Linguistics 726: Mathematical Linguistics

Barbara Partee and Vladimir Borschev
Fall 2006, University of Massachusetts, Amherst

Ling726 2006 home | description | homework | book errata | links | LING 726 2004 Website


Time and Place:  Tu, Th 1:00 – 2:15, Herter 640.

Instructors: Barbara Partee and Vladimir Borschev

Added attraction: With guest lectures by Chris Potts, Rajesh Bhatt and possibly others.

Required text:      Partee, ter Meulen and Wall (1990 - Corrected first edition, 1993 or later), Mathematical Methods in Linguistics, Dordrecht: Kluwer: Student edition (paperback). Available at discounted price from Partee. Other readings will be xeroxed and/or made available for download.

Office Hours and Contact Information:

Barbara H. Partee: by appointment; a regular time could be set up by arrangement.

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

Office: So. College 317;  545-0885

Home phone: 549-4501


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.)

       This course was offered in Fall 2001 and 2004; the syllabus, lecture notes, homeworks, and answers to homeworks from those years are all on the web: you can find the site through Partee’s website by going to her ‘teaching’ page. The Fall 2006 course will be similar, with some differences mostly in the choices of linguistic applications to discuss, and mostly in the second half of the course.

       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 some 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.) Some examples of simple algebras will include semilattices, lattices, and Boolean algebras: in the second part of the course we may consider applications to ‘feature algebras’, Link’s semantics for mass and plural, event algebras, type-shifting, and/or 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.

Basic plan: Cover selected main topics from (Partee et al. 1993): set theory, logic, algebra, grammars and automata, with somewhat more algebra than is that book, and bring in “enrichment topics” of current interest that presuppose some mathematical background. See the syllabi from Ling 726 in 2004 and Ling 726 in 2001 for precedents; the basics will be similar, with most of the variation in the enrichment topics. If students have particular requests, we’ll try to accommodate them.  Note: If you don’t already own that textbook, you can buy it from me at author’s discount price.

       The second goal of the course, to be pursued throughout, but probably more in the second half, will be to explore linguistic applications of these basic notions and to work on more specific mathematical and logical tools that may be needed/useful in particular linguistic research paradigms.

       The course has no specific prerequisites, but it would be good to have had either a logic course or Ling 409 or some other math course(s) before. The pace and workload will be more demanding than that of Linguistics 409, which will next be offered in Fall 2007. Students who would benefit from starting out with a slightly slower-paced course with more emphasis on the fundamentals should take 409 and/or a logic course before taking 726. If in doubt, consult instructor.

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 new obligatory written homework after November 14. 

      A note for those concerned about workload: if you do your homework regularly and do "redos" promptly, 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 a 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.