In the course of this blog I am going to use a variant of Eiffel called

“Modern Eiffel”. This blog entry gives a short outline of the language

elements. Although “Modern Eiffel” is a full blown language with all

sorts of imperative elements, mutable reference type objects, multiple

inheritance, dynamic bind etc. this blog entry concentrates only on a

small subset: immutable types and inductive types. The reason:

immutable/inductive types are a very important concept to verify

statically the correctness of a program. More advanced features will be

introduced if necessary.

Skeleton of a class

Modern Eiffel is an object oriented language. All types are defined by

classes (as in other object oriented languages like C++, C#, java,

scala, etc.). The skeleton of a class looks like

class MY_CLASS -- class name create creator1(a:A) ... creator2 ... feature some_function(b:B): RT ... some_command(c:C) ... all(i:INT) some_property(i) proof ... end CHAR = CHARACTER -- type alias invariant ... -- consistency conditions end

Comments begin with “–” and span to the end of the line.

Classes and types are always written in uppercase. Each class can have

creators by which objects of the corresponding type can be constructed.

The features of a class are either functions (i.e. routines which return

a value) or commands (i.e. routines which change the state of the

object) or assertions or type aliases.

Basic types

There are some basic types. These are types with built-in functions and

built-in value representation. The basic types are BOOLEAN, NAT

(unsigned integers of a certain size), INT (signed integers of a certain

size). Beside their built-in functions and value representation the

basic types are nothing special. They are defined with classes as well.

immutable class BOOLEAN feature implication alias "=>" (o:BOOLEAN): BOOLEAN external "built_in" end negated alias "not": BOOLEAN external "built_in" end conjuncted alias "and" (o:BOOLEAN): BOOLEAN external "built_in" end disjuncted alias "or" (o:BOOLEAN): BOOLEAN external "built_in" end ... end

Functions can have operator aliases for unary and binary operators. I.e.

having variables a,b,c of type BOOLEAN the expressions

not a -- highest precedence a and b a or b a => b -- lowest precedence

have the expected meaning. The most important boolean operator is the

implication “=>”. It is right associative i.e. “a => b => c” is parsed

as “a => (b => c)”. The binary operators “and” and “or” are left

associative. All binary boolean operators are not strict (short

circuited) i.e. the second operand is only evaluated if necessary to

determine the truth value of the expression.

Default inheritance and equality

A class can inherit features from other classes. For this blog entry

inheritance plays a minor role. I just mention the fact that each class

has some implicit inheritance. A class T with no inheritance clause is

treated as if it were defined as

class T inherit COMPARABLE ... end

The parent class COMPARABLE defines equality. It has the outline

class COMPARABLE feature is_equal(o:like Current): BOOLEAN external "built_in" ensure all(x:like Current) x=x all(x,y: like Current) x=y => y=x all(x,y,z: like Current) x=y => y=z => x=z -- "=" has higher precedence than "=>" end end

The feature “is_equal” is used in Modern Eiffel to define equality. It

is a built-in function which guarantees properties which are expected

from an equivalence relation. Reflexivity (every entity is equal to

itself), symmetry and transitivity. In Modern Eiffel the postconditions

of built-in routines are treated as axioms, i.e. properties which do not

need any proof. Note the transitivity law written as

x=y => y=z => x=z

Usually one would expect the transitivity law written as

x=y and y=z => x=z

Both forms are equivalent. This will be proved later. If possible we

write all properties with the implication operator, because implication

is more powerful and practical for proofs.

Each class inheriting from COMPARABLE can redefine (i.e. overwrite) the

feature “is_equal”. The postconditions are inherited but in case of

redefinition they are no longer treated as axioms; they have to be

proved to be true.

Inductive types

Simple inductive types

People with experience in imperative languages like C++, C#, Java, etc.

are not acustomed to use inductive types. People having experience with

functional languages like Haskell, ML, OCAML, COQ, Isabelle know

inductive types well. This chapter is written for people not knowing

inductive types. The others might just be interested to learn how

inductive types are presented in Eiffel syntax.

Inductive data types are best explained with some examples. The simplest

inductive type is similar to an enumeration type in other languages. We

can define a class COLOR like

case class COLOR create red green blue end

The keyword “case” introduces an inductive class. The class COLOR has

just three creators to define objects of type color. E.g. we can use

this class to define variable of type COLOR.

r: COLOR := red g: COLOR := green ...

The assigment operator “:=” can be used to initialize variables.

In a similar fashion we can define a class WEEKDAY

case class WEEKDAY create monday thuesday wednesday ... sunday feature next: WEEKDAY ... end

The class WEEKDAY has a feature “next” in order to calculate the next

weekday. The definition of “next” looks like

next: WEEKDAY = inspect Current case monday then thuesday case thuesday then wednesday ... case sunday then monday end

Inductive types allow case expression to make distinctions on how the

object has been constructed. The keyword “Current” returns the current

object. For each creator a cause clause is given. The meaning should be

quite clear. It is like C’s select statement.

The above form is the short form of a function definition. The function

“next” does not have any precondition. Therefore the short form is

appropriate. In the case that a function has preconditions, the long

form must be used.

next: WEEKDAY require ... -- precondition(s) ensure Result = inspect Current case monday then thuesday ... end end

The keyword “Result” indicates the return value of the function. The

postcondition “Result = … ” specifies the return value of the function.

Inductive types with stored values

Objects of the very simple inductive types like COLOR and WEEKDAY just

store as its value the way they have been constructed. But objects of

inductive type can store other values as well.

Let us consider the following class

case class OPTIONAL[G] create none value(value: G) end

This class is generic. A generic class is a type constructor. It can

construct the types OPTIONAL[INT], OPTIONAL[BOOLEAN], OPTIONAL[COLOR],

OPTIONAL[OPTIONAL[CHARACTER]], …

The class has just two creators: “none” and “value”. The object

n: OPTIONAL[INT] := none

has no value and the object

i7: OPTIONAL[INT] := value(7)

has the value 7. The name “value” is used for the creator and for an

attribute. The attribute “value” is an optional attribute. It is

available only if the object has been created with the “value” creator.

The access of the attribute “value” is illegal, if the corresponding

object has not been created with the “value” creator. In many case the

attribute is not accessed directly. Usually the attribute is accessed

better through pattern matching expressions.

In order to introduce pattern matching we use a contrived example of a

function.

value_plus_100(o:OPTIONAL[INT]) = inspect o case none then -1 case value(i) then i+100 end

The function “value_plus_100 is contrived, because one would not write

such a function in Modern Eiffel. The inspect expression distinguishes

the two possible creators for an object of type OPTIONAL[INT].

The first case clause is for objects created with the “none” creator.

Since the “none” creator has no arguments, no value can be extracted in

that case.

The second case clause is for objects which have been created with the

“value” creator. This creator has one argument which can be retrieved

and attached to a fresh local variable with name “i”. This variable is

availalbe only on the right hand side of this case clause to calculate

the return value.

The function “value_plus_100” can be written in a form which accesses

the attribute “value” directly.

value_plus_100(o:OPTIONAL[INT]) = inspect o case none then -1 case value(_) then o.value + 100 end

The wild card “_” is used to supress the creations of a fresh local

variable. But the compiler derives the fact that in this case clause the

object “o” has been created with the “value” creator. Therefore the

access to the attribute “value” is valid within this clause.

Inductive types with recursive structure

The full expressive power of inductive types unfolds with recursively

defined inductive types. We can use an inductive type to define natural

numbers.

case class UNARY_NATURAL create zero succ(pred: UNARY_NATURAL) feature plus alias "+" (o:UNARY_NATURAL): UNARY_NATURAL = ... times alias "*" (o:UNARY_NATURAL): UNARY_NATURAL = ... power alias "^" (o:UNARY_NATURAL): UNARY_NATURAL = ... ... end

The class UNARY_NATURAL is highly inefficient to do arithmetics. But it

is a good vehicle to model natural numbers and use this model to prove

properties of very efficient implementations of natural numbers.

The class UNARY_NATURAL has tow creators. One to create the number 0 and

one to create the successor of an already created number.

The recursive structure of the type can be used to define the functions

“plus”, “times” and “power” recursively.

plus alias "+" (o:UNARY_NATURAL): UNARY_NATURAL = inspect Current case zero then o case succ(p) then succ(p+o) end times alias "*" (o:UNARY_NATURAL): UNARY_NATURAL = inspect Current case zero then zero case succ(p) then o + p*o end power alias "^" (o:UNARY_NATURAL): UNARY_NATURAL = inspect o case zero then succ(zero) case succ(p) then Current * Current^p end

Recursive functions for inductive types defined in this fashion have the

nice property that the compiler can verify the termination of the

recursion. In order to see this let us look at the recursive branch of

the definition of “+”

case succ(p) then succ(p+o)

This branch is entered only if the current object has been created with

the “succ” creator using an already existing number, the predecessor of

the current object. The left hand side of the case expression attaches

the predecessor of the current object to the fresh local variable “p”.

The right hand side

succ(p+o)

uses the predecessor to make the recursive call “p+o”, i.e. it calls the

function “plus” recursively. This recursive call will finally terminate

i.e. end up in a branch “case zero”, because the very first creator of

any UNARY_NATURAL number must have been the “zero” creator. This is the

only way to create a UNARY_NATURAL if no other UNARY_NATURAL is available.

Another classic inductive type is defined by the class LIST.

case class LIST[G] create nil cons alias "::" (first:G; tail:LIST[G]) feature ... end

A list is either empty or is an element (the first element) followed by

another list (the tail of the newly created list).

If a creator has two arguments, a binary operator can be used as an

alias. All binary operators ending with “::” are right associative.

Therefore the following initialization of variables is well defined

lst: LIST[INT] := 1::2::3::nil -- parsed as 1::(2::(3::nil)) -- equivalent to cons(1,cons(2,(cons(3,nil)))

Since lists are very expressive on its own and can serve as models an

extra blog entry will be dedicated to lists.

Summary

This blog has introduced some basic language elements of Modern Eiffel.

Some next blog entries will concetrate more on lists and on how to prove

assertions within Modern Eiffel.

Keep in mind that all sorts of mutable structures and imperative

elements are possible in Modern Eiffel. But in order to verify SW

immutable types and especially inductive types are very important.

Therefore this blog will first scrutinize immutable types.