2.1. Segments, features, and inventories

Figure 1 shows the use of the interactive IPA chart. Here, the user has clicked on four IPA symbols [θ s ʃ ʂ], which are now highlighted, and the program displays their differences in the matrix shown in the lavender pop-out window. In this case, the segments differ only in the features [anterior], [distributed], and [strident], so only these features are presented in the matrix. Below that, it lists all of the features these segments have in common. By default, the Phonomaton employs the feature set of Hayes Reference Hayes2009, although the user can modify the feature sets of segments as well as introduce new segments (see Section 2.1).

Figure 1.

Comparison of segments without an inventory.

The Phonomaton can also compare segments in conjunction with an inventory. An inventory is declared by listing a set of phones in the preamble (e.g. ‘Inventory: a i u p t k’) or simply by selecting segments in the IPA chart and clicking the ‘Make inventory’ button.

When an inventory has been declared, the segments it contains are highlighted in the chart with a blue outline, as seen in Figure 2. At this point, clicking any group of segments shows their distinctive features if they constitute a natural class within the inventory, in addition to the standard feature comparison. In Figure 2, the user has clicked three segments, [t θ s], and the lavender window shows that this is a natural class within the inventory with distinctive features [−voice, +anterior].

Figure 2.

Comparing segments within an inventory that form a natural class.

If no set of features uniquely distinguishes the group of selected segments, the software notes that fact and suggests the closest match containing all of the selected segments. For instance, given the same inventory as in Figure 2, the user has clicked [d n θ ð s] in Figure 3. The lavender box now informs us that this collection of segments does not form a natural class and suggests the closest match, which in this case involves adding [z] to the set.

Figure 3.

Comparing segments within an inventory that do not form a natural class.

In addition to selecting segments directly on the IPA chart, the user can select segments by their features using the interface shown in Figure 4. Upon selection of a set of feature specifications, the relevant segments are highlighted in the IPA chart.

Figure 4.

Selecting segments by feature.

2.1.1. Custom segments

The Phonomaton also allows the user to create new segments based on a custom set of feature specifications. This facility can also be used to create segments that are underspecified for particular features. To take a concrete example, obstruent voicing in Turkish suggests an analysis in which stops in the underlying stratum are underspecified for voice, whereas those in loan strata are specified. Inkelas (Reference Inkelas1995) posits the set of stops in 1, where the ‘archiphonemes’ in 1a lack a voice feature altogether, and the fully specified sets in 1b and 1c have a negative and positive value for the [voice] feature, respectively.

The user can define segments in the preamble of the rules using the syntax shown in 2.Footnote ¹⁰

The declaration in 2 creates a new segment /T/ that is identical to /t/ except that it lacks the [voice] feature. ‘Define’ can create completely novel segments with any feature set, and one can also redefine the feature sets of existing segments by using ‘Redefine’. For a language without contrastive voicing in stops, we could thus redefine stops such as /p t k/ as lacking a [voice] feature entirely.Footnote ¹¹ As discussed below in Section 2.4, ‘feature-filling’ rules can subsequently target segments with underspecified features to provide them with a value in a particular environment.Footnote ¹²

2.2. Phonological rules

Beyond functioning as a feature calculator, the strength of the Phonomaton lies in its ability to execute ordered rules and to derive surface forms from underlying representations. Phonological rules are written in the traditional SPE notation: target → change / environment (optionally preceded by a rule name), and the software executes them in the order they are written. The user can type rules from scratch in the phonological rules field or make use of the sample rules, shown in Figure 5, which are automatically pasted into a derivation when clicked and can be modified manually from there. The sample rule collection contains simple segmental rules as well as more complex ones for syllabification, mora assignment, footing, and stress, which can be experimented with freely.

Figure 5.

The library of sample rules.

A simple derivation with two of the sample rules applying to user-generated representations is shown in Figure 6.

Figure 6.

A sample derivation.

Figure 6 shows two ordered rules that operate on the right edge of the word (symbolized by ‘>’; see Section 2.6 for the full set of boundary symbols), with three underlying representations. The resulting derivation is shown at the top, where light blue cells show intermediate representations and blank cells represent the failure of a rule to apply.

2.3. The target and change

The user can specify the target, change, and environment of a rule directly in IPA symbols and the common abbreviations C (for [−syllabic] segments), V (for [+syllabic] segments), and X (for all segments, excluding boundary symbols), as in 3a. All parts of a rule can also refer to features, as in 3b.

Rules can make use of traditional feature notation, exemplified in 3b, but the Phonomaton also introduces several novel methods for relating to features that go beyond textbook treatments. The symbol ± for a feature in the change means ‘change only if necessary to match a phone in the inventory’ (or, if there is no specified inventory, in the entire IPA). This specification is useful in cases in which a single phonological process affecting one or more features implies an incidental change in a different feature only in a subset of the target segments. For instance, a process of intervocalic lenition may effect the following set of changes: p → ɸ, t → s, k → x. Such lenition in featural terms is a change to [+continuant, +delayed_release]. However, for the second segment, /t/, there is an additional change from [−strident] to [+strident], a feature that is incompatible with the bilabial and velar places of articulation. In this situation, we could write the rule as in 4a, which produces the derivation in 4b.

The change to [+strident] is necessary only in the case of /t/, because no segment differs from /t/ only in being [+continuant, +delayed_release] without also being [+strident]. (The other candidate for a continuant here is [θ], but this segment differs in being [+distributed].)

Although ± causes a change only when necessary, a feature change can also be defeasible, taking place only when possible, given a particular inventory. Defeasible changes are signaled by the symbols ⊕ and ⊖ preceding a feature, which mean ‘change to plus or minus, respectively, if the resulting segment is contained in the inventory’.Footnote ¹³ This restriction is the default behavior when a rule makes reference to only a single feature. For instance, the rule in 5a affects only the segments /s/ and /t/ in 5b because the other segments targeted have no counterpart differing only in being [−anterior]. We do not expect the change to create an impossible segment in these other cases; rather, we expect that the rule simply fails to apply.

However, asserting a defeasible change becomes useful when some partial set of specified changes still takes place even when the full set of changes invoked cannot apply to all of the relevant segments. For instance, a process of lenition turns voiceless stops into voiced fricatives, as shown in 6, but if the inventory language lacks a voiced uvular fricative [ʁ], lenition to a fricative takes place without voicing when applying to uvular stops.

Such a process can often be modeled by two rules, in this case, a spirantization rule and a subsequent voicing rule, the second of which would not apply to the uvular because of the gap in the inventory. But such a two-step approach can fail by incorrectly merging segments that should remain distinct. For instance, the language may have underlying /ɸ/ and /x/, which do not voice intervocalically under the same process, yet the underlying spirants would not be differentiated from derived ones after the spirantization rule. In this scenario, the rule in 7 obtains exactly the correct result for 6, effecting the change from stops to fricatives across the board but adding [+voice] only when possible, as signaled by [⊕voice] in the change.

Another approach to facts such as these makes use of feature deletion. Noncontrastive features can be removed from the calculation entirely using the ‘Delete’ command, as in 8. Here, surface segments may differ in [voice], but rules cannot refer to this feature, nor are differences in [voice] visible to the grammar.

Although there are both voiced and voiceless segments in the surface inventory in 8, noncontrastive voicing can now be treated as simply being concomitant with [+continuant] or [+delayed release]. Deleting the voice feature can therefore also derive the alternation in 6 above. This case exemplifies one way in which the Phonomaton facilitates comparison between competing solutions.

2.4. Underspecification and feature-filling rules

As discussed in Section 2.1 above, the user can declare underspecified segments as part of the preamble. Now we demonstrate how rules can target underspecified values so that they provide feature values only where none previously exist. First, we introduce an underspecified segment using Define, as shown in 9a, by copying the features of /t/ and deleting the voice specification. In 9b, we formulate a rule that is strictly ‘feature filling’, as signaled by ⊗ in the target.Footnote ¹⁴ The polarity ⊗ matches only an underspecified feature value (that is, neither + nor −).

Figure 7 shows a derivation based on 9. Here, intervocalic voicing applies only to /T/, which is underspecified for [voice], and not to its voiceless counterpart /t/. Similarly, final devoicing applies only to /T/, and not to its voiced counterpart /d/. Underspecification in conjunction with ⊗ provides us with an elegant way to model alternations of the type seen in Turkish voicing, discussed in Section 2.1, where loan strata contain the contrastively voiced stops /t/ and /d/, but where voicing is underspecified (and therefore predictable) in /T/.

Figure 7.

Derivation with a feature-filling rule.

2.5. Alpha notation and abbreviations

Alpha notation is available through the use of three Greek letters α, γ, and δ, which may also be specified negatively. (We omit β because it is identical to the IPA symbol for the bilabial fricative.) A simple example is given in 10, with a resulting derivation shown in Figure 8. The rule in 10 uses the variable α to assert that a consonant should match the voicing of an immediately following consonant.Footnote ¹⁵ The result is that the /ɡ/ in /aɡta/ assimilates in voice to the following /t/ to yield [akta] and the /p/ in /apdu/ assimilates in the same way to yield [abdu]. Alpha notation is thus able to express feature assimilation as a single process, whether it involves a change to a positive or negative specification of a feature.

Figure 8.

Voice assimilation.

We can also use variables to enforce agreement between the target and the environment. For instance, 11 asserts that a vowel should agree with the following vowel in the feature [round] when both vowels match for height features [high] and [low]. Here we explicitly state the labial feature as ± in the target, so the target’s [labial] value can change if necessary to satisfy rounding assimilation.

Figure 9 shows a derivation resulting from this rule. Because the derivation is decomposed into subrules containing all of the combinatorial possibilities, the exact condition under which the change takes place is displayed clearly. In the first subrule, all instances of α and γ (in the target, change, and environment) are replaced by +; in the second subrule, all instances of α are replaced by +, all instances of γ are replaced by −, and so forth.

Figure 9.

Variables in the target and environment.

The first two forms in Figure 9, /bui/ and /biu/, trigger regressive assimilation because the two adjacent vowels are of similar height. The last two forms, /bue/ and /beu/, however, do not undergo rounding harmony, because the adjacent vowels are of different heights.

2.6. Morphological boundaries

The Phonomaton recognizes several morphological boundaries in the underlying forms, employing the standard symbolic notation of the Leipzig Glossing Rules: - for prefix and suffix boundaries, ~ for reduplicant boundaries, ⟨ ⟩ for infix boundaries, = for clitic boundaries, and white space for word boundaries. Departing from tradition, we provide a boundary symbol ⇒ when an analysis needs to differentiate prefix or proclitic boundaries from suffix or enclitic boundaries. The symbol ≣ is also provided for indicating special morphosyntactic junctures.Footnote ¹⁶ Phonological rules can reference generic word boundaries by # and left and right word boundaries by < and >, respectively. Therefore, one can write a word-final devoicing rule as in 12.

A rule that raises mid vowels before enclitic boundaries and word-finally can be written as in 13. Disjunction is written with the elements separated by the vertical bar character and enclosed in parentheses (see Section 2.8 below).

2.7. Segment indexing

Certain processes, like metathesis, require indexing segments in a rule’s target and change. The Phonomaton employs superscripts (from ¹ to ⁴) to index segments. The rule in 14 matches a sequence of two consonants in which the first is coronal and the second is labial. The rule defines variables by labeling the relevant segments with a superscript; these indices appear again in the change but without any preceding features or segments. That is, instead of repeating redundant information in the change, as in the common notation C¹C² → C²C¹ for metathesis, the Phonomaton expects C¹C² → ²¹, with the redundancy removed (but see 18 below for an example where features modify an indexed segment).

The result of 14 is to swap a coronal consonant with a following labial consonant, as shown in Figure 10.

Figure 10.

A derivation showing metathesis of coronal and labial stops.

Indexing also assists in rules for total assimilation, reduplication, gemination, and degemination. The Phonomaton requires that indices be associated with a segment or class of segments before they are used as referents. For example, the rule of total assimilation in 15a associates the superscripted index 1 with a segment that matches [+consonantal] in the target and then refers back to that segment in the change with the plain superscript numeral. This would have the effect of changing /nk/ to [kk], /mb/ to [bb], and so forth. A formulation of the same rule as in 15b would fail because it invalidly employs an index in the change before the environment where it is defined.

Similarly, a rule of vowel copying, often written in a form like 16b, runs afoul of the requirement that the target define each index. Such rules can easily be reformulated as in 16a, where the entire environment is targeted and copied with a modification. Despite being common practice, it is redundant, and thus disallowed, to repeat the Cs and Vs in the change.

One can write complex rules of reduplication (a morphological operation) using indices in conjunction with simple regular expressions. The rule in 17 reduplicates a word-initial onset followed by the vowel /a/, an example of reduplication with ‘fixed segmentalism’. Figure 11 shows the resulting derivation for three underlying forms.Footnote ¹⁷

Figure 11.

A derivation showing reduplication with fixed segmentalism.

Indexed segments can also undergo featural modifications. Any features specified before an index in the change component of a rule apply to the indexed segment in the output. In 18, the features [−high, −low, −tense] apply to the first instance of a segment indexed by ², that is, the first vowel in the stem. Figure 12 shows sample derivations.

Figure 12.

Reduplication with partial neutralization.

2.8. Regular expression syntax

The Phonomaton allows regular expressions in the target and environment of a rule, the most important component of which are quantifiers. In 19 we present a comparison between the regular expressions for quantifiers used by the Phonomaton (based on Perl’s regex notation) and the familiar SPE notation (Chomsky & Halle Reference Chomsky and Halle1968).

Thus, a rule of vowel lowering that applies at the end of the word can be specified to occur regardless of the presence of a word-final coda, as in 20a, with either a simple word-final coda or no coda at all, as in 20b, with a simple or complex coda, as in 20c, or with a complex coda of two to four consonants, as in the unlikely case of 20d.

These quantifiers can combine with indices to handle complex cases of reduplication. In 21a, we see word-initial CV-reduplication. In 21b, we see reduplication of a word-initial unit containing two vowels and their preceding consonants (a disyllable, in the common case). The index in the target captures the entire expression within the parenthesized group, which is then duplicated in the change. The environment ensures that this process is anchored to the left edge of the word.Footnote ¹⁸

Regular expressions also work in conjunction with various feature specifications and underspecification to model locality and blocking effects. Consider the case of retroflex harmony, where /ʃ/ assimilates to the [−distributed] feature of a retroflex when the retroflex precedes it. We can model this process as in 22, which says that a [−anterior] consonant becomes [−distributed] (retroflex) when preceded anywhere in the word by a [−distributed, −anterior] segment. The symbol X stands for any segment and X* stands for zero or more segments, so the process operates long-distance.

We can also model blocking effects for rules such as these. If we modify the above environment as in 23, the rule will now apply only when the segments intervening between the target and the preceding trigger are [−coronal].

Figure 13 shows a derivation based on this rule; the dorsal consonant /k/ in the first form is transparent for retroflex harmony, but the intervening /s/ in the second form, being coronal, is opaque and thus blocks assimilation.

Figure 13.

Retroflex harmony blocking.

Another relevant feature of regular expressions is grouping and alternation (signaled by brace notation in SPE). If a rule must make reference to segments that cannot be generalized by features, it can express alternation by the vertical bar symbol. For instance, a change that takes place only before the phonemes /s d ɣ/ at the end of a word is expressed as in 24.

2.9. Types of rule application and a brief look at syllabification

The Phonomaton allows for four flavors of rule application. The default is for a rule to apply wherever it can, scanning an underlying or intermediate representation from left to right. But rules can also apply iteratively, in which case the output of the rule can feed another application of the same rule. The default application of a nasalization rule is shown in 26a, while its iterative counterpart is shown in 25b.

The rule in 25a would apply to a representation like /balabon/ to derive [balabõn], whereas that in 25b would derive [bãlãbõn]. The iterative rule in 25b takes the output of its application as its input until no more changes can be made, at which point the derivation proceeds to the next rule.

Other cases require a rule to apply only to its maximal environment, as per the Pāṇini principle (also known as the ‘subset principle’ or ‘elsewhere condition’). Among other cases, this is crucial for syllabifying strings algorithmically, by rules that insert onset and coda boundaries, as exemplified in 26.

The onset rule in 26 states: insert an onset boundary (⟨) preceding a vowel, optionally preceded by (i) an approximant (i.e. trills, taps, laterals, glides), in turn optionally preceded by a nonsonorant, or (ii) a singleton consonant of any type. The embedded parenthesis group captures certain types of CC clusters, such as the ones listed below option I in 27, while excluding others (e.g. ml, rl, nt) that do not fit this template.

Because each part of the embedded group is optional, this part of the pattern could also capture singleton onsets that fit the given feature specifications. However, we would still need option II for a C of any type if there are singleton onsets that do not participate in clusters. For instance, our onset rule in 26 describes a language that allows nasal onsets but does not allow nasals in either component of a CC cluster, because a [+sonorant, −approximant] segment such as m, n, ŋ is not captured by either component of option I.

The coda rule in 26 is simpler and allows for a singleton coda consonant of any type. Crucially, the specification ‘exclusive’ at the end of both rules ensures that the boundaries will be inserted only once at the edge of the maximal string containing the optional segments rather than at every possible point. Otherwise, a UR such as /blabla/ would result in ⟨b⟨l⟨a⟩⟨b⟨l⟨a⟩ instead of the correct ⟨bla⟩⟨bla⟩.Footnote ¹⁹

2.10. Morphological rules

Exercises in phonology courses commonly present a paradigm, which the student must analyze into a set of roots and affixes or other morphophonological processes. One example of such an exercise, based on Serbo-Croatian adjectives, is shown in Table 1 (Kenstowicz & Kisseberth Reference Kenstowicz and Kisseberth1979:74).

Table 1.

Serbo-Croatian adjective paradigm.

Coopting for simplicity’s sake the [tone_high] feature to indicate stress (as the Phonomaton does not treat stress as a segmental feature), we can derive the forms in Table 1 with the phonological rules in 28a and the morphological rules in 28b.

The Phonomaton applies each morphological rule to each underlying representation to create an input form, as shown in Figure 14 for the underlying representations /ledn/, /debel/, and /naɡl/. We see the bare underlying representation, as well, preceding the forms that have undergone morphological rules.Footnote ²⁰

Figure 14.

Serbo-Croatian derivation with morphological rules.

The Phonomaton accepts morphological rules in the same format as phonological rules. For instance, a rule introducing a simple affix converts a null to the phonological form of the affix in a word-edge environment, namely <_ for prefixes and _> for suffixes. This approach also facilitates modeling nonconcatenative processes such as lenition, ablaut, and infixation, as any process that can be written as a phonological rule can be treated as a morphological rule.

With this, we conclude our review of the program’s features and proceed to how it has been implemented in the classroom.

3.1. Pedagogically oriented features

Crucial to its pedagogical function is the Phonomaton’s ability to save and upload analyses. Saving an analysis downloads the data found in each component (underlying representation, phonological rules, and morphological rules) as a single, human-readable text file, with all of the contents of the analysis appearing precisely as in the web interface. Analyses are downloaded and uploaded with a single click without any need to log in or register. Students can use these downloaded analyses as submissions to class exercises for the instructor to check.

An advantage to treating phonological derivations as code is that an analysis can be interspersed with comments that are not executed by the program, following Knuth’s (Reference Knuth1992) popular concept of ‘literate programming’, whereby the programmer writes code as an expository text for a human audience interspersed with executable lines directed at the software. The program explains itself as it goes, preceding the expository comments with a character that tells the program not to execute that particular line. This approach has become routine in reproducible research and can be easily applied to executable linguistic analysis of the type presented here. The following brief example demonstrates the idea with the complex vowel-length alternations of Chimwiini (Kisseberth & Abasheik Reference Kisseberth and Abasheik1974). The comments, not executed by the program, are preceded by %.

When the code above is executed it yields the derivation shown in Figure 15, with the given underlying representations.

Figure 15.

Chimwiini vowel alternations.

Another pedagogically relevant feature of note in Figure 15 is the final row in the table headed by ‘expected’. In the field where underlying representations are entered, the user can optionally add the expected surface form following the representation separated by a double backslash (e.g. kama mpʰaka // kamaː mpʰaka). This inclusion has the effect of displaying the expected form at the bottom of each column and checking it against the actual derived form.Footnote ²¹ If the two match, the expected form is displayed in green, as seen in Figure 15. If they do not match, the expected form is highlighted in red. This feature is also convenient for creating problem sets, which an instructor can present as a blank Phonomaton analysis with expected surface forms; the students’ task is then to provide the underlying representations and rules.

3.2. Implementation

We have received constructive input from two introductory phonology courses where an earlier version of the Phonomaton was employed. We report here from a survey of nineteen students who used the program in the Spring 2024 semester at Queens College (City University of New York), where the first author was the instructor and the third author was the teaching assistant. In addition to the positive comments, we highlight here some of the critical ones as well, since we believe these uncover important issues in the use of technology in the classroom more broadly.

For context, Queens College is part of the City University of New York, a public university with a mission of making higher education accessible to all.Footnote ²² The campus is among the most diverse in the country, and over a third of the students are the first generation in their families to attend college. Within the undergraduate linguistics program, there is also a relatively wide range in the students’ levels of preparation. Some students could compete handily with their peers in the most prestigious departments, while others are not yet comfortable with the technical aspects of linguistic analysis. Optimally, the Phonomaton would lift all boats, aiding both advanced students and those who did not yet have a strong grasp of the fundamentals.

In the semester-long trial, the students submitted all of their work as Phonomaton analyses, so they already knew whether their solutions were formatted correctly and generated the desired results. The instructor’s role in evaluating assignments could then focus on questions of economy, insight, and naturalness, which are higher-level matters often occluded by the effort to get the mechanics right. The instructor also made frequent use of the Phonomaton during lectures in order to show the various effects of rules and their interactions. Overall, the majority of the students reported finding the Phonomaton helpful, as seen in Figure 16, although a considerable number of students had mixed feelings, and four students responded negatively.

Figure 16.

Responses to ‘Was the Phonomaton helpful?’.

When asked about their favorite aspect of the program, eight students responded with reference to finding and calculating features, six students chose the Phonomaton’s ability to check their work, and another six students chose the ability to calculate and display derivations. Several remarks in response to the question, ‘Did you find the Phonomaton program helpful in learning how phonological rules and derivations work? Why or why not? What were the best and worst parts of it?’, made reference to verifying the output of rules and their interactions:

Other comments noted how the program helped them understand feature systems, which involve a steep learning curve, as students have to unlearn descriptive categories, such as stop, fricative, and affricate, in order to learn more abstract features that do not always have easily discernible physical or acoustic correlates (such as [distributed] and [dorsal]):

As for the worst parts of the program, students largely referred to its sensitivity to small errors and the lack of helpful error messages. In the most critical remark to this effect, the student stated that the program actually detracted from their learning because they were so concerned about getting the program to work as they expected. We return to these points below.

During the course, early exercises on features and on the mechanics of rule writing were assigned in two stages. In the first stage, the assignment was to be done by hand and submitted in the normal manner. In the second stage, the students had to complete the same problem using the Phonomaton, from which they downloaded and submitted their analysis. The exercises were not discussed in class until after the second stage. In the exercise on applying rules, students were given a rule (e.g. ə → ∅ / VC _ CV) and asked to apply it to a number of underlying forms. In the exercise on writing rules, they were given three underlying forms with their corresponding surface forms and asked to devise a rule that accounted for the alternations. The before and after assessments for both assignments are shown in Figure 17, where the vertical axis represents the scores out of 100.

Figure 17.

Assessment scores before and after using the Phonomaton.

The scores improved significantly, especially in relation to rule writing, when the students had the opportunity to manipulate and check their analyses using the Phonomaton. Whereas many students struggled with consistent formalization of rules in the first stage, through experimenting with different possibilities, they were able to self-correct to a large extent during the second stage.

A different exercise, adapted from Hayes Reference Hayes2009:101, presented the students with statements like that in 29a, which they were expected to translate into feature-based rules like that in 29b, given a particular inventory of vowels and consonants.

The Phonomaton led to a 30% improvement in the second attempt over the first one, where students had the aid of only a static feature chart. In this case, however, it can be argued that the problem becomes trivial with the help of the Phonomaton, as the program calculates natural classes and featural differences within any subset of segments. The user must only enter the relevant inventory and check each pair to ensure that a particular change in features accounts for the entire alternation. To this point, an anonymous referee asks, ‘What happens if the student needs Phonomaton to identify a natural class? Do they learn this concept if the program does it for them?’. Overall improvements in classroom-based assessments, where the students did not have access to the program, suggest that the program helped them overcome the considerable initial difficulties that the stage-one scores revealed. However, our evaluation was not comprehensive enough to compare classroom-based assessments on exactly the same type of material before and after the students worked with the Phonomaton. Our future evaluations of the program will attempt to pinpoint how the Phonomaton aids progress in independent work, as opposed to overall improvements. We note, though, that the program does not aim to explain phonology any more than a calculator is meant to explain concepts in math.Footnote ²³ For a well-motivated student who independently explores the various possibilities of rule formulation and interaction, we believe the program could serve a broader function, but the Phonomaton is not designed to replace instruction.Footnote ²⁴ One student noted this explicitly in the survey:

The two biggest challenges encountered in applying the software pedagogically were (i) the lack of guidance on malformed rules, and (ii) an unevenness in how the tool was taken up by students at different levels of proficiency. We believe these two challenges are linked. The most common complaint in the evaluations of the program had to do with the unhelpful error messages that malformed rules would trigger. Perhaps in our eagerness for the Phonomaton to handle as much phonological theory as possible, including autosegmental phonology and optimality theory (not discussed here for reasons of space), we skimped on what turned out to be the most important element for novices: detailed guidance on the mechanics of rule writing. Based on the evaluations, we have remedied this problem to a large extent. The program now pinpoints which rule and what part of that rule contains an error, calling out specific problems such as a missing underscore, a missing arrow, a missing bracket, or invoking a feature that doesn’t exist. Most likely as a result of this lacuna in the original trial semester, we found a strong correlation between the students’ overall grades in the course and their assessment of the Phonomaton. Those who achieved higher scores in assessments and in the course as a whole consistently reported finding the Phonomaton more useful. We address this problem with urgency because, if left untreated, it could yield precisely the opposite effect of what we had originally set out to do, namely, to lift all boats, and instead simply exacerbate the gap between those who are already comfortable with mathematical formalism and algorithmic thinking upon entering the course and those who are not. Now that the program offers friendly guidance in rule formulation and allows for a bit more flexibility, we expect a more even response across students at varying levels of proficiency. In particular, we aim for this aspect of the program to have a broader impact in opening the doors to computational thinking for those students who may not take naturally to rule syntax and mathematical formalization.Footnote ²⁵

There are several facilities that have not yet been integrated into the Phonomaton but that would be useful to students in optimizing a working analysis. The program could check not only whether a derivation achieves its desired target, as it does now, but also whether there are redundancies in the rules (e.g. reference to features that do not need to be invoked, or overspecified rule environments). The program could also gauge the complexity of a given analysis and assign a numeric score based on the number of rules and specifications for each rule. Although formal economy metrics of this type have never been widely adopted in theoretical phonology, they could serve an important pedagogical function. We continue to explore these and other potential extensions of the program.