Introduction to Computational Linguistics

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3. Welcome To The Typed Universe 8 The result is an integer, and its value is 89. You can give the function f any term that evaluates to an integer. It may contain function symbols, but they must be defined already. Now, the functions * and + are actually predefined. You can look them up in the manual. It tells you that their type is int -> int -> int. Thus, they are functions from integers to functions from integers to integers. Hence (2 * x) is an integer, because x is an integer and 2 is. Likewise, (2 * x) + 3 is an integer. Likewise, if α and β are types, so is α•β. This is the product type. It contains the set of pairs 〈x, y〉 such that x is type α and y of type β: (17) M α•β = M α × M β = {〈x, y〉 : x ∈ M α , y ∈ M β } Like function types, product types do not have to be officially created to be used. OCaML’s own notation is ∗ in place of •, everything else is the same. Type in, for example, (’a’, 7) and this will be your response: (18) - : char * int = (’a’, 7);; This means that OCaML has understood that your object is composed of two parts, the left hand part, which is a character, and the right hand part, which is a number. It is possible to explicitly define such a type. This comes in the form of a type declaration: (19) # type prod = int * int;; type prod = int * int The declaration just binds the string prod to the type of pairs of integers, called the product type int * int by OCaML. Since there is nothing more to say, OCaML just repeats what you have said. It means it has understood you and prod is now reserved for pairs of integers. However, such explicit definition hardly makes sense for types that can be inferred anyway. For if you issue let h = (3,4) then OCaML will respond val h : int * int = (3,4), saying that h has the product type. Notice that OCaML will not say that h is of type prod, even if that follows from the definition. There is often a need to access the first and second components of a pair. For that there exist functions fst and snd. These functions are polymorphic. For every product type α • β, they are defined and fst maps the element into its first component, and snd onto its second component.
3. Welcome To The Typed Universe 9 An alternative to the pair constructor is the constructor for records. The latter is very useful. It is most instructive to look at a particular example: (20) type car = {brand : string; vintage : int; used : bool};; This defines a type structure called car, which has three components: a brand, a vintage and a usedness value. On the face of it, a record can be replicaed with products. However, notice that records can have any number of entries, while a pair must consist of exactly two. So, the record type car can be replicated by either string * (int * bool) or by (string * int) * bool. But there are also other differences. One is that the order in which you give the arguments is irrelevant. The other is that the names of the projections can be defined by yourself. For example, you can declare that mycar is a car, by issuing (21) # let mycar = {brand = "honda"; vintage = 1977;} used = false};; This will bind mycar to the element of type car. Moreover, the expression mycar.brand has the value honda. (To communicate that, it has to be enclosed in quotes, of course. Otherwise it is mistaken for an identifier.) It is not legal to omit the specification of some of the values. Try this, for example: (22) # let mycar = {brand = "honda"};; Some record field labels are undefined: vintage used;; This is not a warning: the object named ”mycar” has not been defined. (23) # mycar.brand;; Unbound value mycar If you look carefully, you will see that OCaML has many more type constructors, some of which are quite intricate. One is the disjunction, written |. We shall use the more suggestive ∪. We have (24) M α∪β = M α ∪ M β Notice that by definition objects of type α are also objects of type M α∪β . Hence what we said before about types keeping everything distinct is not accurate. In
Page 1 and 2: Introduction to Computational Lingu
Page 3 and 4: 2. Practical Remarks Concerning OCa
Page 5 and 6: 3. Welcome To The Typed Universe 5
Page 7: 3. Welcome To The Typed Universe 7
Page 11 and 12: 4. Function Definitions 11 4 Functi
Page 13 and 14: 5. Modules 13 can find out why the
Page 15 and 16: 5. Modules 15 let length l = length
Page 17 and 18: 6. Sets and Functors 17 write it do
Page 19 and 20: 7. Hash Tables 19 to actually see t
Page 21 and 22: 8. Combinators 21 can be used on an
Page 23 and 24: 9. Objects and Methods 23 of type
Page 25 and 26: 10. Characters, Strings and Regular
Page 31 and 32: 11. Interlude: Regular Expressions
Page 37 and 38: 12. Finite State Automata 37 Suppos
Page 39 and 40: 12. Finite State Automata 39 There
Page 41 and 42: 12. Finite State Automata 41 The pr
Page 43 and 44: 12. Finite State Automata 43 Proof.
Page 45 and 46: 13. Complexity and Minimal Automata
Page 51 and 52: 14. Digression: Time Complexity 51
Page 53 and 54: 14. Digression: Time Complexity 53
Page 55 and 56: 15. Finite State Transducers 55 Con
Page 57 and 58: 15. Finite State Transducers 57 And
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16. Finite State Morphology 59 Then
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17. Using Finite State Transducers
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18. Context Free Grammars 67 18 Con
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18. Context Free Grammars 69 symbol
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18. Context Free Grammars 71 which
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19. Parsing and Recognition 73 Z
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19. Parsing and Recognition 75 numb
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20. Greibach Normal Form 77 The red
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20. Greibach Normal Form 79 Definit
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20. Greibach Normal Form 81 Proposi
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20. Greibach Normal Form 83 nonterm
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21. Pushdown Automata 85 pushdown.
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21. Pushdown Automata 87 machine is
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22. Shift-Reduce-Parsing 89 differe
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23. Some Metatheorems 91 If the loo
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23. Some Metatheorems 93 This proof
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23. Some Metatheorems 95 a decompos
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Introduction to Computational Linguistics

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