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(ll r(   f  c $ʚ A@]`ʚ  <RCD  The Movie< M/( /  H̚ )@"`$ *]`̚P "< Alan Prince. 2009. Thanks to Birgit Alber, Jane Grimshaw, Alison Prince, Adrian Brasoveanu, and Brett Hyde for valuable suggestions and comments which have formed and reformed this piece. Thanks to Bruce Tesar for much enlightening background discussion.<  NΚ/@?"< @`NNN?N ]`Κ <Formatted for Mac - Excel 2004<   HϚ)@?"6@`NNN?N ]`Ϛ@ <This file contains no macros.<  <@Ϛ A@"` ] `@Ϛ <RCD  The Movie<!yC  # l`КA@jJ?"6@`NNN?N/0] ``К <last mod 11-11-09<t >@<dd@<7T11yK  next>>Abstract!R1C1 Sheet1 I  c +  dMbP?_*+%"D??U } UD} D} *D} D} D}  D} * D} Dc2'j'jD'`k' '' '@k'Pk' 'WH ' ' 'k 'k'X''''''X'''Pl'lH'l'0l''''plX'TILLIIIIIIIIIJJJJJJJJJJJJJJJJJJJJJJJJJJJ&TIIIIIIIIIIIIJJJJJJJJJJJJJJJJJJJJJJJJJJJ&TIIIIIIIIIIIIJJJJJJJJJJJJJJJJJJJJJJJJJJJ&TIIIIIIIIIIIIJJJJJJJJJJJJJJJJJJJJJJJJJJJ&TIIIIIIIIIIIIJJJJJJJJJJJJJJJJJJJJJJJJJJJ&TIIIIIIIIIIIIJJJJJJJJJJJJJJJJJJJJJJJJJJJ&TIIIIIIIIIIIIJJJJJJJJJJJJJJJJJJJJJJJJJJJ&TIIIIIIIIIIIIJJJJJJJJJJJJJJJJJJJJJJJJJJJ&TIIIIIIIIIIIIJJJJJJJJJJJJJJJJJJJJJJJJJJJ&T IIIIIIIIIIIIJJJJJJJJJJJJJJJJJJJJJJJJJJJ&T IIIIIIIIIIIIJJJJJJJJJJJJJJJJJJJJJJJJJJJ&T IIIIIIIIIIIIJJJJJJJJJJJJJJJJJJJJJJJJJJJ&T IIIIIIIIIIIIJJJJJJJJJJJJJJJJJJJJJJJJJJJ&T 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(nn ((  ( ( H@A@?"6@ NNN?N !]`@ <Abstract Recursive Constraint Demotion (RCD) is introduced, and with it, the basic concepts of OT ranking theory. RCD is an efficient method for obtaining a ranking from data, with the added bonus that it identifies those cases where no successful ranking exists (Tesar & Smolensky 1993, Tesar 1995). RCD is treated here as a method of organizing a tableau, which makes its workings transparent and easy to follow. We start from an unordered display of ranking information and step-by-step rearrange it into an ordered, stratified ranking. The exposition can be accessed at various levels, ranging from the basic visual grok to detailed consideration of the underlying principles. An effort has been made to define terms and to comment on points of interest. Some discussion of the foundational work by Tesar, Smolensky, and Prince is provided, with links to the the original sources. No knowledge of OT is presupposed, but a certain amount of prior exposure will probably smooth the way. The file is an Excel workbook, which comes in three versions: two formatted for PC: Excel 2003, Excel 2007; and one formatted for Mac Excel 2004. There are no macros in these files. < Fm ( NA@?"6@ NNN?N ]`  (<Copyright Notice RCD-The Movie may be freely used for classroom and other academic and research purposes. No fees of any kind may be charged for its use. It may not be sold or rented, nor may any part of it be sold or rented, nor may it or any part of it be included in any document or object intended to be sold or rented. 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F7!dAtE"`&e.0 =88|7wI& ʫT˯0vJ}2*n(<5$JNb{Mlk ]uJW\V!+/RzFI|W@bk R] zHfNc5׹YؖbI]X RU)Td6^p]q_ElSԟ%We*K M/do^nGv?Fg/lOqvpr}(7jrKrCfll9> symbols on the sheets to move backward and forward. - Keyboarders: to go forward one sheet: PC: ctrl+PgDown Mac: Option+right-arrow to go back one sheet: PC: ctrl+PgUp Mac: option+left-arrow 2. Zoom - To re-set the size of the stuff in the Excel window - First select all sheets Right-click the Name Tab at the bottom of the page (Mac Mouse: ctrl+click Name Tab) [Left-]click "Select All Sheets" - Then set the zoom from the main menu bar up top: View>Zoom Make your choice; nb. "Custom" at bottom - Unselect all, by moving to another sheet. 3. Hide the Ribbon (2007 only) - ctrl + F1 toggles the Ribbon. 4. Row/Column Headers - If you wish to see the row & column numbers (row letters), 2003: Tools>Options>Row and Column Headers 2007: View>Show/Hide>Headings or Page Layout>Sheet Options>Headings>View Mac: ???? - if you wish to un-see them, ditto. 5. Adjust a text box - Click on the border or in the box - To resize: hover your pointer on the border over one of the little o's or boxes so that a doubleheaded arrow [Mac: box with 2 arrowheads] appears. Drag. - To move: hover your pointer on the border so that a cruciform symbol appears, composed of two doubleheaded arrows [Mac: hand]. 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RCD is an astonishingly efficient method of extracting a ranking from violation data  a set of rankings, in the typical case. It may be that there is no successful ranking, no ranking that dismisses all forms desired to be suboptimal, because of contradictory requirements in the data. RCD will let you know, and in addition will provide some insight into the specific region of the data where inconsistency exists. RCD does not squeeze every last piece of ranking information out of the data: that is the secret of its efficiency. RCD identifies a set of rankings that will work, if any exist, but it is not guaranteed to find every such ranking. It gives a set of sufficient ranking conditions, but does not expel from that set those conditions that are not necessary. For that, you must turn to the rather more labor-intensive Fusional Reduction (FRed), which is computed by the RUBOT functions of OTWorkplace. <P D# $ & @     A ff  c $M A@]`MH <Basic references: Tesar & Smolensky 1993. ROA-2. 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<lllp(  l  s *+ A@ !]`+P+ X<Q RCD  The Idea Tesar's key insight was this: avoid necessity -- determining where constraints must be ranked; instead ask this much simpler question: which constraints can be ranked at the top? Given the answer, dismiss the data filtered out by having those constraints in top position. Group these top-rankable constraints into a stratum, the first. Now ask the question again, if data remains. This time we are asking it of a reduced set of data, from which the first-round losers have been removed. This is a hopeful sign: we have reduced the size of the problem, but it is still the same kind of problem, and therefore one that we know how to approach. Once again when we receive an answer, we can dismiss all the data ruled out by these now-rankable constraints. We form these into a second stratum, placed right below the first. These are constraints which could not sit in undominated position on the first round. But once the top-stratum constraints have filtered out their losers, these second stratum constraints are free to declare their preferences. If data remains, we yet again face the very same kind of situation we started out with, except with (yet again) fewer candidates to consider. We respond accordingly: exactly as before. 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(llt(  f  c $" A@ $]`"! <Representing RCD Here we take a concrete approach, which makes it easy to grasp visually what's going on. We view RCD as a way of sorting a table of data, rearranging it so that the order of constraints and the order of data reflects the emerging ranking structure and its action on the candidates. To do this, we will work entirely within the Comparative Tableau (CT) format, a simple transformation of the Violation Tableau (VT) in which the basic violation data is recorded. The CT explicitly displays the performance of each candidates we want to be optimal each 'desired optimum' or 'des.opt' as measured against that of its competitors, taken one by one. OT is all about such comparisons: the desired optimum must do better than each of its competitors if it is to attain optimality. Ranking is the one tool we have to make sure that this happens. Explicitly marking the results of each optimum-suboptimum contest gives us the representation we need to figure out the rankings. <    N'A@?"6@ NNN?N ]`'P' # <$RCD in Outline RCD has two basic steps: [1] Identify and consolidate the constraints that can be ranked in the highest possible position [2] Identify and consolidate the data handled by those constraints We then repeat this process on the remaining unresolved data, until we're done.< < #>@@)))7V))yK <<'RCD- preview 2'!A1V))yK >>'RCD- preview 4'!A1 Sheet3 I  cH  dMbP?_*+%"D??U } D}  D} UD} D} *D}  D} D} UD} D}  D} Dc11j1jo1`k1jM111@k1Pk1 1WH 1 1 1k 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(ll(    N^A@?"6@ NNN?N ]`^P\ <7RCD  The Calculation Our strategy is to work through RCD, step by step, on an interesting example. The goal is to develop an understanding of how it works, and why. If you want to perform RCD on your data, you could apply these techniques by hand, cutting and pasting repeatedly to make the required re-arrangements. Or you could abandon the risk of carpal tunnel syndrome, use OTWorkplace and simply push a button. You will be guaranteed an instantaneous and correct result, in the same format that is used throughout this discussion.<    H`A@?"6@ NNN?N ]``^ <In the next few sheets, we will look at the key background notions that underlie the representation of RCD and the analysis of ranking in general.< >@@)))7 yK  OTWorkplaceyK lhttp://equinox.rutgers.edu/people/faculty/prince.htmlV))yK <<'RCD- preview 3'!A1V))yK >>'RCD- preview 5'!A1 Sheet61 I  c,</0  dMbP?_*+%"D??U } D}  D} *D} D} D} D} D} D}  D} Dc11j@1j1`k@1jhF1k511k1kB1Pk$ 1pC 1k 1@k 1l 1PkB1hkH1@1k1k(1kD1k1@11k11d10n@/1 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(ll(  f  c $i A@ ,+]`ih <The Elementary Ranking Condition (ERC) When a constraint examines a comparison between two candidates, one desired to be better than the other, it sees only one of three things, regardless of how many violations are involved. W - the constraint favors the desired Winner. It has fewer violations. L - the constraint favors the desired Loser. It has fewer violations. e - nothing, neither way: the constraint is neutral: both have the same number of violations. Notation: we will write e as a blank cell in what follows. A constraint distinguishes two competitors if it assigns a polar value W,L of their competition. A list of these values, organized by constraint, gives rise to the Elementary Ranking Condition: ERC: Some constraint assessing W must dominate all constraints assessing L. Recall the definition of better-than-on-a-constraint-hierarchy, which lies behind the notion of optimality. One candidate is better than another on a given hierarchy iff the highest-ranked constraint that distinguishes them finds in its favor. (That is: awards W to the competition, where the favored candidate is the one designated as the desired optimum.) To be optimal is to win every such competition: to beat all comers. To be the best. The ERC, as stated above, directly reflects the fact that the highest-ranked constraint distinguishing two competitors is the constraint that decides between them. That decisive constraint had better favor the desired winner (W) or it's just not going to win. An L in that deciding position would present a fatal obstacle to the success of the hierarchy. As far as a given competition goes, any of the W's can perform the duty of defending the desired optimum from the constraints that favor its competitor (L's).< ( g i     0 s= ! \X E} d   )9 q   ZlA@jJ?"6@`NNN?N- /]`lpj "`<The Comparative Tableau (CT) shows the ERCs  ranking information hidden in the violation data.< `>@@0007 yK  Prince 2002byK Phttp://roa.rutgers.edu/view.php3?id=742 yK  Prince 2002ayK Phttp://roa.rutgers.edu/view.php3?id=627 yK  Prince 2006yK Rhttp://roa.rutgers.edu/view.php3?id=1247B00yK TOC TOC!R1C1V))yK <<'RCD- preview 4'!A1V))yK >>'RCD- preview 6'!A1yK Optimality Defined'Optimality Defined'!R1C1 Sheet4 I  d@S'euw  dMbP?_*+%"d??U } U} *} *}  } } * }  }  }  } } } } } U }  } *} }  } U} U} } d22j2j2`k2j 22@2@k@ 2Pk@ 2  2TWH 2h! 2x@ 2k 2k222222@2@!2 2T2|Pl 2l@H2l20l2222pl2jILLIIIIIIIIIIIIQQQQQJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJ1 IINIQQQQQJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJ1 IINIQQQQQJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJ1 IINIQQQQQJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJ1 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Here's an example from the analysis we will be analyzing (see Prince & Smolensky 1993/2004, ch. 7). <(     T@(kA@?"6@ NNN?N ]`@~ 0<From observation & analysis, we know that the winning output from /kaN/ is .kaN.KA., as in candidate # 7.1. " What ranking(s) will deliver this desired optimum as truly optimal? We assume no prior ranking: this is just a list of constraints, and we are trying to figure out what rankings are needed. " A first step in the right direction: let's ensure that the desired optimum beats its one listed competitor (since it must beat them all, listed or not). So: compare the two  <0 tE6 F <N6 O   H@A@?"6@ NNN?N&]`@ +<WWe have annotated the desired loser's row, indicating the outcome of its local contests with the des.opt. " The des.opt Wins locally on the constraint Algn, because the desired loser does Worse on violations. " But the desired Loser does better locally on DepC and on NoCoda (Less violations). The ERC tells us, and we believe it: if the des.opt is to win globally, Align (W) must dominate both NoCoda (L) and DepC (L). % Take down the scaffolding and we are left with a CT, which tells us what we need to know and nothing more. < O B o  W F " #  0 !   +  T(kA@jJ?#" @`]` r <sThe order of columns here does not reflect ranking order. This is just an arbitrary listing of the constraints.< i*r  B`A@?"@")/]``  < This CT spells out the entire ranking content of the specific piece of data at hand. Given that kAN.KA is the desired output from /kaN/, and that ka.NA. is desired to lose to it: Algn (W) must dominate both DepC (L) and NoCoda (L).<^ Fj bu A{  = W =  49*C9 9  >@@)))7yK  P&S 93/04yK Phttp://roa.rutgers.edu/view.php3?id=845LyK WTF? 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 8 8 T(kA@?"6@ NNN?N s]`p (< Same story, different pictures The standard CT gives its ranking information in the W's and L's and blanks, not in the order of columns. <( !  8 T(kA@jJ?#" @` ]` 0<The order of columns here does not reflect ranking order. These are just arbitrary listings of the constraints. Look to the W's, L's, and blanks for the content.<0$ si*  8 BA@?"@! &]` )x<* Any of these CTs spells out the entire ranking content of the specific piece of data at hand. In every ranking in which the desired winner beats the desired loser: Algn (W) dominates both DepC (L) and NoCoda (L).<xBa  u 49n 0=9 &9 '( ) 8 HA@?"6@ NNN?N S ]` 6 <7These CTs are equivalent. Same constraints, same data:< a K 6 8 H A@?"6@ NNN?N ]`  e<fAnd so on, through the entire 362, 880 rearrangements of the 9 constraint columns. 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In Lardil, other ranking requirements happen to winnow this grand disjunction. In the whole-language context, only f:DepV does the job. But here we see only a single unit of ranking data, highly disjunctive in its content. Other data de-confounds it.<   o  T஛pGA@?"6@ NNN?N]`஛ p<qRCD will have no trouble whatever dealing with such complexities, and will not even try to parse them logically.< p  T#pGA@?"6@`NNN?N]` " <It is a mere accident of fate (or of cut-and-paste) that f:DepV stands helpfully as the first polar value (W,L) in the left-right sweep. There is absolutely no presupposition here that an order of constraints has been established. < (    T`pGA@jJ?"6@`NNN?N ]`` i0<jThe standard CT gives its ranking information in the W's and L's and blanks, not in the order of columns.<09 <A D i  TPpGA@?"6@ NNN?N!#]`P C<DColumn order simply cannot give general ranking information anyway!< C>@@'''7L""yK WTF? 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(ll*(    Z›pGA@?"6@ NNN?N ]`› \<ERC Talk Just as tres personae sunt una tantum res, so too (what we have called) the ERC falls into three parts: " The Data: winner & loser: the WL-pair. " The List of values: W,L, e. " The Interpretation. SomeW dominates every L. We want the (desired) winner to be better than the (desired) loser. The interpretation tells us exactly under what conditions this will happen. Exploiting the notorious flexibility of the human cognitive system, we have referred to both the List and its Interpretation under the same name: ERC. They stand in 1:1 correspondence, so the possibility of harm is limited. More precision can be obtained by referring to the list as the ERC vector, saving ERC for the interpretation itself, which is after all the ranking condition we seek. But we shall not usually be so careful, leaving context and the onlooker's good sense to disambiguate. A valuable feature of ERC vectors is that they can be manipulated directly, without reference to their interpretation, to give us everything we want to know about ranking. We will see something of this in RCD. < O A 3 ~D  .  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& >X < HכA@?"6@ NNN?N % ]`כ t<uWe could, for example, arrange the columns into one of orders that the ERCs are happy with. < t < H0ڛA@?"6@ NNN?N ]`0ڛ R<SThe heavy vertical lines indicate that we are thinking of this tableau as ordered.< R < NݛA@?"6@ NNN?N]`ݛp @<AHere's a simple one-ERC CT, so ordered: < ~@  < H`ߛA@?"6@ NNN?N ] ``ߛ (<Each such row must lead with W. The first constraint that cares must favor the des.opt over its competitor. If not, we will have imposed an order contrary to the demands of the very ERC we are looking at.<(^    < HA@?"6@ NNN?N ] `P  <This particular order is not viable for Lardil as a whole. It is merely one of the 120,960 that satisfy the demands of this data pair.<  %   < HA@?"6@ NNN?N" (] ` $H<RCD enables us to convert any standard CT into rankings that work, whenever they exist. 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It might look something like this:< r 0^xx  60 A@ ]`0@ "-P<.The first column gives a numbering system that relates the ERCs to the the candidate numbers. The Input column cites the input form. The Winner column gives the desired optimum. The Loser column gives the competitor. Each row contains, under the relevant constraint, its assessment of the competition.<PLb g.ib'#-  B0pG A@jJ"` +]`0 vX<Question of the hour " Does a successful ranking exist  one that dismisses all suboptimal competitors? Not obvious to the naked eye. There are L's and W's scattered across the tableau. 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To make the problem more apparent, let's color the W's and L's.< ] L $ ZA@jJ?"6@`NNN?N ]`0 $_<Can we impose a ranking on this mess  an order on the columns so that every row leads with W?< _>@@)))7V))yK <<'RCD- preview 9'!A1X))yK >>'RCD- preview 11'!A1 Sheet6 I  by"u;K'\o]  dMbP?_*+%"D??U } D}  D} UD} D}  D} U D} D} D} U D} D} *D} D} D} D} D}  D} D} D Column 14b11j1j1`k1j111@k 1Pk1 1WH 1 1 1k 1k1"11111 1"111Pl 1lH1l10l 1!111pl1hILLIIIIIIIIIIIJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJI0hIIIIIIIIIIIIIIJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJI0hIIIIIIIIIIIIIIJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJI0hIIIIIIIIIIIIIIJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJI0hIIIIIIIIIIIIIIJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJI0hIIIIIIIIIIIIIIJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJI0 y Gz G{ | 2` -_ 0 -^ -]  @  0  -\  [L IJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJ/ cr L? 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Nothing has been done to the data beyond re-arrangement: some rows and columns have been shifted around. Everything works out as desired:< R   Bp pG A 8c"`$ &]`p   R<SRCD provides the key tool for doing this kind of re-arrangement algorithmically. <$ Rrr  0@ pG A@ ]`@   v <If we now take the order of constraint columns to be a ranking order, we have a workable hierarchy. Various other orders will also succeed  imagine shuffling the first three constraints, or even dropping the first down to the very bottom of the order. Without looking into them, we know for sure that a successful ranking exists: we've got one right in front of us.< %" _ v  HA@?"6@ NNN?N "]`  w <xThe desired winner wins in each row on the first constraint (left to right) that says something about the competition.< . 2w>@@)))7lyK  For example'Other Rankings'!R1C1X))yK <<'RCD- preview 10'!A1F))yK >> 'Lardil'!A1 Sheet7 I  c]hk]  dMbP?_*+%"D??U } D} *D} D} D} UD} UD} UD} D} D} D} U D} D} D}  D} D}  D} D} D} *D} Dc1+j+j +`k+j+k++k+kB+ +WH +k +@k +l +PkB+hkH+@+k+k+k+k+@++k+++0n@+ lO+Pl8+@l+L+'O+`l@H dZCSSJJJJJJJJJJJJJJJJJJJJJJJJJJ* c b a ` _  ^ ]   \    [   Z@nSSJJJJJJJJJJJJJJJJJJJJJJJJJJ* Y X e ~  @D nSSJJJJJJJJJJJJJJJJJJJJJJJJJJ* W~ ?~ ? ~  ?F nSSJJJJJJJJJJJJJJJJJJJJJJJJJJ*\nSSJJJJJJJJJJJJJJJJJJJJJJJJJJ* V U e~ ?  ?? 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All the constraints they discuss are included here, except ONS. A couple of the listed candidates are harmonically bounded, notated as HB in the Remarks column. Aficionados might want to keep track of their fate.<! F (  `pGA3 8c?"6@`NNN?N F]` z<{Tesar first presented RCD at the Rutgers Optimality Workshop in October 1993, using these very data, to general amazement.< Rz (  `PpGA3@jJ?"6@`NNN?N]`P   <APC Users: If you have OTWorkplace installed, you can press the RCD button right now, to get an RCD report based on this page. RCD The Button will insert a couple of sheets after this one. By resuming (at the sheet "Lardil-nmd") you will see how the RCD report was obtained, step-by-step.< x w  >@@,,,7N))yK >>'Lardil-nmd'!A1X))yK <<'RCD- preview 11'!A1B!!yK HB ?HB!R1C1B,,yK TOC TOC!R1C1 Sheet8 I  c;  dMbP?_*+%"D??U } UD}  D} *D} D} D}       !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~UD} UD} UD} D} D} D} U D} *D} D}  D} D}  D} D} UD}  Dc11j1j 1`k1j11pM1@k1Pk1 1WH 1 1 1k 1k1111111111Pl1lH1l10l1@111pl1@ dfLLvvvvvvvvvvlv0  c b a ` _  ^ ]    \    [  mZJ0  Y X e ~  @N v0  W~ ?~  ? ~  ?P v0hv0  V U e~  ?  ?? N v0  T?? ~  ?N v0  S~ ?~  ?   ~  ?N v0hv0  R Q e ~ ? ~ ?P v0   P ~ ?  ~ ?N v0h v0  O N e ~ ?  ~ ? ~ ?N v0   M ~ ?  ~ ?N v0  L~ ?~  ?T v0hv0  K J e ~ ?~  ?   ~  ?N v0  I ~ ? ~  ?  ?? N v0hv0  H G e ~ ?~  ?T v0  F ~ ?~ ?  ~  ? ~  ?N v0hv0  E D e ~ ?  ~  ? ~  ?N v0  C~ ?~  ?T v0hv0  B A e~  ?  ?? N v0  @~ ?V v0hv0  ? > e~  ?  @? N v0  =?? ~  ?  ?? N v0  <~ ?~  ?  ?? 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(nnP0(  0 0 <P0pG A@"` k]`P0.  <This is the CT we get straight from the violation data. For convenience of the ensuing discussion, the constraints are presented in a slightly different order than in the first rendition of the CT. The initial order of constraints and data makes no difference whatever to the operation of RCD, but it can influence how easy it is to see what's going on. There are 1020 arrangements of this tableau. Any one of them is fine as a starting point. 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The fusion is in the detached new row at the bottom. Fusion is a logical operation that combines information from a set of ERCs. It produces an ERC which records key properties of the set. Unlike conjunction, it does not retain every property of its source: it cannot, since the full set describes a many-layered hierarchy and the ERC can only distinguish between possible dominators (W) and the necessarily dominated (L), along with the irrelevant (e or blank). Nevertheless, fusion gives us exactly the information we need to analyze the logic of ERC sets  the logic of ranking, which RCD plays on. Fusion combines values pairwise in the following way. We write X for any value W,L, or e. fus(X, L) = L any fusion with L yields L fus(X, e) = X e is the identity: fusion of self with e yields self. fus(W,W) = W W fused with W is W. (Indeed fus(X,X)=X.) In addition, we want fus(X,Y) = fus(Y,X), Y any value, so that order of fusion (as with other familiar logical operations) doesn't matter. Fusion detects whether a constraint column contains L. No L-containing constraint can stand at the front of a hierarchy. But any other kind of constraint, fusing to W or to e (if all e's), can. That's exactly what RCD needs to know: who can be first. <XQ 8 .= n <u y7  - . "  z0 1 E_ tp q ~   " /  > " ^   > ? 0G I N Q ~ B ) 4 HCA@?"6@ NNN?N"]`CA E<FPrince 2002a. ROA-500. Prince 2002b. ROA-562. Prince 2006. 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