Draft of May 20, 2006, comments welcome
t-orders
Arto Anttila and Curtis Andrus
StanfordUniversity
Abstract
t-order constructor is a Windows program that computes the implicational universals (t-orders) hidden in optimality-theoretic factorial typologies (Prince and Smolensky 1993/2004). The input to the program is a factorial typology computed by OTSoft (Hayes, Tesar, and Zuraw 2003). Theoutput isa Hasse diagram that displays the implicational universals in visual format that is easy for the linguist to understand.These diagrams have an important application in the study of quantitative variation: they spell out the universal limits of quantitative variation permitted by the constraint set. These limits hold true for several theories of variation, including Multiple Grammars, Partially Ordered Grammars, and Stochastic Optimality Theory.
1. Linguistic variation
We start by a simple example of linguistic variation. The example comes from English phonology, but our point is a general one. Similar examples could easily be drawn from other languages as well as from other subfields of linguistics (morphology, syntax, semantics).
In many dialects of English, word-final /t,d/ is variably deleted, as shown in (1):
(1)It cost ~ cos’ five dollars.(t before a consonant)
It cost ~ cos’ us five dollars.(t before a vowel)
That’s how much it cost ~ cos’.(t before a pause)
Whether t,d-deletion applies or not depends on various factors, in particular the quality of the following segment. It is well known that t,d-deletion is more common before consonants than before pauses and vowels (see Coetzee 2004: 218 and references there). Typically, the process applies variably in all three environments depending on the speaker, speech rate, lexical item, etc., but the quantitative tendency is robust and clear. This is shown by the cross-dialectal data in (2).
(2)t,d-deletion data from five dialects (Coetzee 2004: 218)
_C_V_##
Chicano English (Los Angeles) n3,6931,5741,024
(Santa Ana 1991:76, 1996:66)% deleted624537
Tejano English (San Antonio)n1,738974564
(Bayley 1995:310)% deleted622546
AAE (Washington, DC)n14320237
(Fasold 1972:76)% deleted762973
Jamaican mesolect (Kingston)n1,252793252
(Patrick 1991:181)% deleted856371
Trinidadian acrolectn224316
(Kang 1994:157)% deleted812131
Neu datan814495--
(Neu 1980:45)% deleted3616--
The quantitative generalization that holds across all five dialects is stated in (3):
(3)(a) In all dialects, the deletion rate is highest in _C.
(b)Deletion rates in _V and _## may occur in either order.
Any satisfactory analysisshould thus make the following predictions:
(4)(a) Exclude dialects where t/d-deletion rate is higher in _{V, ##} than in _C.
(b)Include dialects where t/d-deletion rate is higher in _V than in _##.
(c)Include dialects wheret/d-deletion rate is higher in _## than in _V.
If we look at the quantitative facts in onedialect only, wesee three different percentages, e.g. 62-45-37, but cannot tell which differences are deep and which superficial. If we look at the quantitative facts across several dialects, we learn that the first number is always higher than the second and the third, and that the latter two can occur in either order. A satisfactory theory ofvariation should make a distinction between these two types of quantitative patterns: (i) quantitative universals that do not vary across dialects, presumably because they are hard-wired in universal grammar; (ii) quantitative particularsthat are subject to cross-dialectal variation.t/d-deletion provides a simple example of this distinction.We now proceed to illustrate how this distinction emerges from Optimality Theory.
2. An optimality-theoretic analysis of t/d-deletion
Optimality Theory (OT, Prince and Smolensky 1993/2004) is a theory of constraint interaction in generative grammar. For a clear exposition of the theory, seeKager 1999. At the heart of Optimality Theory there are three important assumptions: (i) grammars of natural languages consist of constraints that make potentially conflicting structural demands; (ii) conflicts among constraints are resolved by strict ranking; (iii) constraints are universal, rankings are language-specific.
We now outline an optimality-theoretic analysis of t,d-deletion originally proposed by Kiparsky (1993). Foralternative analyses, see Reynolds 1994 and Coetzee 2004:214-329. The analysis builds on five universal constraints:
(5)Constraints ont,d-deletion (Kiparsky 1993):
*ComplexAvoid consonant clusters within a syllable.
OnsetSyllables have onsets.
ParseSegments belong to syllables.
Align-Left-WordSyllables cannot straddle word boundaries.
Align-Right-PhrasePhrase-final consonants are also syllable-final.
The tableau in (6) shows one possible ranking of the five constraints. This ranking predicts no deletion in the prevocalic environment. Instead, the t is resyllabified as the onset of the following word. Deletion is predicted to occur in both preconsonantal and prepausal environments. The theoretical assumption is that t is deleted if it is not parsed as part of a syllable. Syllables are marked by square brackets.
(6)Sample ranking. Winners: cost us (no deletion), cos’ me (deletion), cos’ (deletion)
Inputs / Outputs / *Complex / Onset / Align-L-W / Align-R-P / Parsecost us / (a) [cost][us] / *! / *
(b) [cos]t[us] / *! / *
(c) [cos][tus] / *
cost me / (a) [cost][me] / *!
(b) [cos]t[me] / *
(c) [cos][tme] / *! / *
cost / (a) [cost] / *!
(b) [cos]t / * / *
3. What are t-orders?
What kinds of dialects does the analysis predict to be possible in principle? This can be figured out by computing the factorial typologyof the five constraints with the aid of OTSoft (Hayes, Tesar, and Zuraw 2003).The program will consider all the 5! = 120 permutations of the 5 constraints and work out the predicted output pattern in each case. The result is displayed in (7).t-deletion is highlighted in grey.
(7)Factorial typology
There were 6 different output patterns.
Output #1 Output #2 Output #3 Output #4
/cost us/: [cost][us] [cos]t[us] [cos]t[us] [cos][tus]
/cost me/: [cost][me] [cos]t[me] [cos]t[me] [cost][me]
/cost/: [cost] [cost] [cos]t [cost]
Output #5 Output #6
/cost us/: [cos][tus] [cos][tus]
/cost me/: [cos]t[me] [cos]t[me]
/cost/: [cost] [cos]t
The factorial typology in (7) reveals the following prediction that holds true of all dialects:
(8)Universal:t-deletion before a vowel or a pause implies t-deletion before a consonant, but not vice versa.
This is an implicational universalof the sort familiar from typological literature: if a pattern p occurs in a language, so will pattern q. This implicational universal correctly captures the asymmetry between the preconsonantal environment and the other two environments. However, this is not quite sufficient: all the dialects in (2)are variable and the implicational universal comes out quantitatively, not categorically.We need something morein order to talk about variation and quantitative patterns.
Here we adopt a particularly simple theory of variation: the Multiple Grammars Theory. Its basic assumptions are stated in (9).
(9)The Multiple Grammars Theory of variation:
(a)Variation arises from multiple grammars within/across individuals.
(b)The number of grammars predicting an output is proportional to the
frequency of occurrence of this output.
Assume an individual whose mental grammarconsistsof three grammars of types 1, 5, 6 in the factorial typology: At the moment of speaking, the individual reachesinto the grammar urn and selects a grammar at random. In the long run, the following quantitative pattern will emerge: no deletion before vowels, 67% of deletion before consonants, and 33% of deletion before pauses.
(10)Sample grammar: {#1, #5, #6}
Output #1 Output #5Output #6Del. rate
/cost us/: [cost][us] [cos][tus] [cos][tus] 0/3
/cost me/: [cost][me] [cos]t[me] [cos]t[me]2/3
/cost/: [cost] [cost][cos]t1/3
It is now easy to see that a universal quantitative prediction is being made. Given the implicational universal in (8), the following quantitative universal must also hold:
(11)Quantitative universal: t-deletion rate isat least as high before a consonant than beforea vowel or a pause.
The implicational universal states that if t is deleted anywhere, it is deleted before a consonant. Since this statement holds for all individual dialects, it will hold quantitatively for all combinations of dialects. In other words, it is never possible to pick a combination of grammars that would result in more t,d-deletion before vowels and pauses than before consonants. This is the empirical observation in (3a). In contrast, no implicational universal is predicted to hold between the vowel and pause environments. In dialect 2, t,d-deletion occurs before a vowel, but not before a pause. In dialect 6 the opposite situation holds. This means that the two environments can occur in either order quantitatively as well. This is the empirical observation in (3b).
4. What is the T-Order Constructor?
The problem with factorial typologies is that they are hard for humans to understand. The typology in(7) is small and relatively easy to figure out. However, in any analysis of a realistic size, factorial typologies tend to be much larger and the number of output dialects often runs in the hundreds. OTSoft does a good job computing factorial typologies, but the result tends to be useless because the typology is too complex for the linguist to understand. What is needed, then, is a way to make factorial typologies understandable for humans. In this case, we need a way to figure out all the implicational universals hidden in a factorial typology. This problem is solved by the T-Order-Constructor:
The T-Order-Constructor takes a factorial typology as input and returns all the implicational universals implicit in it in a visual format that is easy for the linguist to understand.
The present version of the programreads factorial typology files produced by OTSoft (Hayes, Tesar, and Zuraw 2003). The output is a Hasse diagram that displays the implicational universals. For the t,d-deletion grammar in (6), T-Order-Constructor produces the graph in (12). We call such graphs t-orders.
(12)t-order
Every optimality-theoretic grammar has a t-order. The t-order is linguistically interesting because it describes two kinds of universals:
- qualitativeuniversals: The qualitative distribution of variants should never conflict with the t-order. For example, the t-order excludes dialects with deletion before a vowel (cos’ again), but not before a consonant (*cos’ me). Such dialects cannot be derived by any constraint ranking.
- quantitative universals: The quantitative distribution of variants should never conflict with the t-order. For example, the t-order excludes dialects where the rate of deletion before a vowel (cos’ again) exceeds the rate of deletion before a consonant (cos’ me). Such dialects cannot be derived by any combination of constraint rankings.
Both the qualitative and quantitative universals are hard-wired in the constraints and thus ranking-independent. All other quantitative relationships are language-particular, ranking-dependent, and subject to cross-dialectal variation. Thesequantitative universals hold true for several theories of variation, including Multiple Grammars, Partially Ordered Grammars, and Stochastic Optimality Theory. This is because in all these theories the factorial typology is the same.
In conclusion, we now know that the constraint set in (5)indeed derives the empirical universals in (3) and excludes unnatural dialects with more t-deletion in cost us and cost than in cost me.
5. About the program
The T-Order Constructor was programmed by Curtis Andrus in the Python programming language in the spring of 2006. The program can be downloaded from
.
The T-Order Constructor requires Graphvis software to be installed in order to draw t-order graphs. More information on Graphvis software can be found at the following URL:
To use the program, take the following steps.
- Download and unzip the program file.
- Runthe program by clicking on tordergui.exe.
- Open the factorial typology file conventionally named MYFILEDraftOutput.txt.
- Select the desired user options (see below) and click on “Generate T-Order”.
The program will give you the option of saving the result into a text file (“Save T-Order”) or into a graph file (“Save Graph”). The graph files are read with Graphvis software. The program can output the t-order graph to any supported file format.
The program allows the following user options:
Save pair possibilities: Writes <input, output> pairs and implications into a text file.
Collapse cycles: Removes cycles, i.e. <input,output> pairs that imply each other, and collapses the nodes into a box. The program computes the transitive reduction (i.e. the t-order with all unecessary edges removed) using the property that the t-order is the transitive closure of a relation (i.e. all edges implied by transitivity are in the t-order).
Split Disjoint Graphs: Splits the t-order graph into disjoint components and output each component to a separate file. [Recommended, or else the graphs may get too large.]
1