a little pre-talk review a little pre-talk review Regular Relation (of strings) Regular Relation (of strings) � Relation: like a function, but multiple outputs ok � Can weight the arcs: → vs. → � Regular: finite-state � a → {} b → {b} a: ε ε ε ε a: ε ε ε ε � Transducer: automaton w/ outputs � aaaaa → {ac, aca, acab, acabc} b:b b:b � b → ? a → ? a:a a:a a:c a:c � aaaaa → ? ?:c ?:c � How to find best outputs? b:b b:b ?:b � For aaaaa? ?:b � Invertible? � For all inputs at once? � Closed under composition? b: ε ε ε ε ?:a b: ε ε ε ε ?:a Directional Constraint Synopsis: Fixing OT’s Pow er Evaluation in OT � Consensus: Phonology = regular relation E.g., composition of little local adjustments (= FSTs) Jason Eisner U. of Rochester � Problem: Even finite-state OT is worse than that Global “counting” (Frank & Satta 1998) � Problem: Phonologists want to add even more Try to capture iterativity by Gen. Alignment constraints � Solution: In OT, replace counting by iterativity August 3, 2000 – COLING - Saarbrücken Each constraint does an iterative optimization What Is Optimality Theory? Outline � Review of Optimality Theory � Prince & Smolensky (1993) � The new “directional constraints” idea � Alternative to stepwise derivation � Linguistically: Fits the facts better � Stepwise winnowing of candidate set � Computationally: Removes excess power such that different constraint Gen . . . orders yield different languages Constraint 1 � Formal stuff Constraint 2 3 � The proposal Constraint input � Compilation into finite-state transducers output � Expressive power of directional constraints 1
Filtering, OT-style A Troublesome Example �� = candidate violates constraint twice Input: bantodibo Constraint 1 Constraint 2 Constraint 3 Constraint 4 Harmony Faithfulness Candidate A � � ��� Candidate B ban.to.di.bo �� � � � Candidate C � � Candidate D ben.ti.do.bu ��� � ���� Candidate E �� � � ban.ta.da.ba Candidate F ��� �� ��� � bon.to.do.bo �� � constraint would prefer A, but only “Majority assimilation” – impossible with FST - allowed to break tie among B,D,E - and doesn’t happen in practice! Outline An Artificial Example � Review of Optimality Theory Candidates have 1, 2, 3, 4 violations of NoCoda � The new “directional constraints” idea � Linguistically: Fits the facts better � Computationally: Removes excess power NoCoda � Formal stuff ban.to.di.bo � � � The proposal ban.ton.di.bo �� � Compilation into finite-state transducers ban.to.dim.bon � Expressive power of directional constraints ��� ban.ton.dim.bon ���� An Artificial Example An Artificial Example Add a higher-ranked constraint Imagine splitting NoCoda into 4 syllable-specific constraints This forces a tradeoff: ton vs. dim.bon NoCoda σ 1 σ 2 σ 3 σ 4 C NoCoda C ban.to.di.bo ban.to.di.bo � � � � ban.ton.di.bo ban.ton.di.bo �� �� � � ban.to.dim.bon ban.to.dim.bon ��� ��� ban.ton.dim.bon ban.ton.dim.bon ���� ���� 2
An Artificial Example An Artificial Example Imagine splitting NoCoda into 4 syllable-specific constraints For “right-to-left” evaluation, reverse order ( σ 4 first) Now ban.to.dim.bon wins - more violations but they’re later NoCoda NoCoda σ 1 σ 2 σ 3 σ 4 σ 4 σ 3 σ 2 σ 1 C C ban.to.di.bo ban.to.di.bo � � � � ban.ton.di.bo ban.ton.di.bo � � � � � ban.to.dim.bon ban.to.dim.bon � � � � � � � ban.ton.dim.bon ban.ton.dim.bon � � � � � � � � Outline When is Directional Different? � Review of Optimality Theory � The crucial configuration: � The new “directional constraints” idea σ 1 σ 2 σ 3 σ 4 � Linguistically: Fits the facts better ban.to.di.bo � � Computationally: Removes excess power ban.ton.di.bo � � ban.to.dim.bon � Formal stuff � � � � � The proposal � Forced location tradeoff � Compilation into finite-state transducers solve location conflict � Can choose where to violate, by ranking locations but must violate somewhere � Expressive power of directional constraints (sound familiar?) � Locations aren’t “orthogonal” When is Directional Different? When is Directional Different? � The crucial configuration: � But usually locations are orthogonal: σ 1 σ 2 σ 3 σ 4 σ 1 σ 2 σ 3 σ 4 ban.to.di.bo ban.to.di.bo � � � ban.ton.di.bo ban.ton.di.bo � � � � ban.to.dim.bon ban.to.dim.bon � � � � � � � � But if candidate 1 were available … � Usually, if you can satisfy σ 2 and σ 3 separately, you can satisfy them together � Same winner under either counting or directional eval. (satisfies everywhere possible) 3
Linguistic Hypothesis Test Cases for Directionality � Q: When is directional evaluation different? � Prosodic groupings � A: When something forces a location tradeoff. � Syllabification In a CV(C) language, /CVCCCV/ needs epenthesis � Hypothesis: Languages always resolve these cases directionally. V V [CV.CVC.CV] [CVC.CV.CV] Iraqi Arabic Cairene Arabic R-to-L syllabification L-to-R syllabification Analysis: NoI nsert is Analysis: NoI nsert is evaluated L-to-R evaluated R-to-L Test Cases for Directionality Test Cases for Directionality � Prosodic groupings � Prosodic groupings � Syllabification � Syllabification [CV.CVC.CV] vs. [CVC.CV.CV] [CV.CVC.CV] vs. [CVC.CV.CV] � Footing � Footing σ(σσ)(σσ) vs. (σσ)(σσ)σ With binary footing, σσσσσ must have lapse � Floating material � Lexical: σ(σσ)(σσ) (σσ)(σσ)σ � Tone docking ban.tó.di.bo vs. ban.to.di.bó R-to-L Parse- σ σ σ σ L-to-R Parse- σ σ σ σ � Infixation grumadwet vs. gradwumet (σσ)σ(σσ) � Stress “end rule” (bán.to)(di.bo) vs. (ban.to)(dí.bo) unattested � OnlyStressFootHead, HaveStress » NoStress (L-R) � Harmony and OCP effects Generalized Alignment Outline � Review of Optimality Theory � Phonology has directional phenomena � [CV.CVC.CV] vs. [CVC.CV.CV] - both have 1 coda, 1 V � The new “directional constraints” idea � Directional constraints work fine � Linguistically: Fits the facts better � But isn’t Generalized Alignment fine too? � Computationally: Removes excess power � Ugly � Non-local; uses addition � Formal stuff � Not well formalized � The proposal � Measure “distance” to “the” target “edge” � Compilation into finite-state transducers � Way too powerful � Expressive power of directional constraints � Can center tone on word, which is not possible using any system of finite-state constraints (Eisner 1997) 4
Computational Motivation Why OTFS > FST? � Directionality not just a substitute for GA � It matters that OT can count � Also a substitute for counting � HeightHarmony » HeightFaithfulness � Input: to.tu.to.to.tu � Frank & Satta 1998: can both be implemented � Output: to.to.to.to.to by weighted FSAs OTFS > FST vs. tu.tu.tu.tu.tu (Finite-state OT is more powerful prefer candidate with than finite-state transduction) fewer faithfulness violations � Majority assimilation (Bakovi ć 1999, Lombardi 1999) � Beyond FST power - fortunately, unattested Making OT=FST: Proposals Why Is OT > FST a Problem? � Approximate by bounded constraints � Consensus: Phonology = regular relation � Frank & Satta 1998, Karttunen 1998 � OT supposed to offer elegance, not power � Allow only up to 10 violations of NoCoda � Yields huge FSTs - cost of missing the generalization � FSTs have many benefits! � Another approximation � Generation in linear time (with no grammar constant) � Gerdemann & van Noord 2000 � Comprehension likewise (cf. no known OTFS algorithm) � Exact if location tradeoffs are between close locations � Invert the FST � Allow directional and/or bounded constraints only � Apply in parallel to weighted speech lattice � Directional NoCoda correctly disprefers all codas � Intersect with lexicon � Handle location tradeoffs by ranking locations � Compute difference between 2 grammars � Treats counting as a bug, not a feature to approximate Outline Tuples � Violation levels aren’t integers like ��� � Review of Optimality Theory � They’re integer tuples , ordered lexicographically � The new “directional constraints” idea � Linguistically: Fits the facts better � Computationally: Removes excess power � Formal stuff NoCoda σ 1 σ 2 σ 3 σ 4 � The proposal ban.ton.di.bo 1 1 0 0 � � � Compilation into finite-state transducers ban.to.dim.bon 1 0 1 1 � � � � � Expressive power of directional constraints ban.ton.dim.bon 1 1 1 1 � � � � 5
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