1. Types Type = Why? a set of values
operations that are allowed on these values.
Why?
To generate better code, with less runtime overhead
To avoid runtime errors
To improve expressiveness (see overloading)

2. Type Type system for a programming language = Actions set of types AND
rules that specify how a typed program is allowed to behave
Actions
Type checking
Given an operation and an operand of some type, determine whether the operation is allowed on that operand
Type inference
Given the type of operands, determine
the meaning of the operation
the type of the operation
OR, without variable declarations, infer type from the way the variable is used

3. Issues in typing Does the language have a type system?
Untyped languages (e.g. assembly) have no type system at all
When is typing performed?
Static typing: At compile time
Dynamic typing: At runtime
How strictly are the rules enforced?
Strongly typed: No exceptions
Weakly typed: With well-defined exceptions
Type equivalence & subtyping
When are two types equivalent?
What does "equivalent" mean anyway?
When can one type replace another?

4. Components of a type system
Built-in types
Rules for constructing new types
Where do we store type information?
Rules for determining if two types are equivalent
Rules for inferring the types of expressions

5. Component: Built-in types
These are the basic types. Usually,
integer
usual operations: standard arithmetic
floating point
character
character set generally ordered lexicographically
usual operations: (lexicographic) comparisons
boolean
usual operations: not, and, or, xor

6. Component: type constructors
Arrays
array(I,T) denotes the type of an array with elements of type T, and index set I
multidimensional arrays are just arrays where T is also an array
operations: element access, array assignment, products
Strings
bitstrings, character strings
operations: concatenation, lexicographic comparison
Products
Groups of multiple objects of different types
essentially, Cartesian product of types (useful in functions later)
Records (structs)
Groups of multiple objects of different types where the elements are given specific names.
How is a recursive type definition handled?

7. Component: type constructors
Pointers
addresses
operations: arithmetic, dereferencing, referencing
issue: equivalency
Function types
A function such as "int add(real, int)" has type realintint

8. Component: type equivalence
Name equivalence
Two types are equivalent only when they have the same name.
Why?
Loose vs. strict
Structural equivalence
Two types are equivalent when they have the same structure
Should records require member name identity to be equivalent?
Does the order of record fields matter?
How about Array(int, 1..10) and Array(int, 1..20). Should we consider the bounds or just the element type?
How about recursively defined types?
Example
C uses structural equivalence for structs and name equivalence for arrays/pointers.

9. Component: type equivalence
Type coercion
If x is float, is x=3 acceptable?
Disallow
Allow and implicitly convert 3 to float.
"Allow" but require programmer to explicitly convert 3 to float
How do we convert?
Reinterpret bit sequence
Build new object
What should be allowed?
float to int ?
int to float ?
What if both types are equally general?
What if multiple coercions are possible?
Consider 3 + "4" in a language that can convert strings to integers.

10. Overloading
Same operation name but different effect on different types
For an overloaded function f
resolve arguments
look up f in symbol table to get list of all visible "versions"
ignore those with wrong parameter types.

11. A simple type checker
We can use an attribute grammar to implement a simple type checker.
Enum {E.type = integer}
EE1+E2 {E.type = (E1.type == integer && E2.type == integer)?
integer : error}
EE1[E2] {E.type = (E1.type == array(int, T) && E2.type == integer)?
T : error}
E*E1 {E.type = (E1.type == pointer(T))? T : error}
EE1(E2) {E.type = (E1.type == (ST) && E2.type == S)? T : error
S if E then S1 {S.type = (E.type == boolean)? void : error}

12. Inference rules
The formal notation for type checking/inference is rules of inference
They have the form:
If we can prove the hypotheses are all true in the existing environment, then the conclusion is also true.
For example: In English: If expressions E1 and E2 both have type int, then E1+E2 also has type int.
Environment Hypotheses
Environment Conclusions
A E1: int A E2: int
A E1 + E2 : int

13. Inference rules
The inference rules give us templates that describe how to type various expressions.
We can use the templates to prove that an expression has a valid type (and find what that type is)
The construct parallels the parse tree.
Short example:
Type the expression x+a[i] under the assumption that x is an char, a[i] is an array of chars and i is an int.
The environment is A={x:char, a:array(int, char), i:char}
A E1: array(T,S) , A E2: T
A E1: char , A E2: char
[ARRAY]
[ADD]
A E1 + E2 : char
A E1[E2] : S

14. Inference rules A |– a: array(int,char) A |– i:int A |– x:char
A |– a[i]: char
[ADD]
A |– x+a[i]: char

15. Types Where do we store type information? The symbol table
In general, the symbol table contains information about
variables
functions
class names
type names
temporary variables
etc.
Do we need a separate symbol table for types?

16. Symbol Tables
What kind of information is usually stored in a symbol table?
type
storage class
size
scope
stack frame offset
register

17. Symbol Tables How is a symbol table implemented? array
simple, but linear LookUp time
However, we may use a sorted array for reserved words, since they are generally few and known in advance.
tree
O(lgn) lookup time if kept balanced
hash table
most common implementation
O(1) LookUp time

18. Revisiting hash tables (cs311)
use array of size m to store elements
given key k (the identifier name), use a function h to compute index h(k) for that key
collisions are possible
two keys hash into the same slot.
Hash functions
A good hash function
is easy to compute
avoids collisions (by breaking up patterns in the keys and uniformly distributing the hash values)

19. Revisiting hash tables (cs311)
Hash functions
A common hash function is h(k) = m*(k*c- k*c), for some constant 0<c<1
In English
multiply the key k by the constant c
Take the fractional part of k*c
Multiply that by size m
Take the floor of the result
A good value for c:

20. Revisiting hash tables (cs311)
Different elements may still hash into the same slot. How do we resolve collisions?
Chaining
Put all the elements that collide in a chain (list) attached to the slot.
Insert/Delete/Lookup in expected O(1) time
However, this assumes that the chains are kept small.
If the chains start becoming too long, the table must be enlarged and all the keys rehashed.

21. Revisiting hash tables (cs311)
Different elements may still hash into the same slot. How do we resolve collisions?
Open addressing
Store all elements within the table
The space we save from the chain pointers is used up to make the array larger.
If there is a collision, probe the table in a systematic way to find an empty slot.
If the table fills up, we need to enlarge it and rehash all the keys.
Open addressing with linear probing
Probe the slots in a linear manner
Simple but Bad: results in clustering (long sequences of used slots build up very fast)

22. Revisiting hash tables (cs311)
Open addressing with double hashing
Use a second hash function.
The probe sequence is: (h(k) + i*h2(k) ) mod m, with i=0, 1, 2, ...
Good performance
Since we use a second function, keys that originally collide will subsequently have different probe sequences.
No clustering
A good choice for h2(k) is p-(k mod p) where p is a prime less than m

23. Scope issues Declaration before use promotes one-pass compiling
Block-structured languages allow nested name scopes.
Usual visibility rules
Only names created in the current or enclosing scopes are visible
When there is a conflict, the innermost declaration takes precedence.
What about "same-level" declarations?

24. Scope issues
One idea is to have a global symbol table and save the scope information for each entry.
When an identifier goes out of scope, scan the table and remove the corresponding entries
We may even link all same-scope entries together for easier removal.
Careful: deleting from a hash table that uses open addressing is tricky
We must mark a slot as Deleted, rather than Empty, otherwise later LookUp operations may fail.
Alternatively, we can maintain a separate, local table for each scope.

25. Structure tables Where should we store struct field names?
Separate mini symbol table for each struct
Conceptually easy
Separate table for all struct field names
We need to somehow uniquely map each name to its structure (e.g. by concatenating the field name with the struct name)
No special storage
struct field names are stored in the regular symbol table.
Again we need to be able to map each name to its structure.

26. Static vs. Dynamic scope
What we've seen so far is static scope: Scoping follows the structure of the program
An alternative is dynamic scope, where scoping follows the execution path.
Example:
int i = 1;
void func() {
cout << i << endl; }
int main () {
int i = 2;
func();
return 0;
If C++ used dynamic
scoping, this would
print out 2, not 1.