown in (17).</Paragraph>
    <Paragraph position="22"> &lt;File evaluation of the optimal path in the transducer is (O,-l).</Paragraph>
    <Paragraph position="23"> pruning Having determined the harnmny value of the optimal path to the final state, and others, it only remains to remove suboptimal paths. As it happens, this can be easily done by removing all arcs which cannot occur in an optimal path  If the sum of the harmony ot' an arc, and the harmony of the optinml path to the state it comes from, is less than the harmony of the state the arc goes to, then that means that there is a lnore optimal path to the second state which will always be preferred. Consequently this arc can never be part of an optimal path. It is, therefore, saliz and appropriate to delete it.</Paragraph>
    <Paragraph position="24"> The complexity, in the number of comparisons performed, of this algorithm is identical to the number of arcs in the transducer. This is of lower order than the wurstcase complexity for LABEl.NODES, so the complexity of the combined algorithm is still o(\[ares\[ log \[stctte.sl) con,parisons. null It is not immediately obvious that the only paths which can be formed by the remaining arcs are optimal. This is, howevel; the case.</Paragraph>
    <Paragraph position="25"> (t 9) Theorem. After tile application of LAIIELNODES and PRUNE there arc no non-oi}timal paths from the start state to any state.</Paragraph>
    <Paragraph position="26"> Proof. By induction on the length of the path.</Paragraph>
    <Paragraph position="27"> P(n) = Alier the application of LABt~LN{/DES and PP, UNE there is no non-minimal path of length n from tile initial state to any other state.</Paragraph>
    <Paragraph position="28"> Base case. P(0) is trivially true, as there is only a unique path of length zero.</Paragraph>
    <Paragraph position="29"> Step. Assume P(k) is true. Suppose we have a non-optimal path of length k + 1. By the assumption, this must consist of an optimal path of length k followed by a non-optinml arc a fi'om M to N. c~ would have been deleted unless harmony(M)+harmony(a)&gt;hannony(N). But, by theorem (16), harnlony(N) is the harmony of the optimal path to N. So harlnony(M)+hannony(a)&lt;harmony(N), and the path must be optimal. This contradicts our supposition, and so P(k + 1) is true.</Paragraph>
    <Paragraph position="30"> The theorem follows by induction. \[\] Consequently, the only paths fiom the initial state to tim linal state will he optimal and deline optimal candidates. The regnlar expression corresponding to the culled automaton describing the syllabifications of alqalamu appears in (20). It includes only a single candklate sylhtbilication of the sequence.</Paragraph>
    <Paragraph position="31">  The work described in this paper was based oil the Optimal ity Theory of Prince and Smolensky (I 993), making three additional assumptions:  1. All constl'aints are binary, or can be recast as binary constraints. This seems to be true of all constraints used by P&amp;S.</Paragraph>
    <Paragraph position="32"> 2. That the initial set of candidates, the OUtlmt of GEN, is a reguhtr set which can be specified by a finite-state automaton.</Paragraph>
    <Paragraph position="33"> 3. Each constraint cml be implemented as a reguhu'tran null sdncer which determines the list of marks lor each candidate.</Paragraph>
    <Paragraph position="34"> On the basis of those assumptions, the following developments were made: * Transducers were defined wlfich computed not just a single constraint, but an ordered hierarchy of collstraints. null * An algorithm for a product operation on these trunsducers was given. With this operation transducers representing constraints coukl be applied to sets of candidates, and also be combined into transducers representing collections of constraints.</Paragraph>
    <Paragraph position="35"> * Algorithms were presented for - tinding the harmony of the optimal candidate in a transducer, and - culling all non-optimal paths li'om a transducer. * These algorithms were proved to fullill their goals. * The worst-case complexity o f the combined algorithna in terms of harmony comparisons was found to be less than o(\[arcsl log \[.~t.zr;sl), for a given transducer. Using the assumptions and algorithms given here, there are three stages to computing a derivation in OT:  1. Specify the regular class of candidates as an automatoil. null 2. Build tip the product of this automaton with the tran null sducers of each constraint in decreasing order of priority. null  3. Cull subol)timal paths.</Paragraph>
    <Paragraph position="36">  There are at three lllOl'C points worth noting, lqrstly, the constraints in a hierarchy can be precompiled into a single transducer, l';ach application to a set of candidates then only requires a single product operation tollowed by a cull. Secondly, casting tim output of GI,:N and all constraints as reguhu' means that, al all stages in a derivation, the set of candidates is regular. This is because the output of the product and ctllling operations are reguhlr-- - Ix)lh return autonlata.</Paragraph>
    <Paragraph position="37"> Finally, this speeilication of OT in te,ms of reguhu' sets Zllld \[inite-state iltltonlata opens the way for more rigorous exploration of the differences between OT and declarative phonological theories, such ;.is One-Level Phonology (Bird and l illison 1994), which is a constraint-based phonology that defines inviolable constraints with automata.</Paragraph>
  </Section>
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