1. Overview: rethinking typology

One of the central arguments in favour of Optimality Theory (OT; Prince & Smolensky Reference Prince and Smolensky1993, Reference Prince and Smolensky2004) has been its ability to capture cross-linguistic variation with language-particular rankings of a universal constraint set (Con). Research in the more than 30 years since OT’s inception has capitalised on this potential through an ongoing and iterative process in which new constraints are proposed (based on particular phenomena in particular languages) and the resulting typological predictions are tested against the currently-known set of attested patterns (e.g., Pater’s Reference Pater, Kager, Hulst and Zonneveld1999 study of the typology of *NC̥). While this approach to studying typology is dependent on the contents of Con, there is also much to learn from the way OT’s core idea of optimisation over ranked constraints itself organizes and structures the space of possible grammars. In The Mother of All Tableaux, Merchant and Prince demonstrate this broader approach to studying typology through a collection of results in Abstract OT, which puts aside the substance of particular constraints in order to reveal the invariant formal properties of typologies and the grammars they consist of.

2. What is the MOAT?

Given a particular theory of Con, OT’s inherent mechanism for investigating typology is the factorial typology, where factorial invokes the $n!$ possible rankings of a set of n constraints. For example, the set $\{$ A, B, C $\}$ has the following six possible linear orders:

One might think that these rankings would correspond to six possible languages – where a language is the set of candidates selected as optimal by a given ranking – but the number of languages is typically less than $n!$ , as distinct rankings can select the same optima and therefore will be extensionally equivalent. Starting from this well-known fact, Merchant and Prince define a grammar as a set of rankings that generate the same extensional language. For example, the set of rankings in (1) might be grouped into grammars G₁ and G₂, as in (2).

A typology of possible grammars, then, is a partition of the set of $n!$ possible rankings. Typically, the process of determining extensional equivalence relies on a particular constraint set and a particular candidate set, from which the crucial ranking information can be gleaned by examining a Violation Tableau (VT), such as the one in (3).

Unlike a traditional OT tableau, in a VT, the order of the columns does not correspond to a constraint ranking, and no winning candidate is indicated. Rather, the VT tells us how the constraints must be ordered for a particular candidate to win. For example, if cand ₁ is the optimum, then constraint A (which favours cand ₁) must outrank constraint C (which favours cand ₂). But constraint B is indecisive and therefore can be ranked anywhere.

However, the typology of grammars/languages under a particular manifestation of Gen and Con also reflects purely structural properties that are independent of those implementational choices. To access these deeper properties, Merchant and Prince turn to Abstract OT, which extracts from particular VTs the crucial relations among constraints that distinguish one grammar from another. This begins with the construction of a Unitary Violation Tableau (UVT), in which each row corresponds not to a candidate, but to an entire grammar, and all grammars in the typology are represented (Prince Reference Prince2016).

For example, if we reinterpret (3) as a UVT, we see that the first row corresponds to grammar G₁ in (2), or the grammar in which A always outranks C.Footnote ¹ Likewise, the second row corresponds to grammar G₂, or the grammar in which C always outranks A. Going further, each constraint can be represented not in terms of the actual violation scores it assigns, but in terms of how it distinguishes (or doesn’t) the various grammars of the typology. These representations are called the constraint’s Equivalence-augmented Privileged Order (EPO), and the eponymous Mother of All Tableaux (MOAT) is the collective set of EPOs for all constraints.

The key to identifying a constraint’s EPO is the border point pair (BPP), or two rankings that differ minimally but belong to different grammars. For example, in (2), B $\gg $ A $\gg $ C and B $\gg $ C $\gg $ A are a BPP: they differ by a single transposition of the constraints A and C, but that difference is exactly why the former is in G₁ while the latter is in G₂. The relations we can extract from this BPP include the following:

(4a) is an equivalence relation, while (4b) and (4c) are order relations. These relations reflect the filtering of grammars in a manner that parallels the customary filtration of candidates. For example, both grammars include rankings that begin with B; therefore both grammars pass through B’s filter (i.e., as far as B is concerned, they are equivalent). But which grammar survives the next step depends on the relative ranking of A and C: if A is first, then G₂ is out, because it lacks a ranking that begins with B $\gg $ A. And vice versa for C and G₁. A constraint’s EPO then includes all such relations that can be gleaned from every available BPP.Footnote ²

The EPOs thus tell us what relations are invariant across all UVTs for the typology (i.e., what relations are guaranteed despite differences in the actual numerical values). Furthermore, the resulting structure of a typology as reflected in a MOAT and derived from a particular constraint set may be shared with one that is based on an entirely different constraint set (i.e., their MOATs may be isomorphic). This is the potential for Abstract OT to reveal the formal structure of typologies in a manner that is independent of the substantive details of constraints.

In addition, given a collection of grammars we can assess whether it constitutes a well-formed typology using a simple condition on its MOAT: all EPOs (represented as bigraphs) must be acyclic. Lastly, given a typology, distinct grammars can be grouped together to create a coarser typology that reflects broader typological classes (e.g., languages that use the same syllable repair strategy – insertion or deletion – but differ in whether or not onsets are required). The well-formedness of these groupings can again be assessed through the MOAT: merged grammars cannot introduce cycles. These classes can then inform our understanding of typology in a principled manner that is derived entirely from the structure and logic of OT, as opposed to relying solely on our intuitions about what descriptive traits are significant typological features.

3. Structure of the book, audience and accessibility

The book begins with a brief introduction that summarizes the key concepts and the capacity of the MOAT to address three problems: typology, classification and geometry. Chapter 1 then goes through that overview again in more detail, providing useful scaffolding for the more technical content to follow. Chapter 2 defines Elementary Syllable Theory (EST), which serves as the running example to concretely illustrate the concepts of Abstract OT in the remainder of the book. Chapter 3 illustrates the construction of EPOs and the MOAT through examination of BPPs, using EST as a working example. This illustration provides a foundational understanding of these ideas before the technical work of proving the claims about them, which is the subject of Chapter 4.

Chapter 4 is the most substantial chapter of the book, in which the formal definitions and results are presented in extensive detail. Though quite technical, the discussion is often made more intuitive by recalling how the filtration of entire grammars by a constraint ranking mirrors the filtration of candidates that is familiar to anyone who has been introduced to OT. Chapter 5 demonstrates and compares two routes to identifying meaningful typological classes by creating a coarser typology. The concrete route is to build a new MOAT starting from a restricted Gen; the abstract route is to merge grammars in the original MOAT, subject to the well-formedness condition that merging cannot introduce cycles. While in some cases, these routes converge on the same typology, in other cases Abstract OT can access classes that could not be instantiated concretely. Chapter 6 offers another perspective on when grammars can be merged using the join operation introduced by Merchant (Reference Merchant2008), which works from the logic of Elementary Ranking Condition (ERC) grammars (Prince Reference Prince2002a,Reference Princeb). Lastly, Chapter 7 recasts these ideas in terms of geometry by showing that a typology is a permutohedron in which grammars are connected and spherically convex regions. Here, convexity – based on earlier work by Riggle (Reference Riggle2012) – means every point on the shortest path for a grammar is also part of that grammar, where distance is measured in terms of how many constraints follow the point of transposition that distinguishes two permutations.

There is no way around it: the book is dense in terms of notation and terminology, and the discussion is technical and abstract (which is actually the point, of course). But the authors do go to great lengths to make the presentation accessible, with the aforementioned scaffolding, the use of familiar and simple constraints from syllable theory, and periodic reminders of how the fundamental idea of optimization as candidate filtration underlies everything. It is a challenging read (perhaps ideal fare for an advanced graduate seminar to tackle), but potentially essential for OT practitioners to gain a deeper understanding of their chosen formalism. And for anyone with an appreciation for formal methods, it provides another opportunity to recognize that once we find the right way to step back from the details, what remains is quite elegant in its simplicity.

4. Looking outward

The book most definitely assumes prior knowledge of OT, and it is furthermore starkly self-contained in the sense that there is no attempt at theory comparison aside from the implicit argument that ranked constraints are what enable this approach to studying typology. The fact that in OT the actual number of constraint violations does not matter is what allows typologies to be reduced to a set of order and equivalence relations. Yet that same fact has also been the target of arguments for weighted constraints over ranked ones, as weighted sums of violations have been shown to be necessary to capture a range of phenomena, including gradient well-formedness (Coetzee & Pater Reference Coetzee and Pater2008), gang effects (Pater Reference Pater2009) and cumulativity (Shih Reference Shih2017; Breiss & Albright Reference Breiss and Albright2022). This comparison is not so much an omission, though, as an opportunity in the form of questions raised (again implicitly) that are primed for engagement in subsequent work.

For example, how does Abstract OT’s purely structural conception of typology interact or intersect with other factors of interest, most notably learnability (Heinz Reference Heinz2010; Stanton Reference Stanton2016) and the aforementioned effects of gradience? Can these structural properties address concerns about the computational complexity of optimisation, raised most recently by Lamont (Reference Lamont2021a,Reference Lamontb) (though see also Hao Reference Hao2024 for a more favourable interpretation). Where are the points of overlap in these sources of explanation for typology, and where do they either complement or contradict each other? Altogether, there is great potential here to reinvigorate typological discussions across competing approaches.

While the content of Con remains an open question, meaningful progress on our understanding of typology can still be made given the formal tools offered by Abstract OT and thoroughly demonstrated in this book. The Mother of All Tableaux is also an invitation for all phonologists to think more deeply about typology and how we approach it, as well as to recognise the added value of studying the formal structure of grammars in any theory.