Mathematical Methods in Linguistics

1. Mathematical Methods in Linguistics

2. Basic Concepts of Set Theory

3. What Is a Set? • An abstract collection of distinct object (its members) • Can have (almost) anything as a member, including other sets • May be small (even empty) or large (even infinite) FST - Torbjörn Lager, UU

4. Specification of Sets • List notation (enumeration) • Diagram • Predicate notation • Recursive rules • For an example, see page 9 in MML FST - Torbjörn Lager, UU

5. Identity and Cardinality • Identity • {Torbjörn Lager} = {x | x is the teacher in C389} • Cardinality • |A| means "the number of elements in the set A" FST - Torbjörn Lager, UU

6. The Member and Subset Relations • a  A means "a is a member of the set A" • A  B means "every element of A is also an element of B" • A  B means "every element of A is also an element of B and there is at least one element of B which is not in A" • a  B means a  B does not hold • A  B means A  B does not hold FST - Torbjörn Lager, UU

7. Powerset • The powerset of a set A is the set of all subsets of A • E.g the powerset of {a,b} is {{a,b},{a},{b},Ø} FST - Torbjörn Lager, UU

8. Union and Intersection • The union of two sets A and B, written A  B, is the set of all objects that are members of either the set A or the set B (or both) • The intersection (sv: "snittet") of two sets A and B, written A  B, is the set of all objects that are members of both the set A and the set B FST - Torbjörn Lager, UU

9. Difference and Complement • The difference between two sets A and B, written A-B, is all the elements of A which are not also elements of B • The complement of a set A and B, written A', is all the elements which are not in A • A complement of a set is always relative to a universe U. It also holds that A' = U-A FST - Torbjörn Lager, UU

10. Set Theoretic Equalities • See page 18 in MML FST - Torbjörn Lager, UU

11. Relations and Functions

12. Ordered Pairs and Cartesian Products • The Cartesian product (sv: "kryssprodukten") of A and B, written A  B, is the set of pairs <x,y> such that x is an element in A and y is an element in B FST - Torbjörn Lager, UU

13. Functions: Domain and Range Domain Range meta tro 4 få klo 5 ta 1 3 jul feg 6 se mat 2 be ful rop FST - Torbjörn Lager, UU

14. A Function • A set of pairs • Each element is in the domain is paired with just one element in the range • A subset of a Cartesian product A  B can be called a function just in case every member of A occurs exactly once a the first element in a pair FST - Torbjörn Lager, UU

15. Functions (cont'd) Domain Range meta tro 4 få klo 5 ta 3 jul feg 6 se mat 2 be ful rop FST - Torbjörn Lager, UU

16. Properties of Relations page 39-53 in MML this part is optional

17. Lecture 2:Logic and Formal Systems

18. Basic Concepts of Logic and Formal Systems FST - Torbjörn Lager, UU

19. Statement Logic FST - Torbjörn Lager, UU

20. Predicate Logic FST - Torbjörn Lager, UU

21. Lecture 3:Knowledge and Meaning Representation

22. Lecture 4:English as a Formal Language

23. Compositionality FST - Torbjörn Lager, UU

24. Lambda Abstraction FST - Torbjörn Lager, UU

25. Lecture 5:Finite Automata, Regular Languages and Type 3 Grammars

26. Lecture 6:Pushdown Automata, Context Free Grammars and Languages Lecture 6:Feature Structures and Equations

27. Lecture 7:Feature Structures and Unification-Based Grammars