The Mother of all Tableaux; Merchant; Prince - Equinox Publishing

Nazarré Merchant [+–]

Eckerd College, St Petersburg, Florida

Nazarré Merchant is a Professor of Mathematics at Eckerd College in St. Petersburg, Florida, and has a Ph.D. in linguistics from Rutgers University.

Alan Prince [+–]

Independent Scholar

Alan Prince has worked on prosody, prosodic morphology, cognitive science of language, and Optimality Theory while teaching at several universities in the United States.

An Optimality Theoretic grammar arises from the comparison of candidates over a set of constraints, oriented toward obtaining certain of those candidates as optimal. The typology of a specified system collects its grammars, encompassing all total domination orders among the posited constraints. Considerable progress has been made in understanding the internal structure of Optimality Theoretic grammars but, in this book, we move up a level from grammar to typology, probing the structure that emerges from the most basic commitments of the theory.

Comparison is once again central: a constraint viewed at the typological level rates entire grammars against each other. From this perspective, the constraint goes beyond its familiar role as an engine of comparison based on quantitative penalties and instead takes the form of a more abstract order and equivalence structure. This “Equivalence-augmented Privileged Order” (EPO) can be presented as a kind of enriched Hasse diagram. The collection of the EPOs, one for each constraint, forms the MOAT, the “Mother of All Tableaux”. The EPOs of a typology’s unique MOAT are respected in every violation tableau associated with it.

With the MOAT concept in place, it becomes possible to understand exactly which sets of disjoint grammars constitute valid typologies. This finding provides the conditions under which grammars of a given typology can merge to produce another, simpler typology and thereby abstract away informatively from various differences between them. Geometrically, the MOAT concept enables us to show, following the insights of Jason Riggle, that the grammars of a typology neatly partition its representation on the permutohedron into connected, spherically convex regions.

Discussion proceeds along both concrete and abstract lines, facilitating access for readers across a wide range of interests.

Series: Advances in Optimality Theory

Table of Contents

Prelims

Acknowledgements xi

Nazarré Merchant,Alan Prince

Eckerd College, St Petersburg, Florida

Nazarré Merchant is a Professor of Mathematics at Eckerd College in St. Petersburg, Florida, and has a Ph.D. in linguistics from Rutgers University.

Independent Scholar

Alan Prince has worked on prosody, prosodic morphology, cognitive science of language, and Optimality Theory while teaching at several universities in the United States.

Introduction

Introduction [+–] 1-4

Nazarré Merchant,Alan Prince

Eckerd College, St Petersburg, Florida

Nazarré Merchant is a Professor of Mathematics at Eckerd College in St. Petersburg, Florida, and has a Ph.D. in linguistics from Rutgers University.

Independent Scholar

Alan Prince has worked on prosody, prosodic morphology, cognitive science of language, and Optimality Theory while teaching at several universities in the United States.

system, grammar, typology are defined. The three key problems are stated: (1) What defining characteristics must all Violation Tableaux (VT) representations of a typology share? (2) When does a set of grammars constitute an OT typology? (3) When does a subset of the grammars in a typology form a coherent class that can be described as a kind of super-grammar? An order and equivalence structure on grammars, the ‘Mother of All Tableaux’ (MOAT), is introduced as a graphical structure similar to a Hasse diagram, and it is shown in broad strokes how it resolves these problems.

Chapter 1

Overview [+–] 5-48

Nazarré Merchant,Alan Prince

Eckerd College, St Petersburg, Florida

Nazarré Merchant is a Professor of Mathematics at Eckerd College in St. Petersburg, Florida, and has a Ph.D. in linguistics from Rutgers University.

Independent Scholar

Alan Prince has worked on prosody, prosodic morphology, cognitive science of language, and Optimality Theory while teaching at several universities in the United States.

1.1 Language, Grammar, Typology 1.1.1 Concrete OT 1.1.2 Abstract OT 1.2 One Tableau Suffices 1.3 Problems, Problems, Problems 1.3.1 Typological Equivalence 1.3.2 Compatibility of Grammars within a Typology 1.3.3 Classification

Chapter 2

The EST Typology [+–] 49-59

Nazarré Merchant,Alan Prince

Eckerd College, St Petersburg, Florida

Nazarré Merchant is a Professor of Mathematics at Eckerd College in St. Petersburg, Florida, and has a Ph.D. in linguistics from Rutgers University.

Independent Scholar

Alan Prince has worked on prosody, prosodic morphology, cognitive science of language, and Optimality Theory while teaching at several universities in the United States.

The Elementary Syllable Theory (EST) typology is laid out. It will be studied in the following chapters, providing the examples through which techniques and ideas will be worked out in concrete detail.

Chapter 3

EST: The MOAT and its EPOs [+–] 61-89

Nazarré Merchant,Alan Prince

Eckerd College, St Petersburg, Florida

Nazarré Merchant is a Professor of Mathematics at Eckerd College in St. Petersburg, Florida, and has a Ph.D. in linguistics from Rutgers University.

Independent Scholar

Alan Prince has worked on prosody, prosodic morphology, cognitive science of language, and Optimality Theory while teaching at several universities in the United States.

The concept of the MOAT is developed through analysis of the EST. From the key notion of a Border Point Pair, order and equivalence relations between grammars emerge, generalizing the notion of constraint and determining all the possible numerical representations of a typology in Violation Tableaux.

Chapter 4

Analysis of the MOAT [+–] 91-236

Nazarré Merchant,Alan Prince

Eckerd College, St Petersburg, Florida

Nazarré Merchant is a Professor of Mathematics at Eckerd College in St. Petersburg, Florida, and has a Ph.D. in linguistics from Rutgers University.

Independent Scholar

Alan Prince has worked on prosody, prosodic morphology, cognitive science of language, and Optimality Theory while teaching at several universities in the United States.

The MOAT is related to another structure, which derives directly from the set of all VTs giving rise to a typology. From this relation, we deduce the main result: that the MOAT uniquely characterizes a typology, uncovering a number of properties of OT grammars along the way. We go on to examine the logical structure that inheres in Border Point Pairs, introducing the ercoid, which provides the hidden microstructure from which the three-valued logic of OT emerges. This is by far the most technical chapter, though the formal ideas involved are of a kind that most linguistics will have encountered. The argument unfolds in a way intended to make the basics available to those who do not wish to see every detail.

Chapter 5

Working out the MOAT: Jump to the CSys [+–] 237-255

Nazarré Merchant,Alan Prince

Eckerd College, St Petersburg, Florida

Nazarré Merchant is a Professor of Mathematics at Eckerd College in St. Petersburg, Florida, and has a Ph.D. in linguistics from Rutgers University.

Independent Scholar

Alan Prince has worked on prosody, prosodic morphology, cognitive science of language, and Optimality Theory while teaching at several universities in the United States.

We return to the concrete via scrutiny of a self-contained subsystem of the EST, the CSys, which deals with the fate of consonants, illustrating how the MOAT functions as a tool of typological analysis. We derive the MOAT and show that simple graphical operations on the MOAT parallel union of grammars in the typology, characterizing the result, when licit, as a generalization of the original typology, revealing aspects of its structure.

Chapter 6

Compatibility and the Join [+–] 257-276

Nazarré Merchant,Alan Prince

Eckerd College, St Petersburg, Florida

Nazarré Merchant is a Professor of Mathematics at Eckerd College in St. Petersburg, Florida, and has a Ph.D. in linguistics from Rutgers University.

Independent Scholar

Alan Prince has worked on prosody, prosodic morphology, cognitive science of language, and Optimality Theory while teaching at several universities in the United States.

Discussion then takes a mildly abstract turn toward the issue of coexistence of grammars within a single typology. Simple examples are presented which illustrate how the MOAT determines the typological compatibility of grammars, and how (graphical) MOAT structure relates to the (algebraic) join of grammars into super-grammars.

Chapter 7

Geometry [+–] 277-328

Nazarré Merchant,Alan Prince

Eckerd College, St Petersburg, Florida

Nazarré Merchant is a Professor of Mathematics at Eckerd College in St. Petersburg, Florida, and has a Ph.D. in linguistics from Rutgers University.

Independent Scholar

Alan Prince has worked on prosody, prosodic morphology, cognitive science of language, and Optimality Theory while teaching at several universities in the United States.

We conclude with a discussion of the remarkably well-behaved geometry of OT typologies, which gives a concrete sense to the notion of the Border Point Pair from which the MOAT derives. We show how a typology is represented on the permutohedron, a well-studied figure where each vertex corresponds to a linear order. We then show this representation reduces to the typohedron, a structure in which each vertex represents an entire grammar, leading to a visually accessible representation of the overall structure of a typology. We complete the discussion by offering proofs of several striking results announced by Jason Riggle, showing that when distance is defined between linear orders, grammars are convex regions analogous to disks and balls in the familiar Euclidean world.

Appendices

Appendix I: Leg Set Partition of EST [+–] 329

Nazarré Merchant,Alan Prince

Eckerd College, St Petersburg, Florida

Nazarré Merchant is a Professor of Mathematics at Eckerd College in St. Petersburg, Florida, and has a Ph.D. in linguistics from Rutgers University.

Independent Scholar

Alan Prince has worked on prosody, prosodic morphology, cognitive science of language, and Optimality Theory while teaching at several universities in the United States.

What is the large-scale structure relating grammars that emerges from the low-level interactions inside them? This volume resolves certain fundamental aspects of this question for OT, preparing the way for a yet deeper understanding of the theory. Any OT system specifies the candidates, candidate sets, and constraints at play, and from these a typology of the system follows: the set of its admitted grammars. An OT grammar arises from the comparison of candidates over a set of constraints. An OT typology, as an unexpected consequence, compares entire grammars over the same set of constraints. At this level, a constraint is revealed as an abstract order and equivalence structure relating grammars rather than linguistic forms, which we call an EPO, an ‘Equivalence-augmented Privileged Order’. The EPO has a simple graphical representation, generalizing the Hasse diagram. The collection of the EPOs, each one representing a single constraint, forms the MOAT, the ‘Mother of All Tableaux’. The unique MOAT of a typology is instantiated in every violation tableau that gives rise to it. With this new characterization of ‘typology’ in hand, we can pose and answer fundamental questions about the structure imposed by OT on its grammars. (1) Typological Status. When does a family of grammars constitute an OT typology? Since a typology must have a well-formed MOAT, we can assess whether a given partition of the set of all rankings constitutes an OT typology. Surprisingly, dividing this set into individually well-formed grammars is not guaranteed to produce a legitimate typology. Failures are detected by the appearance of cycles in the EPO graphs of the MOAT. Concomitantly, we can determine which VT representations are equivalent in the sense that they yield the same typology. (2) Classification. How are the grammars of a typology similar and how are they different? Within a typology, MOAT structure determines whether a collection of grammars can be classified together as a kind of super-grammar, one that retains their shared linguistic patterns while abstracting away from their differences. This contributes to the foundations of the Classification Program of Alber & Prince (2015, 2017). (3) Representation. What formal structures provide analytically useful information about the organization of typologies? The MOAT arises from a notion of adjacency between rankings, which correlates with the way that rankings filter candidate sets. Adjacency leads to a natural graphical interpretation. Each typology is associated with a geometric figure that represents the relations between its grammars: its typohedron. Super-grammars appear as connected regions on the typohedron. The MOAT brings out symmetries between constraints, and these appear on the typohedron as symmetries between super-grammar regions. Such patterns transparently reveal global aspects of typological form which may be obscure or even invisible in the microstructure of constraints and candidates where the project of analysis begins.

Appendix II: EST: SKBs and Hasse Diagrams [+–] 331-332

Nazarré Merchant,Alan Prince

Eckerd College, St Petersburg, Florida

Nazarré Merchant is a Professor of Mathematics at Eckerd College in St. Petersburg, Florida, and has a Ph.D. in linguistics from Rutgers University.

Independent Scholar

Alan Prince has worked on prosody, prosodic morphology, cognitive science of language, and Optimality Theory while teaching at several universities in the United States.

What is the large-scale structure relating grammars that emerges from the low-level interactions inside them? This volume resolves certain fundamental aspects of this question for OT, preparing the way for a yet deeper understanding of the theory. Any OT system specifies the candidates, candidate sets, and constraints at play, and from these a typology of the system follows: the set of its admitted grammars. An OT grammar arises from the comparison of candidates over a set of constraints. An OT typology, as an unexpected consequence, compares entire grammars over the same set of constraints. At this level, a constraint is revealed as an abstract order and equivalence structure relating grammars rather than linguistic forms, which we call an EPO, an ‘Equivalence-augmented Privileged Order’. The EPO has a simple graphical representation, generalizing the Hasse diagram. The collection of the EPOs, each one representing a single constraint, forms the MOAT, the ‘Mother of All Tableaux’. The unique MOAT of a typology is instantiated in every violation tableau that gives rise to it. With this new characterization of ‘typology’ in hand, we can pose and answer fundamental questions about the structure imposed by OT on its grammars. (1) Typological Status. When does a family of grammars constitute an OT typology? Since a typology must have a well-formed MOAT, we can assess whether a given partition of the set of all rankings constitutes an OT typology. Surprisingly, dividing this set into individually well-formed grammars is not guaranteed to produce a legitimate typology. Failures are detected by the appearance of cycles in the EPO graphs of the MOAT. Concomitantly, we can determine which VT representations are equivalent in the sense that they yield the same typology. (2) Classification. How are the grammars of a typology similar and how are they different? Within a typology, MOAT structure determines whether a collection of grammars can be classified together as a kind of super-grammar, one that retains their shared linguistic patterns while abstracting away from their differences. This contributes to the foundations of the Classification Program of Alber & Prince (2015, 2017). (3) Representation. What formal structures provide analytically useful information about the organization of typologies? The MOAT arises from a notion of adjacency between rankings, which correlates with the way that rankings filter candidate sets. Adjacency leads to a natural graphical interpretation. Each typology is associated with a geometric figure that represents the relations between its grammars: its typohedron. Super-grammars appear as connected regions on the typohedron. The MOAT brings out symmetries between constraints, and these appear on the typohedron as symmetries between super-grammar regions. Such patterns transparently reveal global aspects of typological form which may be obscure or even invisible in the microstructure of constraints and candidates where the project of analysis begins.

Appendix III: EST.CSys: SKBs and Hasse Diagrams [+–] 333

Nazarré Merchant,Alan Prince

Eckerd College, St Petersburg, Florida

Nazarré Merchant is a Professor of Mathematics at Eckerd College in St. Petersburg, Florida, and has a Ph.D. in linguistics from Rutgers University.

Independent Scholar

Alan Prince has worked on prosody, prosodic morphology, cognitive science of language, and Optimality Theory while teaching at several universities in the United States.

What is the large-scale structure relating grammars that emerges from the low-level interactions inside them? This volume resolves certain fundamental aspects of this question for OT, preparing the way for a yet deeper understanding of the theory. Any OT system specifies the candidates, candidate sets, and constraints at play, and from these a typology of the system follows: the set of its admitted grammars. An OT grammar arises from the comparison of candidates over a set of constraints. An OT typology, as an unexpected consequence, compares entire grammars over the same set of constraints. At this level, a constraint is revealed as an abstract order and equivalence structure relating grammars rather than linguistic forms, which we call an EPO, an ‘Equivalence-augmented Privileged Order’. The EPO has a simple graphical representation, generalizing the Hasse diagram. The collection of the EPOs, each one representing a single constraint, forms the MOAT, the ‘Mother of All Tableaux’. The unique MOAT of a typology is instantiated in every violation tableau that gives rise to it. With this new characterization of ‘typology’ in hand, we can pose and answer fundamental questions about the structure imposed by OT on its grammars. (1) Typological Status. When does a family of grammars constitute an OT typology? Since a typology must have a well-formed MOAT, we can assess whether a given partition of the set of all rankings constitutes an OT typology. Surprisingly, dividing this set into individually well-formed grammars is not guaranteed to produce a legitimate typology. Failures are detected by the appearance of cycles in the EPO graphs of the MOAT. Concomitantly, we can determine which VT representations are equivalent in the sense that they yield the same typology. (2) Classification. How are the grammars of a typology similar and how are they different? Within a typology, MOAT structure determines whether a collection of grammars can be classified together as a kind of super-grammar, one that retains their shared linguistic patterns while abstracting away from their differences. This contributes to the foundations of the Classification Program of Alber & Prince (2015, 2017). (3) Representation. What formal structures provide analytically useful information about the organization of typologies? The MOAT arises from a notion of adjacency between rankings, which correlates with the way that rankings filter candidate sets. Adjacency leads to a natural graphical interpretation. Each typology is associated with a geometric figure that represents the relations between its grammars: its typohedron. Super-grammars appear as connected regions on the typohedron. The MOAT brings out symmetries between constraints, and these appear on the typohedron as symmetries between super-grammar regions. Such patterns transparently reveal global aspects of typological form which may be obscure or even invisible in the microstructure of constraints and candidates where the project of analysis begins.

Appendix IV: Notation [+–] 335-338

Nazarré Merchant,Alan Prince

Eckerd College, St Petersburg, Florida

Nazarré Merchant is a Professor of Mathematics at Eckerd College in St. Petersburg, Florida, and has a Ph.D. in linguistics from Rutgers University.

Independent Scholar

Alan Prince has worked on prosody, prosodic morphology, cognitive science of language, and Optimality Theory while teaching at several universities in the United States.

What is the large-scale structure relating grammars that emerges from the low-level interactions inside them? This volume resolves certain fundamental aspects of this question for OT, preparing the way for a yet deeper understanding of the theory. Any OT system specifies the candidates, candidate sets, and constraints at play, and from these a typology of the system follows: the set of its admitted grammars. An OT grammar arises from the comparison of candidates over a set of constraints. An OT typology, as an unexpected consequence, compares entire grammars over the same set of constraints. At this level, a constraint is revealed as an abstract order and equivalence structure relating grammars rather than linguistic forms, which we call an EPO, an ‘Equivalence-augmented Privileged Order’. The EPO has a simple graphical representation, generalizing the Hasse diagram. The collection of the EPOs, each one representing a single constraint, forms the MOAT, the ‘Mother of All Tableaux’. The unique MOAT of a typology is instantiated in every violation tableau that gives rise to it. With this new characterization of ‘typology’ in hand, we can pose and answer fundamental questions about the structure imposed by OT on its grammars. (1) Typological Status. When does a family of grammars constitute an OT typology? Since a typology must have a well-formed MOAT, we can assess whether a given partition of the set of all rankings constitutes an OT typology. Surprisingly, dividing this set into individually well-formed grammars is not guaranteed to produce a legitimate typology. Failures are detected by the appearance of cycles in the EPO graphs of the MOAT. Concomitantly, we can determine which VT representations are equivalent in the sense that they yield the same typology. (2) Classification. How are the grammars of a typology similar and how are they different? Within a typology, MOAT structure determines whether a collection of grammars can be classified together as a kind of super-grammar, one that retains their shared linguistic patterns while abstracting away from their differences. This contributes to the foundations of the Classification Program of Alber & Prince (2015, 2017). (3) Representation. What formal structures provide analytically useful information about the organization of typologies? The MOAT arises from a notion of adjacency between rankings, which correlates with the way that rankings filter candidate sets. Adjacency leads to a natural graphical interpretation. Each typology is associated with a geometric figure that represents the relations between its grammars: its typohedron. Super-grammars appear as connected regions on the typohedron. The MOAT brings out symmetries between constraints, and these appear on the typohedron as symmetries between super-grammar regions. Such patterns transparently reveal global aspects of typological form which may be obscure or even invisible in the microstructure of constraints and candidates where the project of analysis begins.

End Matter

Bibliography 339-343

Nazarré Merchant,Alan Prince

Eckerd College, St Petersburg, Florida

Nazarré Merchant is a Professor of Mathematics at Eckerd College in St. Petersburg, Florida, and has a Ph.D. in linguistics from Rutgers University.

Independent Scholar

Alan Prince has worked on prosody, prosodic morphology, cognitive science of language, and Optimality Theory while teaching at several universities in the United States.

Definitions, Facts, Remarks, Lemmas, Theorems and Corollaries 345-350

Nazarré Merchant,Alan Prince

Eckerd College, St Petersburg, Florida

Nazarré Merchant is a Professor of Mathematics at Eckerd College in St. Petersburg, Florida, and has a Ph.D. in linguistics from Rutgers University.

Independent Scholar

Alan Prince has worked on prosody, prosodic morphology, cognitive science of language, and Optimality Theory while teaching at several universities in the United States.

Index 351-356

Nazarré Merchant,Alan Prince

Eckerd College, St Petersburg, Florida

Nazarré Merchant is a Professor of Mathematics at Eckerd College in St. Petersburg, Florida, and has a Ph.D. in linguistics from Rutgers University.

Independent Scholar

Alan Prince has worked on prosody, prosodic morphology, cognitive science of language, and Optimality Theory while teaching at several universities in the United States.

Reviews

This important book is privileged within the 'Abstract OT' research program in that it ties together many key pursuits and results within that program: ERC representations and unitary violation tableaux via Minkowski summation (Prince), the FRed algorithm (Brasoveanu & Prince), the join (Merchant), the classification program (Alber & Prince), and the geometric properties of typologies and of the grammars within them (Merchant & Riggle). This is the first monograph-length work published within Abstract OT, and the Advances in Optimality Theory series is its rightful place.
Professor Eric Bakovic, University of California, San Diego