**Linguistics 726 – Mathematical Linguistics
(really: mathematics for and in linguistics)**

syllabus | course description
| lectures | homework
| book errata | links | readings
| LING 726 2001 Website

**
Time and Place:
Tu, Th 1:00 – 2:15, (Room being negotiated – watch for announcements of room
change from initially assigned Herter 114. We start in Herter 114.)**

**
Instructors:
Barbara Partee and Vladimir Borschev**

**
Added attraction:
With some guest lectures by Chris Potts and Rajesh Bhatt!**

**Required text**: Partee, ter Meulen and Wall (1990 -
Corrected first edition, 1993 or later), *Mathematical Methods in Linguistics*,
Dordrecht: Kluwer: Student edition (paperback). Available from Partee.

Other readings will be xeroxed and/or made available for download.

Other books useful for supplementary topics will be suggested, and we will work out how to make them available.

**
Office Hours and
Contact Information:**

Barbara H. Partee: by appointment

Vladimir Borschev: by appointment; a regular time could be set up by arrangement.

Office: So. College 222; 545-0885

Home phone: 549-4501

E-mail: partee@linguist.umass.edu, borschev@linguist.umass.edu

**Current schedule is
here.**

**
Course
description: **

(Note: a more accurate title for this course might be “mathematics in and for linguistics”; the term “mathematical linguistics” now mostly refers to ‘formal language theory’. But this course title is on the books, and it’s not unreasonable in principle.)

The first goal of the course is to strengthen students’ math background in the areas most widely relevant to linguistic theorizing: linguists in all subfields are concerned with “structures” and their formal properties. “Structures” are in general algebras; “theorizing” generally involves positing axioms (in some logic) and studying properties of the models that satisfy those axioms (this is what model theory is about). The algebraic notions of isomorphism and homomorphism formalize the notion of "same structure". Other basic background notions include elementary set theory and first-order logic. When we formalize the syntax and semantics of propositional and first-order logic, we can illustrate what it means to say “compositionality can be formalized as a homomorphism between a syntactic algebra and a semantic algebra”. We will look at the ingredients of the notion of ‘trees’, and practice figuring out the consequences of varying different parameters in the definition. We will include a bit of automata theory and formal language theory in order to be able to talk about the Chomsky hierarchy of languages (finite-state languages, context-free languages, etc.), and we will look at the nature of debates about formal properties of human languages, including issues concerning the premises on which some of the arguments are constructed. Some examples of simple algebras will include semilattices, lattices, and Boolean algebras. We’ll also include a bit about “model-theoretic syntax”, something both Volodja and Chris Potts have worked on. In the later part of the course, which is open to joint decisions with the class about what to include, we can consider applications to ‘feature algebras’, Link’s semantics for mass and plural, event algebras, type-shifting, and OT. The first part of 726 is similar to the undergraduate course 409, but we will go a little faster in 726 and there will be more opportunity for exploration of application to topics of particular interest to participants.

The second goal of the course will be to explore linguistic applications of various mathematical notions and to work on more specific mathematical and logical tools that may be needed/useful in particular linguistic research paradigms. The following are some examples. Some may be of enough interest for us to look at together; other topics might be pursued by individual students or subgroups, depending on interest. (i) OT makes some use of lattice theory, and we can study a chapter of Andries Coetzee’s 2004 dissertation that grew out of a project he did for this course; (ii) Godehard Link's work uses lattice structures for the semantics of mass and plurals and Link, Bach, Krifka, Landman and others have extended such structures to the eventuality domain and for mappings between the entity domain and the eventuality domain. (iii) Angelika Kratzer has noted some interesting structural parallels among modality, OT, and pragmatics, all based on formal work on ways of revising inconsistent sets of premises and relating to the partitioning of sets of premises. (And the notion of giving priority to the more specific rule over the less specific may be part of this family of notions.) (iv) An algebraic perspective on categorial grammars links syntax, semantics, type theory, logical deduction (via the Lambek calculus), and the type-shifting exploited in Ades and Steedman processing models. (v) A number of researchers have explored the use of finite-state transducers in modeling various linguistic processes; this is something Rajesh Bhatt has included in his courses at the University of Texas (see http://www.cs.utexas.edu/users/bhatt/lin386m/ ). Lauri Karttunen has done interesting work in this area that would be good to study. (vi) Unification-based grammar systems (LFG, GPSG, HPSG, and unification-based categorial grammars) build on feature structures that have much in common with phonological feature structures, and a glimpse at such structures and the algebra of unification may be of general interest. --- These are just sample possibilities.

The course has no specific prerequisites, but the pace and workload will be a bit more demanding than that of Linguistics 409, which will next be offered in Fall 2005. Students who would benefit from a slightly slower-paced course with more emphasis on the fundamentals might take 409 rather than 726. If in doubt, consult us.

**Requirements: **First two-thirds: frequent written
homework, with “first try” and “redo” to give you a chance to build your
mathematical muscles. Last third: a lightening of the overall workload, with
optional continued homeworks or an optional individual or team project such as
working through some research paper(s) or book sections that require some
mathematical tools, in consultation with the instructors. In the last part,
there will be more emphasis on readings and discussions concerning linguistic
applications: attendance and participation will become relatively obligatory in
that last part, but written work substantially lighter. No written homework in
December.

A note for those concerned about workload: if you do your homework more or less regularly, then in the last several weeks you don’t have to do any more – you can spend more time at the end of the semester on courses that need a term paper. But you can also have the option of doing less of the regular homework and doing some project – this choice would be good for those who already have some math background. And most of the homeworks themselves will include choices in which problems to do, so that you can find problems that suit your level.