Opacity and rule ordering

Can we always recover a rule’s conditioning environment from the surface form? A phonological process is transparent when we can. It is opaque when an ordered derivation hides that environment (Kiparsky 1973).

Four interactions

Suppose that two rules $R_1$ and $R_2$ apply to the same representation.

In a feeding order, $R_1$ creates an environment in which $R_2$ applies.
In a bleeding order, $R_1$ removes an environment in which $R_2$ would otherwise apply.
In a counterfeeding order, $R_2$ applies before the rule that would have created its environment. The surface contains an apparent environment in which $R_2$ failed to apply.
In a counterbleeding order, $R_2$ applies before the rule that would have removed its environment. The surface preserves the result of $R_2$ even though its environment is no longer visible.

Feeding and bleeding interactions are transparent. Counterfeeding and counterbleeding interactions are opaque because the surface form alone does not show why the earlier rule did or did not apply.

Canadian raising and flapping

Canadian raising provides a standard case of counterbleeding. In a simplified statement, /aɪ/ raises before a voiceless consonant, while intervocalic /t/ later becomes [ɾ].

\[ \begin{aligned} R_{\text{raise}} &: aɪ \rightarrow ʌɪ\;/\;\_[-\text{voice}] \\ R_{\text{flap}} &: t \rightarrow ɾ\;/\;V\_V \end{aligned} \]

Applied to /raɪtər/, raising produces [rʌɪtər] and flapping then produces [rʌɪɾər]. The surface [ɾ] is not voiceless, so the environment that conditioned raising is absent from the output. Flapping would have bled raising if it had applied first; the reverse order is thus counterbleeding.

Composition preserves the derivation

Let $T_{\text{raise}}$ and $T_{\text{flap}}$ be transducers compiled from the two rules. Their ordered composition is

\[T_{\text{flap}}\circ T_{\text{raise}}.\]

The composed FST relates /raɪtər/ directly to [rʌɪɾər], but its construction factors through the intermediate [rʌɪtər]. In this sense, the intermediate representation is removed from the external interface without being removed from the analysis.

Question

Would $T_{\text{raise}}\circ T_{\text{flap}}$ describe the same relation?

Answer

No. That composition applies flapping first. Once /t/ has become [ɾ], the simplified raising rule no longer sees a voiceless consonant and does not apply.

The fact that the opaque mapping can be represented by an FST is not exceptional. A finite sequence of rational rules remains rational under composition, whether the ordering is transparent or opaque (Kaplan and Kay 1994). Opacity may make the linguistic analysis harder to infer from surface forms, but it does not by itself require a more powerful transducer.

The practical cost is machine size. Composition may create many product states, and a grammar with several ordered rules may need determinization, minimization, or lazy composition to remain usable (Mohri 1997; Mohri et al. 2002). This is a computational issue about representing the relation, distinct from the formal question of whether the relation is rational.

References

Kaplan, Ronald M., and Martin Kay. 1994. “Regular Models of Phonological Rule Systems.” Computational Linguistics 20 (3): 331–78. https://aclanthology.org/J94-3001/.

Kiparsky, Paul. 1973. “Abstractness, Opacity, and Global Rules.” In Three Dimensions of Linguistic Theory, edited by Osamu Fujimura. TEC.

Mohri, Mehryar. 1997. “Finite-State Transducers in Language and Speech Processing.” Computational Linguistics 23 (2): 269–311. https://aclanthology.org/J97-2003/.

Mohri, Mehryar, Fernando Pereira, and Michael Riley. 2002. “Weighted Finite-State Transducers in Speech Recognition.” Computer Speech & Language 16 (1): 69–88. https://doi.org/10.1006/csla.2001.0184.

--- title: Opacity and rule ordering bibliography: ../../references.bib --- Can we always recover a rule's conditioning environment from the surface form? A phonological process is **transparent** when we can. It is **opaque** when an ordered derivation hides that environment [@kiparsky1973abstractness]. ## Four interactions Suppose that two rules $R_1$ and $R_2$ apply to the same representation. - In a **feeding** order, $R_1$ creates an environment in which $R_2$ applies. - In a **bleeding** order, $R_1$ removes an environment in which $R_2$ would otherwise apply. - In a **counterfeeding** order, $R_2$ applies before the rule that would have created its environment. The surface contains an apparent environment in which $R_2$ failed to apply. - In a **counterbleeding** order, $R_2$ applies before the rule that would have removed its environment. The surface preserves the result of $R_2$ even though its environment is no longer visible. Feeding and bleeding interactions are transparent. Counterfeeding and counterbleeding interactions are opaque because the surface form alone does not show why the earlier rule did or did not apply. ## Canadian raising and flapping Canadian raising provides a standard case of counterbleeding. In a simplified statement, /aɪ/ raises before a voiceless consonant, while intervocalic /t/ later becomes [ɾ]. $$ \begin{aligned} R_{\text{raise}} &: aɪ \rightarrow ʌɪ\;/\;\_[-\text{voice}] \\ R_{\text{flap}} &: t \rightarrow ɾ\;/\;V\_V \end{aligned} $$ Applied to /raɪtər/, raising produces [rʌɪtər] and flapping then produces [rʌɪɾər]. The surface [ɾ] is not voiceless, so the environment that conditioned raising is absent from the output. Flapping would have bled raising if it had applied first; the reverse order is thus counterbleeding. ## Composition preserves the derivation Let $T_{\text{raise}}$ and $T_{\text{flap}}$ be transducers compiled from the two rules. Their ordered composition is $$T_{\text{flap}}\circ T_{\text{raise}}.$$ The composed FST relates /raɪtər/ directly to [rʌɪɾər], but its construction factors through the intermediate [rʌɪtər]. In this sense, the intermediate representation is removed from the external interface without being removed from the analysis. ::: {.callout-caution} ## Question Would $T_{\text{raise}}\circ T_{\text{flap}}$ describe the same relation? ::: ::: {.callout-tip collapse=true} ## Answer No. That composition applies flapping first. Once /t/ has become [ɾ], the simplified raising rule no longer sees a voiceless consonant and does not apply. ::: The fact that the opaque mapping can be represented by an FST is not exceptional. A finite sequence of rational rules remains rational under composition, whether the ordering is transparent or opaque [@kaplan1994regular]. Opacity may make the linguistic analysis harder to infer from surface forms, but it does not by itself require a more powerful transducer. The practical cost is machine size. Composition may create many product states, and a grammar with several ordered rules may need determinization, minimization, or lazy composition to remain usable [@mohri_fst_1997; @mohri_weighted_2002]. This is a computational issue about representing the relation, distinct from the formal question of whether the relation is rational.