Principles of Object-Oriented Software Development

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Abstract data types

The history of programming languages may be characterized as the genesis of increasingly powerful abstractions to aid the development of reliable programs.

Abstract data types

abstraction and types
algebraic specification
modules versus classes
types as constraints

Additional keywords and phrases: control abstractions, data abstractions, compiler support, description systems, behavioral specification, implementation specification

slide: Abstract data types

In this chapter we will look at the notion of abstract data types, which may be regarded as an essential constituent of object-oriented modeling. In particular, we will study the notion of data abstraction from a foundational perspective, that is based on a mathematical description of types. We start this chapter by discussing the notion of types as constraints. Then, we look at the (first order) algebraic specification of abstract data types, and we explore the trade-offs between the traditional implementation of abstract data types by employing modules and the object-oriented approach employing classes. We conclude this chapter by exploring the distinction between classes and types, as a preparation for the treatment of (higher order) polymorphic type theories for object types and inheritance in the next chapter.

Abstraction and types

subsections:

Abstraction in programming languages
Foundational perspectives -- types as constraints
Objectives of typed OOP

The concern for abstraction may be regarded as the driving force behind the development of programming languages (of which there are astoundingly many). In the following we will discuss the role of abstraction in programming, and especially the importance of types. We then briefly look at what mathematical means we have available to describe types from a foundational perspective and what we may (and may not) expect from types in object-oriented programming.

Abstraction in programming languages

In [Shaw84], an overview is given of how increasingly powerful abstraction mechanisms have shaped the programming languages we use today. See slide 8-abstraction.

Abstraction -- programming methodology

control abstractions -- structured programming
data abstraction -- information hiding

The kind of abstraction provided by ADTs can be supported by any language with a procedure call mechanism (given that appropriate protocols are developed and observed by the programmer). [DT88]

slide: Abstraction and programming languages

Roughly, we may distinguish between two categories of abstractions: abstractions that aid in specifying control (including subroutines, procedures, if-then-else constructs, while-constructs, in short the constructs promoted by the school of structured programming in their battle against the goto); and abstractions that allow us to hide the actual representation of the data employed in a program (introduced to support the information hiding approach, originally advocated in [Parnas72a]). Although there is clearly a pragmatic interest involved in developing and employing such abstractions, the concern with abstraction (and consequently types) is ultimately motivated by a concern with programming methodology and, as observed in [DT88], the need for reliable and maintainable software. However, the introduction of language features is also often motivated by programmers' desires for ease of coding and naturalness of expression. In the same vein, although types were originally considered as a convenient means to assist the compiler in producing efficient code, types have rapidly been recognized as a way in which to capture the meaning of a program in an implementation independent way. In particular, the notion of abstract data types (which has, so to speak, grown out of data abstraction) has become a powerful device (and guideline) to structure large software systems. In practice, as the quotation from [DT88] in slide 8-abstraction indicates, we may employ the tools developed for structured programming to realize abstract data types in a program, but with the obvious disadvantage that we must rely on conventions with regard to the reliability of these realizations. Support for abstract data types (support in the sense as discussed in section paradigms) is offered (to some extent) by languages such as Modula-2 and Ada by means of a syntactic module or package construct, and (to a larger extent) by object-oriented languages in the form of object classes. However, both realizations are of a rather ad hoc and pragmatic nature, relying in the latter case on the metaphor of encapsulation and message passing. The challenge to computer science in this area is to develop a notion of types capturing the power of abstract data types in a form that is adequate both from a pragmatic point of view (in the sense of allowing efficient language support) and from a theoretical perspective (laying the foundation for a truly declarative object-oriented approach to programming).

A foundational perspective -- types as constraints

Object-oriented programming may be regarded as a declarative method of programming, in the sense that it provides a computation model (expressed by the metaphor of encapsulation and message passing) that is independent of a particular implementation model. In particular, the inheritance subtype relation may be regarded as a pure description of the relations between the entities represented by the classes. Moreover, an object-oriented approach favors the development of an object model that bears close resemblance to the entities and their relations living in the application domain. However, the object-oriented programming model is rarely introduced with the mathematical precision characteristic of descriptions of the other declarative styles, for example the functional and logic programming model. Criticizing, [DT88] remark that {\em OOP is generally expressed in philosophical terms, resulting in a proliferation of opinions concerning what OOP really is}. From a type-theoretical perspective, our interest is to identify abstract data types as elements of some semantic (read mathematical) domain and to characterize their properties in an unambiguous fashion. See slide 8-mathematical.

Abstract data types -- foundational perspective

unambiguous values in some semantic domain

Mathematical models -- types as constraints

algebra -- set oriented
second order lambda calculus -- polymorphic types
constructive mathematics -- formulas as types

slide: Mathematical models for types

There seems to be almost no limit to the variety and sophistication of the mathematical models proposed to characterize abstract data types and inheritance. We may make a distinction between first order approaches (based on ordinary set theory) and higher order approaches (involving typed lambda calculus and constructive logic). The algebraic approach is a quite well-established method for the formal specification of abstract data types. A type (or sort) in an algebra corresponds to a set of elements upon which the operations of the algebra are defined. In the next section, we will look at how equations may be used to characterize the behavioral aspects of an abstract data type modeled by an algebra. Second order lambda calculus has been used to model information hiding and the polymorphism supported by inheritance and templates. In the next chapter we will study this approach in more detail. In both approaches, the meaning of a type is (ultimately) a set of elements satisfying certain restrictions. However, in a more abstract fashion, we may regard a type as specifying a constraint. The better we specify the constraint, the more tightly the corresponding set of elements will be defined (and hence the smaller the set). A natural consequence of the idea of types as constraints is to characterize types by means of logical formulas. This is the approach taken by type theories based on constructive logic, in which the notion of formulas as types plays an important role. Although we will not study type theories based on constructive logic explicitly, our point of view is essentially to regard types as constraints, ranging from purely syntactical constraints (as expressed in a signature) to semantic constraints (as may be expressed in contracts). From the perspective of types as constraints, a typing system may contribute to a language framework guiding a system designer's conceptualization and supporting the verification (based on the formal properties of the types employed) of the consistency of the descriptive information provided by the program. Such an approach is to be preferred (both from a pragmatic and theoretical point of view) to an ad hoc approach employing special annotations and support mechanisms, since these may become quite complicated and easily lead to unexpected interactions.

Formal models

There is a wide variety of formal models available in the literature. These include algebraic models (to characterize the meaning of abstract data types), models based on the lambda-calculus and its extensions (which are primarily used for a type theoretical analysis of object-oriented language constructs), algebraic process calculi (which may be used to characterize the behavior of concurrent objects), operational and denotational semantic models (to capture structural and behavioral properties of programs), and various specification languages based on first or higher-order logics (which may be used to specify the desired behavior of collections of objects). We will limit ourselves to studying algebraic models capturing the properties of abstract data types and objects (section algebra), type calculi based on typed extensions of the lambda calculus capturing the various flavors of polymorphism and subtyping (sections flavors--self-reference), and an operational semantic model characterizing the behavior of objects sending messages (section behavior). Both the algebraic and type theoretical models are primarily intended to clarify the means we have to express the desired behavior of objects and the restrictions that must be adhered to when defining objects and their relations. The operational characterization of object behavior, on the other hand, is intended to give a more precise characterization of the notion of state and state changes underlying the verification of object behavior by means of assertion logics. Despite the numerous models introduced there are still numerous approaches not covered here. One approach worth mentioning is the work based on the pi-calculus. The pi-calculus is an extension of algebraic process calculi that allow for communication via named channels. Moreover, the pi-calculus allows for a notion of migration and the creation and renaming of channels. A semantics of object-based languages based on the pi-calculus is given in [Walker90]. However, this semantics does not cover inheritance or subtyping. A higher-order object-oriented programming language based on the pi-calculus is presented in [PRT93]. Another approach of interest, also based on process calculi, is the object calculus (OC) described in [Nier93]. OC allows for modeling the operational semantics of concurrent objects. It merges the notions of agents, as used in process calculi, with the notion of functions, as present in the lambda calculus. For alternative models the reader may look in the {\tt comp.theory} newsgroup to which information concerning formal calculi for OOP is posted by Tom Mens of the Free University, Brussels.

Objectives of typed OOP

Before losing ourselves in the details of mathematical models of types, we must reflect on what we may expect from a type system and what not (at least not currently). From a theoretical perspective our ideal is, in the words of [DT88], to arrive at a simple type theory that provides a consistent and flexible framework for system descriptions (in order to provide the programmer with sufficient descriptive power and to aid the construction of useful and understandable software, while allowing the efficient utilization of the underlying hardware).

Objectives of typed OOP -- system description

packaging in a coherent manner
flexible style of associating operations with objects
inheritance of description components -- reuse, understanding
separation of specification and implementation
explicit typing to guide binding decisions

slide: Object orientation and types

The question now is, what support does a typing system provide in this respect. In slide 8-objectives, a list is given of aspects in which a typing system may be of help. One important benefit of regarding ADTs as real types is that realizations of ADTs become so-called first class citizens, which means that they may be treated as any other value in the language, for instance being passed as a parameter. In contrast, syntactic solutions (such as the module of Modula-2 and the package of Ada) do not allow this. Pragmatically, the objective of a type system is (and has been) the prevention of errors. However, if the type system lacks expressivity, adequate control for errors may result in becoming over-restrictive. In general, the more expressive the type system the better the support that the compiler may offer. In this respect, associating constructors with types may help in relieving the programmer from dealing with simple but necessary tasks such as the initialization of complex structures. Objects, in contrast to modules or packages, allow for the automatic (compiler supported) initializations of instances of (abstract) data types, providing the programmer with relief from an error-prone routine. Another area in which a type system may make the life of a programmer easier concerns the association of operations with objects. A polymorphic type system is needed to understand the automatic dispatching for virtual functions and the opportunity of overloading functions, which are useful mechanisms to control the complexity of a program, provided they are well understood. Reuse and understanding are promoted by allowing inheritance and refinement of description components. (As remarked earlier, inheritance and refinement may be regarded as the essential contribution of object-oriented programming to the practice of software development.) It goes without saying that such reuse needs a firm semantical basis in order to achieve the goal of reliable and maintainable software. Another important issue for which a powerful type system can provide support is the separation of specification and implementation. Naturally, we expect our type system to support type-safe separate compilation. But in addition, we may think of allowing multiple implementations of a single (abstract type) specification. Explicit typing may then be of help in choosing the right binding when the program is actually executed. For instance in a parallel environment, behavior may be realized in a number of ways that differ in the degree to which they affect locality of access and how they affect, for example, load balancing. With an eye to the future, these are problems that may be solved with a good type system (and accompanying compiler). One of the desiderata for a type system for OOP, laid down in [DT88], is the separation of a behavioral hierarchy (specifying the behavior of a type in an abstract sense) and an implementation hierarchy (specifying the actual realization of that behavior). Separation is needed to accommodate the need for multiple realizations and to resolve the tension between subtyping and inheritance (a tension we have already noted in sections theme and contracts).

Remark

In these chapters we cannot hope to do more than get acquainted with the material needed to understand the problems involved in developing a type system for object-oriented programming. For an alternative approach, see [Palsberg94].

Algebraic specification

Algebraic specification techniques have been developed as a means to specify the design of complex software systems in a formal way. The algebraic approach has been motivated by the notion of information hiding put forward in [Parnas72a] and the ideas concerning abstraction expressed in [Ho72]. Historically, the ADJ-group (see Goguen et al., 1978) provided a significant impetus to the algebraic approach by showing that abstract data types may be interpreted as (many sorted) algebras. (In the context of algebraic specifications the notion of sorts has the same meaning as types. We will, however, generally speak of types.) As an example of an algebraic specification, look at the module defining the data type Bool, as given in slide 8-Bool.

Algebraic specification -- ADT

Bool


  adt bool is
  functions
    true : bool
    false : bool
    and, or : bool * bool -> bool
    not : bool -> bool
  axioms
    [B1]  and(true,x) = x
    [B2]  and(false,x) = false
    [B3]  not(true) = false
    [B4]  not(false) = true
    [B5]  or(x,y) = not(and(not(x),not(y)))
  end

slide: The ADT Bool

In this specification two constants are introduced (the zero-ary functions true and false), three functions (respectively and, or and not). The or function is defined by employing not and and, according to a well-known logical law. These functions may all be considered to be (strictly) related to the type bool. Equations are used to specify the desired characteristics of elements of type bool. Obviously, this specification may mathematically be interpreted as (simply) a boolean algebra.

Mathematical models

The mathematical framework of algebras allows for a direct characterization of the behavioral aspects of abstract data types by means of equations, provided the specification is consistent. Operationally, this allows for the execution of such specifications by means of term rewriting, provided that some (technical) constraints are met. The model-theoretic semantics of algebraic specifications centers around the notion of initial algebras, which gives us the preferred model of a specification. To characterize the behavior of objects (that may modify their state) in an algebraic way, we need to extend the basic framework of initial algebra models either by allowing so-called multiple world semantics or by making a distinction between hidden and observable sorts (resulting in the notion of an object as an abstract machine). As a remark, in our treatment we obviously cannot avoid the use of some logico-mathematical formalism. If needed, the concepts introduced will be explained on the fly. Where this does not suffice, the interested reader is referred to any standard textbook on mathematical logic for further details.

Signatures -- generators and observers

Abstract data types may be considered as modules specifying the values and functions belonging to the type. In [Dahl92], a type T is characterized as a tuple specifying the set of elements constituting the type T and the collection of functions related to the type T. Since constants may be regarded as zero-ary functions (having no arguments), we will speak of a signature $%S$ or $%S_T$ defining a particular type T. Also, in accord with common parlance, we will speak of the sorts $s \e %S$, which are the sorts (or types) occurring in the declaration of the functions in $%S$. See slide 8-signature.

Signature -- names and profiles

$%S$

$ f : s_1 \* ... \* s_n -> s $

Functions -- for $T$

constants -- $c : -> T$
producers -- $g : s_1 \* ... \* s_n -> T$
observers -- $f : T -> s_i$

Type -- generators

$ %S_T = P_T \cup O_T$, $C_T \subset P_T$, $P_T \cap O_T = \emptyset$

slide: Algebraic specification

A signature specifies the names and (function) profiles of the constants and functions of a data type. In general, the profile of a function is specified as

$ f : s_1 \* ... \* s_n -> s $

where $ s_i (i=1..n) $ are the sorts defining the domain (that is the types of the arguments) of the function f, and s is the sort defining the codomain (or result type) of f. In the case $ n=0 $ the function f may be regarded as a constant. More generally, when $ s_1,...,s_n $ are all unrelated to the type T being defined, we may regard f as a relative constant. Relative constants are values that are assumed to be defined in the context where the specification is being employed. The functions related to a data type T may be discriminated according to their role in defining T. We distinguish between producers $ g \e P_T $, that have the type T under definition as their result type, and observers $f \e O_T$, that have T as their argument type and deliver a result of a type different from T. In other words, producer functions define how elements of T may be constructed. (In the literature one often speaks of constructors, but we avoid this term because it already has a precisely defined meaning in the object-oriented programming language C++.) In contrast, observer functions do not produce values of T, but give instead information on some particular aspect of T. The signature $ %S_T $ of a type T is uniquely defined by the union of producer functions $ P_T $ and observer functions $ O_T $. Constants of type T are regarded as a subset of the producer functions $P_T$ defining T. Further, we require that the collection of producers is disjoint from the collection of observers for T, that is $P_T \cap O_T = \emptyset$.

Generators

The producer functions actually defining the values of a data type T are called the generator basis of T, or generators of T. The generators of T may be used to enumerate the elements of T, resulting in the collection of T values that is called the generator universe in [Dahl92]. See slide 8-basis.

Generators -- values of $T$

generator basis -- $G_T = { g \e P_T }$
generator universe -- $GU_T = { v_1, v_2, ... }$

Examples

$G_{Bool} = { t, f }$, $GU_{Bool} = { t, f }$
$G_{Nat} = { 0, S }$, $GU_{Nat} = { 0, S 0, SS 0, ... }$
$G_{Set_{A} } = { \emptyset, add }$, $GU_{Set_{A} = { \emptyset, add(\emptyset,a), ... }$

slide: Generators -- basis and universe

The generator universe of a type T consists of the closed (that is variable-free) terms that may be constructed using either constants or producer functions of T. As an example, consider the data type Bool with generators t and f. Obviously, the value domain of Bool, the generator universe $GU_{Bool}$ consists only of the values t and f. As another example, consider the data type Nat (representing the natural numbers) with generator basis $G_{Nat} = { 0, S }$, consisting of the constant 0 and the successor function $S : Nat -> Nat$ (that delivers the successor of its argument). The terms that may be constructed by $G_{Nat}$ is the set $GU_{Nat} = { 0, S 0, SS 0, ... }$, which uniquely corresponds to the natural numbers ${ 0, 1, 2, ... }$. (More precisely, the natural numbers are isomorphic with $GU_{Nat}$.) In contrast, given a type A with element a, b, ..., the generators of $Set_{A}$ result in a universe that contains terms such as $add(\emptyset,a)$ and $add(add(\emptyset,a),a)$ which we would like to identify, based on our conception of a set as containing only one exemplar of a particular value. To effect this we need additional equations imposing constraints expressing what we consider as the desired shape (or normal form) of the values contained in the universe of T. However, before we look at how to extend a signature $%S$ defining T with equations defining the (behavioral) properties of T we will look at another example illustrating how the choice of a generator basis may affect the structure of the value domain of a data type. In the example presented in slide 8-Seq, the profiles are given of the functions that may occur in the signature specifying sequences. (The notation _ is used to indicate parameter positions.)

Sequences

$ %e : seq T$
empty
$ _ |> _ : seq T \* T -> seq T$
right append
$ _ <| _ : T \* seq T -> seq T$
left append
$ _ \. _ : seq T \* seq T -> seq T$
concatenation
$<< _ >> : T -> seq T$
lifting
$<< _,...,_ >> : T^{n} -> seq T$
multiple arguments

Generator basis -- preferably one-to-one

$G_{seq T } = { %e, |> }$, $GU_{seq T } = { %e, %e |> a, %e |> b, ..., %e |> a |> b, ... }$

$G'_{seq T } = { %e, <| }$, $GU'_{seq T } = { %e, a <| %e , b <| %e, ..., b <| a <| %e, ... }$

$G''_{seq T } = { %e, \. , << _ >> }$, $GU''_{seq T } = { %e, <>, <>, ,..., %e \. %e, ..., %e \. <> , ... }$

Infinite generator basis

$G'''_{seq T } = { %e, << _ >>, << _ , _ >>, ... }$, $GU'''_{seq T } = { %e, <>, <>, ,..., <> , ... }$

slide: The ADT Seq
Dependent on which producer functions are selected to generate the universe of T, the correspondence between the generated universe and the intended domain is either one-to-one (as for G and $G'$) or many-to-one (as for $G''$). Since we require our specification to be first-order and finite, infinite generator bases (such as $G'''$) must be disallowed, even if they result in a one-to-one correspondence. See [Dahl92] for further details.
Equations -- specifying constraints
The specification of the signature of a type (which lists the syntactic constraints to which a specification must comply) is in general not sufficient to characterize the properties of the values of the type. In addition, we need to impose semantic constraints (in the form of equations) to define the meaning of the observer functions and (very importantly) to identify the elements of the type domain that are considered equivalent (based on the intuitions one has of that particular type).

The equivalence relation -- congruence

$x = x$ \zline{reflexivity}

$x = y => y = x$ \zline{symmetry}

$x = y /\ y = z => x = z$ \zline{transitivity}

$x = y => f(...,x,...) = f(...,y,...)$

Equivalence classes -- representatives

abstract elements -- $GU_T /= $

slide: Equivalence
Mathematically, the equality predicate may be characterized by the properties listed above, including reflexivity (stating that an element is equal to itself), symmetry (stating that the orientation of the formula is not important) and transitivity (stating that if one element is equal to another and that element is equal to yet another, then the first element is also equal to the latter). In addition, we have the property that, given that two elements are equal, the results of the function applied to them (separately) are also equal. (Technically, the latter property makes a congruence of the equality relation, lifting equality between elements to the function level.) See slide 8-equivalence. Given a suitable set of equations, in addition to a signature, we may identify the elements that can be proved identical by applying the equality relation. In other words, given an equational theory (of which the properties stated above must be a part), we can divide the generator universe of a type T into one or more subsets, each consisting of elements that are equal according to our theory. The subsets of $GU/=$, that is GU factored with respect to equivalence, may be regarded as the abstract elements constituting the type T, and from each subset we may choose a concrete element acting as a representative for the subset which is the equivalence class of the element. Operationally, equations may be regarded as rewrite rules (oriented from left to right), that allow us to transform a term in which a term $t_1$ occurs as a subterm into a term in which $t_1$ is replaced by $t_2$ if $t_1 = t_2$. For this procedure to be terminating, some technical restrictions must be met, amounting (intuitively) to the requirement that the right-hand side must in some sense be simpler than the left-hand side. Also, when defining an observer function, we must specify for each possible generator case an appropriate rewriting rule. That is, each observer must be able to give a result for each generator. The example of the natural numbers, given below, will make this clear. Identifying spurious elements by rewriting a term into a canonical form is somewhat more complex, as we will see for the example of sets.
Equational theories
To illustrate the notions introduced above, we will look at specifications of some familiar types, namely the natural numbers and sets. In slide 8-Nat, an algebraic specification is given of the natural numbers (as first axiomatized by Peano).

Natural numbers

Nat

functions 0 : Nat S : Nat -> Nat mul : Nat * Nat -> Nat plus : Nat * Nat -> Nat axioms [1] plus(x,0) = x [2] plus(x,Sy) = S(plus(x,y)) [3] mul(x,0) = 0 [4] mul(x,Sy) = plus(mul(x,y),x) end

slide: The ADT Nat
In addition to the constant 0 and successor function S we also introduce a function mul for multiplication and a function plus for addition. (The notation Sy stands for application by juxtaposition; its meaning is simply $S(y)$.) The reader who does not immediately accept the specification in slide 8-Nat as an adequate axiomatization of the natural numbers must try to unravel the computation depicted in slide 8-symbolic.

mul(plus(S 0,S 0),S 0) -[2]-> mul(S(plus(S 0,0)), S 0) -[1]-> mul(SS 0,S 0) -[4]-> plus(mul(SS0,0),SS0) -[3]-> plus(0,SS0) -[2*]-> SS0

slide: Symbolic evaluation
Admittedly, not an easy way to compute with natural numbers, but fortunately term rewriting may, to a large extent, be automated (and actual calculations may be mimicked by semantics preserving primitives). Using the equational theory expressing the properties of natural numbers, we may eliminate the occurrences of the functions mul and plus to arrive (through symbolic evaluation) at something of the form $S^n 0$ (where n corresponds to the magnitude of the natural number denoted by the term). The opportunity of symbolic evaluation by term rewriting is exactly what has made the algebraic approach so popular for the specification of software, since it allows (under some restrictions) for executable specifications. Since they do not reappear in what may be considered the normal forms of terms denoting the naturals (that are obtained by applying the evaluations induced by the equality theory), the functions plus and mul may be regarded as secondary producers. They are not part of the generator basis of the type Nat. Since we may consider mul and plus as secondary producers at best, we can easily see that when we define mul and plus for the case 0 and Sx for arbitrary x, that we have covered all possible (generator) cases. Technically, this allows us to prove properties of these functions by using structural induction on the possible generator cases. The proof obligation (in the case of the naturals) then is to prove that the property holds for the function applied to 0 and assuming that the property holds for applying the function to x, it also holds for Sx. As our next example, consider the algebraic specification of the type $Set_{A}$ in slide 8-Set.

Sets

Set

$G_{Set_{A} = { \emptyset, add }$

$GU_{Set_{A} }$ $ = $ ${ 0, add(0,a), ..., add(add(0,a),a), ... }$

Axioms

[S1] $add(add(s,x),y) = add(add(s,y),x)$
commutativity

[S2] $add(add(s,x),x) = add(s,x)$
idempotence

slide: The ADT Set
In the case of sets we have the problem that we do not start with a one-to-one generator base as we had with the natural numbers. Instead, we have a many-to-one generator base, so we need equality axioms to eliminate spurious elements from the (generator) universe of sets.

Equivalence classes

$ GU_{ Set _{A} }/= $

${ \emptyset }$

${ add(0,a), add(add(0,a),a), ... }$

$...$

${ add(add(0,a),b), add(add(0,b),a), ... }$

slide: Equivalence classes for Set
The equivalence classes of $GU_{Set _{A} }/=$ (which is $GU_{Set _{A} }$ factored by the equivalence relation), each have multiple elements (except the class representing the empty set). To select an appropriate representative from each of these classes (representing the abstract elements of the type $Set_{A}$) we need an ordering on terms, so that we can take the smaller term as its canonical representation. See slide 8-set-equi.
Initial algebra semantics
In the previous section we have given a rather operational characterization of the equivalence relation induced by the equational theory and the process of term rewriting that enables us to purge the generator universe of a type, by eliminating redundant elements. However, what we actually strive for is a mathematical model that captures the meaning of an algebraic specification. Such a model is provided (or rather a class of such models) by the mathematical structures known as (not surprisingly) algebras. A single sorted algebra $|A$ is a structure $ (A,%S) $ where A is a set of values, and $%S$ specifies the signature of the functions operating on A. A multi-sorted algebra is a structure $ |A = ( { A_s }_{s \e S }, %S)$ where S is a set of sort names and $A_s$ the set of values belonging to the sort s. The set S may be ordered (in which case the ordering indicates the subtyping relationships between the sorts). We call the (multi-sorted) structure $|A$ a $%S$ -algebra.

Mathematical model -- algebra

$%S$-algebra -- $ |A = ( { A_s }_{s \e S }, %S )$

interpretation -- $ eval : T_{%S} -> |A $

adequacy -- $ |A |= t_1 = t_2 <=> E |- t_1 = t_2 $

slide: Interpretations and models
Having a notion of algebras, we need to have a way in which to relate an algebraic specification to such a structure. To this end we define an interpretation $eval : T_{%S} -> |A$ which maps closed terms formed by following the rules given in the specification to elements of the structure $|A$. We may extend the interpretation eval to include variables as well (which we write as $ eval : T_{%S}(X) -> |A$), but then we also need to assume that an assignment $%h : X -> T_{%S}(X)$ is given, such that when applying $%h$ to a term t the result is free of variables, otherwise no interpretation in $|A$ exists. See slide 8-algebra.
Interpretations
As an example, consider the interpretations of the specification of Bool and the specification of Nat, given in slide 8-B-N.

Booleans

$ |B = ( { tt, ff }, { \neg, / \/ } )$

$ eval_{|B} : T_{Bool} -> |B = { or |-> \/, and |-> / not |-> \neg }$

Natural numbers

$ |N = ( \nat , { ++ , + , \star } ) $

$ eval_{|N} : T_{Nat} -> |N = { S |-> ++ , mul |-> \star, plus |-> + }$

slide: Interpretations of Bool and Nat
The structure $|B$ given above is simply a boolean algebra, with the operators $\neg$, $ /\ $ and $ \/ $. The functions not, and and or naturally map to their semantic counterparts. In addition, we assume that the constants true and false map to the elements tt and ff. As another example, look at the structure $|N$ and the interpretation $eval_{|N}$, which maps the functions S, mul and plus specified in Nat in a natural way. However, since we have also given equations for Nat (specifying how to eliminate the functions mul and plus) we must take precautions such that the requirement
$ |N |= eval_{|N}(t_1) =_{|N} eval_{|N}(t_2) <=> E_{Nat} |- t_1 = t_2 $
is satisfied if the structure $ |N $ is to count as an adequate model of Nat. The requirement above states that whenever equality holds for two interpreted terms (in $ |N $) then these terms must also be provably equal (by using the equations given in the specification of Nat), and vice versa. As we will see illustrated later, many models may exist for a single specification, all satisfying the requirement of adequacy. The question is, do we have a means to select one of these models as (in a certain sense) the best model. The answer is yes. These are the models called initial models.
Initial models
A model (in a mathematical sense) represents the meaning of a specification in a precise way. A model may be regarded as stating a commitment with respect to the interpretation of the specification. An initial model is intuitively the least committing model, least committing in the sense that it imposes only identifications made necessary by the equational theory of a specification. Technically, an initial model is a model from which every other model can be derived by an algebraic mapping which is a homomorphism.

Initial algebra

$%S E$-algebra -- $|M = ( T_{%S}/=, %S/=)$

Properties

no junk -- $ \A a : T_{%S}/= \E t \dot eval_{|M}(t) = a $

no confusion -- $ |M |= t_1 = t_2 <=> E |- t_1 = t_2 $

slide: Initial models
The starting point for the construction of an initial model for a given specification with signature $%S$ is to construct a term algebra $T_{%S}$ with the terms that may be generated from the signature $%S$ as elements. The next step is then to factor the universe of generated terms into equivalence classes, such that two terms belong to the same class if they can be proven equivalent with respect to the equational theory of the specification. We will denote the representative of the equivalence class to which a term t belongs by $[t]$. Hence $t_1 = t_2$ (in the model) iff $[t_1] = [t_2]$. So assume that we have constructed a structure $ |M = (T_{%S}/=, %S ) $ then; finally, we must define an interpretation, say $eval_{|M} : T_{%S} -> |M$, that assigns closed terms to appropriate terms in the term model (namely the representatives of the equivalence class of that term). Hence, the interpretation of a function f in the structure $|M$ is such that
$f_{|M}([t_1],...,[t_n]) = [ f(t_1,...,t_n) ] $
where $f_{|M}$ is the interpretation of f in $|M$. In other words, the result of applying f to terms $t_1,...,t_n$ belongs to the same equivalence class as the result of applying $f_{|M}$ to the representatives of the equivalence classes of $t1,...,t_n$. See slide 8-initial. An initial algebra model has two important properties, known respectively as the no junk and no confusion properties. The no junk property states that for each element of the model there is some term for which the interpretation in $|M$ is equal to that element. (For the $T_{%S}/=$ model this is simply a representative of the equivalence class corresponding with the element.) The no confusion property states that if equality of two terms can be proven in the equational theory of the specification, then the equality also holds (semantically) in the model, and vice versa. The no confusion property means, in other words, that sufficiently many identifications are made (namely those that may be proven to hold), but no more than that (that is, no other than those for which a proof exists). The latter property is why we may speak of an initial model as the least committing model; it simply gives no more meaning than is strictly needed. The initial model constructed from the term algebra of a signature $%S$ is intuitively a very natural model since it corresponds directly with (a subset of) the generator universe of $%S$. Given such a model, other models may be derived from it simply by specifying an appropriate interpretation. For example, when we construct a model for the natural numbers (as specified by Nat) consisting of the generator universe ${ 0, S 0, SS 0, ... }$ and the operators ${ ++, +, \star }$ (which are defined as $S^n ++ = S^{n+1}$, ${ S^n } \ast { S^m } = S^{ {n \ast m} }$ and $S^n + S^m = S^{n + m}$) we may simply derive from this model the structure $ ({ 0,1,2,... }, { ++, +, \star }) $ for which the operations have their standard arithmetical meaning. Actually, this structure is also an initial model for Nat, since we may also make the inverse transformation. More generally, when defining an initial model only the structural aspects (characterizing the behavior of the operators) are important, not the actual contents. Technically, this means that initial models are defined up to isomorphism, that is a mapping to equivalent models with perhaps different contents but an identical structure. Not in all cases is a structure derived from an initial model itself also an initial model, as shown in the example below.
Example
Consider the specification of Bool as given before. For this specification we have given the structure $ |B $ and the interpretation $ eval_{|B} $ which defines an initial model for Bool. (Check this!)

Structure -- $ |B = ( {{ tt, ff }}, {{ \neg, /\, \/ }} )$

$|B$

$ eval_{|B} : T_{%S_{Bool} -> |B = { or |-> \/, not |-> \neg }$

$ eval_{|B} : T_{%S_{Nat} -> |B = { S |-> \neg, mul |-> / plus |-> xor }$

slide: Structure and interpretation
We may, however, also use the structure $|B$ to define an interpretation of Nat. See slide 8-structure. The interpretation $eval_{|B} : T_{Nat} -> |B$ is such that $eval_{|B}(0) = ff$, $eval_{|B}(Sx) = \neg eval_{|B}(x)$, $eval_{|B}(mul(x,y)) $ $ = eval_{|B}(x) $\hspace{0.2cm} $ /\ $ \hspace{0.2cm} $ eval_{|B}(y)$\hspace{0.3cm} and $eval_{|B}(plus(x,y)) = xor(eval_{|B}(x),eval_{|B}(y))$, where $xor(p,q) = (p \/ q) /\ (\neg ( p /\ q ))$. The reader may wish to ponder on what this interpretation effects. The answer is that it interprets Nat as specifying the naturals modulo 2, which discriminates only between odd and even numbers. Clearly, this interpretation defines not an initial model, since it identifies all odd numbers with ff and all even numbers with tt. Even if we replace ff by 0 and tt by 1, this is not what we generally would like to commit ourselves to when we speak about the natural numbers, simply because it assigns too much meaning.
Objects as algebras
The types for which we have thus far seen algebraic specifications (including Bool, Seq, Set and Nat) are all types of a mathematical kind, which (by virtue of being mathematical) define operations without side-effects. Dynamic state changes, that is side-effects, are often mentioned as determining the characteristics of objects in general. In the following we will explore how we may deal with assigning meaning to dynamic state changes in an algebraic framework. Let us look first at the abstract data type stack. The type stack may be considered as one of the `real life' types in the world of programming. See slide 8-Stack.

Abstract Data Type -- applicative

Stack

functions new : stack; push : element * stack -> stack; empty : stack -> boolean; pop : stack -> stack; top : stack -> element; axioms empty( new ) = true empty( push(x,s) ) = false top( push(x,s) ) = x pop( push(x,s) ) = s preconditions pre: pop( s : stack ) = not empty(s) pre: top( s : stack ) = not empty(s) end

slide: The ADT Stack
Above, a stack has been specified by giving a signature (consisting of the functions new, push, empty, pop and top). In addition to the axioms characterizing the behavior of the stack, we have included two pre-conditions to test whether the stack is empty in case pop or top is applied. The pre-conditions result in conditional axioms for the operations pop and top. Conditional axioms, however, do preserve the initial algebra semantics. The specification given above is a maximally abstract description of the behavior of a stack. Adding more implementation detail would disrupt its nice applicative structure, without necessarily resulting in different behavior (from a sufficiently abstract perspective). The behavior of elements of abstract data types and objects is characterized by state changes. State changes may affect the value delivered by observers or methods. Many state changes (such as the growing or shrinking of a set, sequence or stack) really are nothing but applicative transformations that may mathematically be described by the input-output behavior of an appropriate function. An example in which the value of an object on some attribute is dependent on the history of the operations applied to the object, instead of the structure of the object itself (as in the case of a stack) is the object account, as specified in slide 8-account. The example is taken from [Goguen].

Dynamic state changes -- objects

account

object account is functions bal : account -> money methods credit : account * money -> account debit : account * money -> account error overdraw : money -> money axioms bal(new(A)) = 0 bal(credit(A,M)) = bal(A) + M bal(debit(A,M)) = bal(A) - M if bal(A) >= M error-axioms bal(debit(A,M)) = overdraw(M) if bal(A) < M end

slide: The algebraic specification of an account
An account object has one attribute function (called bal) that delivers the amount of money that is (still) in the account. In addition, there are two method functions, credit and debit that may respectively be used to add or withdraw money from the account. Finally, there is one special error function, overdraw, that is used to define the result of balance when there is not enough money left to grant a debit request. Error axioms are needed whenever the proper axioms are stated conditionally, that is contain an if expression. The conditional parts of the axioms, including the error axioms, must cover all possible cases. Now, first look at the form of the axioms. The axioms are specified as
$fn(method(Object,Args)) = expr$
where fn specifies an attribute function (bal in the case of account) and method a method (either new, which is used to create new accounts, credit or debit). By convention, we assume that $method(Object,...) = Object$ , that is that a method function returns its first argument. Applying a method thus results in redefining the value of the function fn. For example, invoking the method $credit(acc,10)$ for the account acc results in modifying the function bal to deliver the value $bal(acc) + 10$ instead of simply $bal(acc)$ . In the example above, the axioms define the meaning of the function bal with respect to the possible method applications. It is not difficult to see that these operations are of a non-applicative nature, non-applicative in the sense that each time a method is invoked the actual definition of bal is changed. The change is necessary because, in contrast to, for example, the functions employed in a boolean algebra, the actual value of the account may change in time in a completely arbitrary way. A first order framework of (multi sorted) algebras is not sufficiently strong to define the meaning of such changes. What we need may be characterized as a multiple world semantics, where each world corresponds to a possible state of the account. As an alternative semantics we will also discuss the interpretation of an object as an abstract machine, which resembles an (initial) algebra with hidden sorts.
Multiple world semantics
From a semantic perspective, an object that changes its state may be regarded as moving from one world to another, when we see a world as representing a particular state of affairs. Take for example an arbitrary (say John's) account, which has a balance of 500. We may express this as $balance(accountJohn) = 500$ . Now, when we invoke the method credit, as in $credit(accountJohn, 200)$ , then we expect the balance of the account to be raised to 700. In the language of the specification, this is expressed as
$bal(credit(accountJohn,200)) = bal(accountJohn) + 200$
Semantically, the result is a state of affairs in which $bal(accountJohn) = 700$ . In [Goguen] an operational interpretation is given of a multiple world semantics by introducing a database D (that stores the values of the attribute functions of objects as first order terms) which is transformed as the result of invoking a method, into a new database $D'$ (that has an updated value for the attribute function modified by the method). The meaning of each database (or world) may be characterized by an algebra and an interpretation as before. The rules according to which transformations on a database take place may be formulated as in slide 8-multiple.

Multiple world semantics -- inference rules

$<< f(t_1,...,t_n),D >> -> << v, D >>$

attribute

$<< m(t_1,...,t_n),D >> -> << t_1, D' >>$

method

$<< t , D >> -> << t', D' >> => << e(...,t,...), D >> -> < >$

slide: The interpretation of change
The first rule (attribute) describes how attribute functions are evaluated. Whenever a function f with arguments $t_1,...,t_n$ evaluates to a value (or expression) v, then the term $f(t_1,...,t_n)$ may be replaced by v without affecting the database D. (We have simplified the treatment by omitting all aspects having to do with matching and substitutions, since such details are not needed to understand the process of symbolic evaluation in a multiple world context.) The next rule (method) describes the result of evaluating a method. We assume that invoking the method changes the database D into $D'$ . Recall that, by convention, a method returns its first argument. Finally, the last rule (composition) describes how we may glue all this together. No doubt, the reader needs an example to get a picture of how this machinery actually works.

Example - a counter object

object ctr is
ctr

function n : ctr -> nat method incr : ctr -> ctr axioms n(new(C)) = 0 n(incr(C)) = n(C) + 1 end

slide: The object ctr
In slide 8-ctr, we have specified a simple object ctr with an attribute function value (delivering the value of the counter) and a method function incr (that may be used to increment the value of the counter).

Abstract evaluation

-[new]-> -[incr]-> -[incr]-> -[n]-> <2, { C[n:=2] }>

slide: An example of abstract evaluation
The end result of the evaluation depicted in slide 8-ctr-evl is the value 2 and a context (or database) in which the value of the counter C is (also) 2. The database is modified in each step in which the method incr is applied. When the attribute function value is evaluated the database remains unchanged, since it is merely consulted.
Objects as abstract machines
Multiple world semantics provide a very powerful framework in which to define the meaning of object specifications. Yet, as illustrated above, the reasoning involved has a very operational flavor and lacks the appealing simplicity of the initial algebra semantics given for abstract data types. As an alternative, [Goguen] propose an interpretation of objects (with dynamic state changes) as abstract machines. Recall that an initial algebra semantics defines a model in which the elements are equivalence classes representing the abstract values of the data type. In effect, initial models are defined only up to isomorphism (that is, structural equivalence with similar models). In essence, the framework of initial algebra semantics allows us to abstract from the particular representation of a data type, when assigning meaning to a specification. From this perspective it does not matter, for example, whether integers are represented in binary or decimal notation. The notion of abstract machines generalizes the notion of initial algebras in that it loosens the requirement of (structural) isomorphism, to allow for what we may call behavioral equivalence. The idea underlying the notion of behavioral equivalence is to make a distinction between visible sorts and hidden sorts and to look only at the visible sorts to determine whether two algebras $|A$ and $|B$ are behaviorally equivalent. According to [Goguen], two algebras $|A$ and $|B$ are behaviorally equivalent if and only if the result of evaluating any expression of a visible sort in $|A$ is the same as the result of evaluating that expression in $|B$ . Now, an abstract machine (in the sense of Goguen and Meseguer, 1986) is simply the equivalence class of behaviorally equivalent algebras, or in other words the maximally abstract characterization of the visible behavior of an abstract data type with (hidden) states. The notion of abstract machines is of particular relevance as a formal framework to characterize the (implementation) refinement relation between objects. For example, it is easy to determine that the behavior of a stack implemented as a list is equivalent to the behavior of a stack implemented by a pointer array, whereas these objects are clearly not equivalent from a structural point of view. Moreover, the behavior of both conform (in an abstract sense) with the behavior specified in an algebraic way. Together, the notions of abstract machine and behavioral equivalence provide a formalization of the notion of information hiding in an algebraic setting. In the chapters that follow we will look at alternative formalisms to explain information hiding, polymorphism and behavioral refinement.
Decomposition -- modules versus objects

subsections:

Abstract interfaces

Representation and implementation

Adding new generators

Adding new observers

Abstract data types allow the programmer to define a complex data structure and an associated collection of functions, operating on that structure, in a consistent way. Historically, the idea of data abstraction was originally not type-oriented but arose from a more pragmatic concern with information hiding and representation abstraction, see [Parnas72b]. The first realization of the idea of data abstraction was in the form of modules grouping a collection of functions and allowing the actual representation of the data structures underlying the values of the (abstract) type domain to be hidden, see also [Parnas72a]. In [Cook90], a comparison is made between the way in which abstract data types are realized traditionally (as modules) and the way abstract data types may be realized using object-oriented programming techniques. According to [Cook90], these approaches must be regarded as being orthogonal to one another and, being to some extent complementary, deserve to be integrated in a common framework. After presenting an example highlighting the differences between the two approaches, we will further explore these differences and study the trade-offs with respect to possible extensions and reuse of code.

Decomposition -- matrix

data abstraction

nil cons(h,t)

empty(l) true false

head(l) error h

tail(l) error t

Modules -- operation oriented

ADT

organized around observers -- representation hiding

Objects -- data oriented

OOP

organized around generators -- method interface

slide: Decomposition and data abstraction
Recall that abstract data types may be completely characterized by a finite collection of generators and a number of observer functions that are defined with respect to each possible generator. Following this idea, we may approach the specification of a data abstraction by constructing a matrix listing the generators column-wise and the observers row-wise, which for each observer/generator pair specifies the value of the observer for that particular generator. Incidentally, the definition of such a matrix allows us to check in an easy way whether we have given a complete characterization of the data type. Above, an example is given of the specification of a list, with generators nil and cons, and observers empty, head and tail. (Note that we group the secondary producer tail with the observers.) Now, the traditional way of realizing abstract data types as modules may be characterized as operation oriented, in the sense that the module realization of the type is organized around the observers, resulting in a horizontal decomposition of the matrix. On the other hand, an object-oriented approach may be characterized as data oriented, since the object realization of a type is based on specifying a method interface for each possible generator (sub)type, resulting in a vertical decomposition of the matrix. See slide 8-decomposition. Note, however, that in practice, different generators need not necessarily correspond to different (sub)classes. Behavior may be subsumed in variables, as an object cannot change its class/type.
Abstract interfaces
When choosing for the module realization of the data abstraction list in C style, we are likely to have an abstract functional interface as specified in slide mod-face.

Modules -- a functional interface

ADT

typedef int element; struct list; extern list* nil(); extern list* cons(element e, list* l); extern element head(list* l); extern list* tail(list* l); extern bool equal(list* l, list* m);

slide: Modules -- a functional interface
For convenience, the list has been restricted to contain integer elements only. However, at the expense of additional notation, we could also easily define a generic list by employing template functions as provided by C++. This is left as an exercise for the reader. The interface of the abstract class list given in slide obj-face has been defined generically by employing templates.

Objects -- a method interface

OOP

template< class E > class list { public: list() { } virtual ~list() { } virtual bool empty() = 0; virtual E head() = 0; virtual list* tail() = 0; virtual bool operator==(list* m) = 0; };

slide: Objects -- a method interface
Note that the equal function in the ADT interface takes two arguments, whereas the operator== function in the OOP interface takes only one, since the other is implicitly provided by the object itself.
Representation and implementation
The realization of abstract data types as modules with functions requires additional means to hide the representation of the list type. In contrast, with an object-oriented approach, data hiding is effected by employing the encapsulation facilities of classes.
Modules -- representation hiding
Modules provide a syntactic means to group related pieces of code and to hide particular aspects of that code. In slide 8-modules an example is given of the representation and the generator functions for a list of integers.

Modules -- representation hiding

ADT

typedef int element; enum { NIL, CONS }; struct list { int tag; element e; list* next; };

Generators

list* nil() {
nil

list* l = new list; l->tag = NIL; return l; } list* cons( element e, list* l) {
cons

list* x = new list; x->tag = CONS; x->e = e; x->next = l; return x; }

slide: Data abstraction and modules
For implementing the list as a collection of functions (ADT style), we employ a struct with an explicit tag field, indicating whether the list corresponds to nil or a cons. The functions corresponding with the generators create a new structure and initialize the tag field. In addition, the cons operator sets the element and next field of the structure to the arguments of cons. The implementation of the observers is given in slide 8-mod-impl.

Modules -- observers

ADT

int empty(list* lst) { return !lst || lst->tag == NIL; } element head(list* l) {
head

require( ! empty(l) ); return l->e; } list* tail(list* l) {
tail

require( ! empty(l) ); return l->next; } bool equal(list* l, list* m) {
equal

switch( l->tag) { case NIL: return empty(m); case CONS: return !empty(m) && head(l) == head(m) && tail(l) == tail(m); } }

slide: Modules -- observers
To determine whether the list is empty it suffices to check whether the tag of the list is equal to NIL. For both head and tail the pre-condition is that the list given as an argument is not empty. If the pre-condition holds, the appropriate field of the list structure is returned. The equality operator, finally, performs an explicit switch on the tag field, stating for each case under what conditions the lists are equal. Below, a program fragment is given that illustrates the use of the list.
list* r = cons(1,cons(2,nil())); while (!empty(r)) { cout << head(r) << endl; r = tail(r); }
Note that both the generator functions nil and cons take care of creating a new list structure. Writing a function to destroy a list is left as an exercise for the reader.
Objects -- method interface
The idea underlying an object-oriented decomposition of the specification matrix of an abstract type is to make a distinction between the (syntactic) subtypes of the data type (corresponding with its generators) and to specify for each subtype the value of all possible observer functions. (We speak of syntactic subtypes, following [Dahl92], since these subtypes correspond to the generators defining the value domain of the data type. See [Dahl92] for a more extensive treatment.)

Method interface -- list

OOP

template< class E > class nil : public list< E > {
nil

public: nil() {} bool empty() { return 1; } E head() { require( false ); return E(); } list< E >* tail() { require( 0 ); return 0; } bool operator==(list* m) { return m->empty(); } }; template< class E > class cons : public list< E > {
cons

public: cons(E e, list* l) : _e(e), next(l) {} ~cons() { delete next; } bool empty() { return 0; } E head() { return _e; } list* tail() { return next; } bool operator==(list* m); protected: E _e; list* next; };

slide: Data abstraction and objects
In the object realization in slide 8-objects, each subtype element is defined as a class inheriting from the list class. For both generator types nil and cons the observer functions are defined in a straightforward way. Note that, in contrast to the ADT realization, the distinction between the various cases is implicit in the member function definitions of the generator classes. As an example of using the list classes consider the program fragment below.
list<int>* r = new cons<int>(1, new cons<int>(2, new nil<int>)); while (! r->empty()) { cout << r->head() << endl; r = r->tail(); } delete r;
For deleting a list we may employ the (virtual) destructor of list, which recursively destroys the tail of a list.
Adding new generators
Abstract data types were developed with correctness and security in mind, and not so much from a concern with extensibility and reuse. Nevertheless, it is interesting to compare the traditional approach of realizing abstract data types (employing modules) and the object-oriented approach (employing objects as generator subtypes) with regard to the ease with which a specification may be extended, either by adding new generators or by adding new observers.

Adding new generators -- representation

ADT

typedef int element; enum { NIL, CONS, INTERVAL }; struct list { int tag; element e; union { element z; list* next; }; };

Generator

list* interval( element x, element y ) { list* l = new list; if ( x <= y ) { l->tag = INTERVAL; l->e = x; l->z = y; } else l->tag = NIL; return l; }

slide: Modules and generators
Let us first look at what happens when we add a new generator to a data type, such as an interval list subtype, containing the integers in the interval between two given integers. For the module realization of the list, adding an $interval(x,y)$ generator will result in an extension of the (hidden) representation types with an additional representation tag type INTERVAL and the definition of a suitable generator function. To represent the interval list type, we employ a union to select between the next field, which is used by the cons generator, and the z field, which indicates the end of the interval.

Modifying the observers

ADT

element head(list* l) {
head

require( ! empty(l) ); return l->e; // for both CONS and INTERVAL } list* tail(list* l) {
tail

require( ! empty(l) ); switch( l->tag ) { case CONS: return l->next; case INTERVAL: return interval((l->e)+1,l->z); } }

slide: Modifying the observers
Also, we need to modify the observer functions by adding an appropriate case for the new interval representation type, as pictured in slide 8-mod-xx. Clearly, unless special constructs are provided, the addition of a new generator case requires disrupting the code implementing the given data type manually, to extend the definition of the observers with the new case. In contrast, not surprisingly, when we wish to add a new generator case to the object realization of the list, we do not need to disrupt the given code, but we may simply add the definition of the generator subtype as given in slide 8-oop-gen.

Adding new generators

OOP

class interval : public list<int> {
interval

public: interval(int x, int y) : _x(x), _y(y) { require( x <= y ); } bool empty() { return 0; } int head() { return _x; } list< int >* tail() { return (_x+1 <= _y)? new interval(_x+1,_y): new nil<int>; } bool operator==(list@lt;int>* m) { return !m->empty() && _x == m->head() && tail() == m->tail(); } protected: int _x; int _y; };

slide: Objects and generators
Adding a new generator subtype corresponds to defining the realization for an abstract interface class, which gives a method interface that its subclasses must respect. Observe, however, that we cannot exploit the fact that a list is defined by an interval when testing equality, since we cannot inspect the type of the list as for the ADT implementation.
Adding new observers
Now, for the complementary case, what happens when we add new observers to the specification of a data type? Somewhat surprisingly, the object-oriented approach now seems to be at a disadvantage. Since in a module realization of an abstract data type the code is organized around observers, adding a new observer function amounts simply to adding a new operation with a case for each of the possible generator types, as shown in slide 8-mod-obs.

Adding new observers

ADT

int length( list* l ) {
length

switch( l->tag ) { case NIL: return 0; case CONS: return 1 + length(l->next); case INTERVAL: return l->z - l->e + 1; }; }

slide: Modules and observers
When we look at how we may extend a given object realization of an abstract data type with a new observer we are facing a problem. The obvious solution is to modify the source code and add the length function to the list interface class and each of the generator classes. This is, however, against the spirit of object orientation and may not always be feasible. Another, rather awkward solution, is to extend the collection of possible generator subtypes with a number of new generator subtypes that explicitly incorporate the new observer function. However, this also means redefining the tail function since it must deliver an instance of a list with length class. As a workaround, one may define a function length and an extended version of the list template class supporting only the length (observer) member function as depicted in slide 8-oop-obs.

Adding new observers

OOP

template< class E > int length(list< E >* l) {
length

return l->empty() ? 0 : 1 + length( l->tail() ); } template< class E > class listWL : public list<E> {
listWL

public: int length() { return ::length( this ); } };

slide: Objects and observers
A program fragment illustrating the use of the listWL class is given below.

list<int>* r = new cons<int>(1,new cons<int>(2,new interval(3,7))); while (! r->empty()) { cout << ((listWL< int >*)r)->length() << endl; r = r->tail(); } delete r;
Evidently, we need to employ a cast whenever we wish to apply the length observer function. Hence, this seems not to be the right solution. Alternatively, we may use the function length directly. However, we are then forced to mix method syntax of the form $ref->op(args)$ with function syntax of the form $fun(ref,args)$ , which may easily lead to confusion.
Discussion
We may wonder why an object-oriented approach, that is supposed to support extensibility, is at a disadvantage here when compared to a more traditional module-based approach. As observed in [Cook90], the problem lies in the fact that neither of the two approaches reflect the full potential and flexibility of the matrix specification of an abstract data type. Each of the approaches represents a particular choice with respect to the decomposition of the matrix, into either an operations-oriented (horizontal) decomposition or a data-oriented (vertical) decomposition. The apparent misbehavior of an object realization with respect to extending the specification with observer functions explains why in some cases we prefer the use of overloaded functions rather than methods, since overloaded functions allow for implicit dispatching to take place on multiple arguments, whereas method dispatching behavior is determined only by the type of the object. However, it must be noted that the dispatching behavior of overloaded functions in C++ is of a purely syntactic nature. This means that we cannot exploit the information specific for a class type as we can when using virtual functions. Hence, to employ this information we would be required to write as many variants of overloaded functions as there are combinations of argument types. Dynamic dispatching on multiple arguments is supported by multi-methods in CLOS, see [Paepcke93]. According to [Cook90], the need for such methods might be taken as a hint that objects only partially realize the true potential of data abstraction.
Types versus classes
Types are primarily an aid in arriving at a consistent system description. Most (typed) object-oriented programming languages offer support for types by employing classes as a device to define the functionality of objects. Classes, however, have originated from a far more pragmatic concern, namely as a construct to enable the definition and creation of objects. Concluding this chapter, we will reflect on the distinction between types and classes, and discuss the role types and classes play in reusing software through derivation by inheritance. This discussion is meant to prepare the ground for a more formal treatment to be given in the next chapter. It closely follows the exposition given in [WZ88]. Types must primarily be understood as predicates to guide the process of type checking, whereas classes have come into being originally as templates for object creation. It is interesting to note how (and how easily) this distinction may be obscured. In practice, when compiling a program in Java or C++, the compiler will notify the user of an error when a member function is called that is not listed in the public interface of the objects class. As another example, the runtime system of Smalltalk will raise an exception, notifying the user of a dynamic type error, when a method is invoked that is not defined in the object's class or any of its superclasses. Both kinds of errors have the flavor of a typing error, yet they rely on different notions of typing and are based on a radically different interpretation of classes as types. To put types into perspective, we must ask ourselves what means we have to indicate the type of an expression, including expressions that somehow reference a class description. In [WZ88], three attitudes towards typing are distinguished: (1) typing may be regarded as an administrative aid to check for simple typos and other administrative errors, (2) typing may be regarded as the ultimate solution to defining the behavior of a system, or (3) typing may (pragmatically) be regarded as a consequence of defining the behavior of an object. See slide 8-classes. Before continuing, the reader is invited to sort the various programming languages discussed into the three slots mentioned.

Types versus classes

types -- type checking predicates

classes -- templates for object creation

Type specification

syntactically -- signature
(under)

semantically -- behavior

(right)

pragmatically -- implementation
(over)

slide: Types and classes
Typing as an administrative aid is typically a task for which we rely on a compiler to check for possible errors. Evidently, the notion of typing that a compiler employs is of a rather syntactic nature. Provided we have specified a signature correctly, we may trust a compiler with the routine of checking for errors. As languages that supports signature type checking we may (obviously) mention Java and C++. Evidently, we cannot trust the compiler to detect conceptual errors, that is incomplete or ill-conceived definitions of the functionality of an object or collections of objects. Yet, ultimately we want to be able to specify the behavior of an object in a formal way and to check mechanically for the adequacy of this definition. This ideal of semantic types underlies the design of Eiffel, not so much the Eiffel type system as supported by the Eiffel compiler, but the integration of assertions in the Eiffel language and the notion of contracts as a design principle. Pragmatically, we need to rely on runtime (consistency) checks to detect erroneous behavior, since there are (theoretically rather severe) limits on the extent to which we may verify behavioral properties in advance. (Nevertheless, see section types-behavioral for some attempts in this direction.)

Modifications

types -(predicate constraints) $->$ subtypes

classes -(template modification) $->$ subclasses

Varieties of (compatible) modifications

behaviorally -- algebraic, axiomatic

(type)

signature -- type checking

(signature)

name -- method search algorithm

(classes)

slide: Type modifications
Finally, we can take a far more pragmatic view towards typing, by regarding the actual specification of a class as an implicit characterization of the type of the instances of the class. Actually, this is the way (not surprisingly, I would say) types are dealt with in Smalltalk. Each object in Smalltalk is typed, by virtue of being an instance of a class. Yet, a typing error may only be detected dynamically, as the result of not responding to a message. A distinction between perspectives on types (respectively syntactic, behavioral and pragmatic) may seem rather academic at first sight. However, the differences are, so to speak, amplified when studied in the context of type modifications, as for example effected by inheritance. [WZ88] make a distinction between three notions of compatible modifications, corresponding to the three perspectives on types, respectively signature compatible modifications (which require the preservation of the static signature), behaviorally compatible modification (which rely on a mathematical notion of definability for a type) and name compatible modifications (that rely on an operationally defined method search algorithm). See slide 8-refinement.
Signature compatible modifications
The assumption underlying the notion of types as signatures is that behavior is approximated by a (static) signature. Now the question is: to what extent can we define semantics preserving extensions to a given class or object?

Signature compatible modifications

behavior is approximated by signature

Semantics preserving extensions

horizontal -- Person = Citizen + { age : 0..120 }

vertical -- Retiree = Person + { age : 65..120 }

Principle of substitutability

an instance of a subtype can always be used in any context in which an instance of a supertype can be used

subsets are not subtypes

Retiree $\not<_{subtype}$ Person

Read-only substitutability

subset subtypes, isomorphically embedded subtypes

slide: The principle of substitutability
When we conceive of an object as a record consisting of (data and method) fields, we may think of two possible kinds of modifications. We may think of a horizontal modification when adding a new field, and similarly we may think of a modification as being vertical when redefining or constraining a particular field. For example, when we define Citizen as an entity with a name, we may define (at the risk of being somewhat awkward) a Person as a Citizen with an age and a Retiree as a Person with an age that is restricted to the range 65..120. The principle by which we may judge these extensions valid (or not) may be characterized as the principle of substitutability, which may be phrased as: an instance of a subtype can always be used in any context in which an instance of a supertype can be used. Unfortunately, for the extension given here we have an easy counterexample, showing that syntactic signature compatibility is not sufficient. Clearly, a Person is a supertype of Retiree (we will demonstrate this more precisely in section subtypes). Assume that we have a function
set_age : Person * Integer -> Void
that is defined as set_age(p,n) { p.age = n; }. Now consider the following fragment of code:
Person* p = r;
r refers to some Retiree

p->set_age(40);
where we employ object reference notation when calling $set_age$ . Since we have assigned r (which is referring to a Retiree) to p, we know that p now points to a Retiree, and since a Retiree is a person we may apply the function $set_age$ . However, $set_age$ sets the age of the Retiree to 40, which gives (by common standards) a semantic error. The lesson that we may draw from this is that being a subset is no guarantee for being a subtype as defined by the principle of substitutability. However, we may characterize the relation between a Retiree and a Person as being of a weaker kind, namely read-only substitutability, expressing that the (value of) the subtype may be used safely everywhere an instance of the supertype is expected, as long as it is not modified. Read-only substitutability holds for a type that stands in a subset relation to another type or is embeddable (as a subset) into that type. See slide 8-subst.
Behaviorally compatible modifications
If the subset relation is not a sufficient condition for being in a subtype relation, what is? To establish whether the (stronger) substitutability relation holds we must take the possible functions associated with the types into consideration as well. First, let us consider what relations may exist between types. Recall that semantically a type corresponds to a set together with a collection of operations that are defined for the set and that the subtype relation corresponds to the subset relation in the sense that (taking a type as a constraint) the definition of a subtype involves adding a constraint and, consequently, a narrowing of the set of elements corresponding to the supertype. Complete compatibility is what we achieve when the principle of substitutability holds. Theoretically, complete compatibility may be assured when the behavior of the subtype fully complies with the behavior of the supertype. Behavioral compatibility, however, is a quite demanding notion. We will deal with it more extensively in chapter refinement, when discussing behavioral refinement. Unfortunately, in practice we must often rely on the theoretically much weaker notion of name compatibility.

Name compatible modifications

operational semantics -- no extra compile/run-time checks

procedure search(name, module) if name = action then do action elsif inherited = nil then undefined else search(name, inherited)

slide: The inheritance search algorithm

Name compatible modifications
Name compatible modifications approximate behaviorally compatible modifications in the sense that substitutability is guaranteed, albeit not in a semantically verifiable way. Operationally, substitutability can be enforced by requiring that each subclass (that we may characterize as a pragmatic subtype) provides at least the operations of its superclasses (while giving a sensible result on all argument types allowed by its superclasses). Actually, name compatibility is an immediate consequence of the overriding semantics of derivation by inheritance, as reflected in the search algorithm underlying method lookup. See slide 8-search. Although name compatible modifications are by far the most flexible, from a theoretical point of view they are the least satisfying since they do not allow for any theory formation concerning the (desired) behavior of (the components of) the system under development.
Summary
This chapter has presented an introduction to the theoretical foundations of abstract data types. In particular, a characterization was given of types as constraints.

Abstraction and types

1

abstraction -- control and data

abstract data types -- values in a semantic domain

types as constraints -- mathematical models

slide: Section 8.1: Abstraction and types
In section 1, we discussed the notion of abstraction in programming languages and distinguished between control and data abstractions. Abstract data types were characterized as values in some domain, and we looked at the various ways in which to define mathematical models for types.

Algebraic specification

2

signature -- producers and observers

generator universe -- equivalence classes

initial model -- no junk, no confusion

objects -- multiple world semantics

slide: Section 8.2: Algebraic specification
In section 2, we studied the algebraic specification of abstract data types by means of a signature characterizing producers and observers. We discussed the notions of equivalence classes and initial models, which consist of precisely the equivalence classes that are needed. ,p> Also, we looked at the interpretation of objects as algebras, and we discussed a multiple world semantics allowing for dynamic state changes.

Decomposition -- modules versus objects

3

data abstraction -- generators/observers matrix

modules -- operation-oriented

objects -- data-oriented

slide: Section 8.3: Decomposition -- modules versus objects
In section 3, we looked at the various ways we may realize data abstractions and we distinguished between a modular approach, defining a collection of operations, and a data-oriented approach, employing objects.

Types versus classes

4

types -- syntactically, semantically, pragmatically

compatible modifications -- type, signature, class

slide: Section 8.4: Types versus classes
Finally, in section 4, we discussed the differences between a syntactic, semantic and operational interpretation of types, and how these viewpoints affect our notion of refinement or compatible modification.
Questions

Characterize the differences between control abstractions and data abstractions. Explain how these two kinds of abstractions may be embodied in programming language constructs.
How can you model the meaning of abstract data types in a mathematical way? Do you know any alternative ways?
Explain how types may affect object-oriented programming.
Explain how you may characterize an abstract data type by means of a matrix with generator columns and observer rows. What benefits does such an organization have?
How would you characterize the differences between the realization of abstract data types by modules and by objects? Discuss the trade-offs involved.
How would you characterize the distinction between types and classes? Mention three ways of specifying types. How are these kinds related to each other?
How would you characterize behavior compatible modifications? What alternatives can you think of?

slide: Questions

Further reading

There is a vast amount of literature on the algebraic specification of abstract data types. You may consult, for example, [Dahl92].

[] readme course preface 1 2 3 4 5 6 7 8 9 10 11 12 appendix lectures resources

eliens@cs.vu.nl

draft version 0.1 (15/7/2001)

Abstract data types

Abstract data types

Abstraction and types

subsections:

Abstraction in programming languages

Abstraction -- programming methodology

A foundational perspective -- types as constraints

Abstract data types -- foundational perspective

Mathematical models -- types as constraints

Formal models

Objectives of typed OOP

Objectives of typed OOP -- system description

Remark

Algebraic specification

subsections:

Algebraic specification -- ADT

Mathematical models

Signatures -- generators and observers

Signature -- names and profiles

Functions -- for $T$

Type -- generators

Generators

Generators -- values of $T$

Examples

Sequences

Generator basis -- preferably one-to-one

Infinite generator basis

Equations -- specifying constraints

The equivalence relation -- congruence

Equivalence classes -- representatives

Equational theories

Natural numbers

Sets

Axioms

commutativity

idempotence

Equivalence classes

Initial algebra semantics

Mathematical model -- algebra

Interpretations

Booleans

Natural numbers

Initial models

Initial algebra

Properties

Example

Structure -- $ |B = ( {{ tt, ff }}, {{ \neg, /\, \/ }} )$

Objects as algebras

Abstract Data Type -- applicative

Dynamic state changes -- objects

Multiple world semantics

Multiple world semantics -- inference rules

Example - a counter object

ctr

Abstract evaluation

Objects as abstract machines

Decomposition -- modules versus objects

subsections:

Decomposition -- matrix

Modules -- operation oriented

Objects -- data oriented

Abstract interfaces

Modules -- a functional interface

Objects -- a method interface

Representation and implementation

Modules -- representation hiding

Modules -- representation hiding

Generators

nil

cons

Modules -- observers

head

tail

equal

Objects -- method interface

Method interface -- list

nil

cons

Adding new generators

Adding new generators -- representation