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Types. Type = 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). Type. Type system for a programming language = set of types AND

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  • Type =
    • 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)
  • Type system for a programming language =
    • 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
issues in typing
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?
components of a type system
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
component built in types
Component: Built-in types
  • These are the basic types. Usually,
  • integer
    • usual operations: standard arithmetic
  • floating point
    • usual operations: standard arithmetic
  • character
    • character set generally ordered lexicographically
    • usual operations: (lexicographic) comparisons
  • boolean
    • usual operations: not, and, or, xor
component type constructors
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?
component type constructors1
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
component type equivalence
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.
component type equivalence1
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.
  • 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.
a simple type checker
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}

inference rules
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

inference rules1
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



A E1 + E2 : char

A E1[E2] : S

inference rules2
Inference rules

A |– a: array(int,char)

A |– i:int

A |– x:char

A |– a[i]: char


A |– x+a[i]: char

  • 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?
symbol tables
Symbol Tables
  • What kind of information is usually stored in a symbol table?
    • type
    • storage class
    • size
    • scope
    • stack frame offset
    • register
symbol tables1
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
revisiting hash tables cs311
Revisiting hash tables (cs311)
  • Hash tables
    • 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)
revisiting hash tables cs3111
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:
revisiting hash tables cs3112
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.
revisiting hash tables cs3113
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)
revisiting hash tables cs3114
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
scope issues
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?
scope issues1
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.
structure tables
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.
static vs dynamic scope
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;


return 0;


If C++ used dynamic

scoping, this would

print out 2, not 1.